The real numbers include some numbers that are in VV and many that aren't. In what way does that specify VV? That's like saying I can specify the people registered at a hotel this weekend as the human race. Of course everyone at the hotel is human, but humanity includes many people who are not registered at the hotel. — fishfry

That's right, to specify that they are real numbers is to specify, just like to specify that the guests at the hotel are human beings is to specify. The fact that a specification is vague, incomplete, or imperfect does not negate the fact that it is a specification.

How so? — fishfry

I told you how so. You've specified that the set contains real numbers. You are the one who explained to me, that 'set" is logically prior to "number", and that not all sets have numbers as elements. This means that "set" is the more general term. How can you now deny that to indicate that a particular set consists of some real numbers, is not an act of specifying?

And the people at the hotel are humans. As are all the people not at the hotel. If that's all you mean by specification, that all I have to do is name some arbitrary superset of the set in question, then every set has a specification. If that's what you meant, I'll grant you your point. But it doesn't seem too helpful. It doesn't tell me how to distinguish members of a set from non members. — fishfry

Good, you now accept that every set has a specification. Do you also agree now that this type of specification, which "doesn't tell me how to distinguish members of a set from non members", is simply a bad form of specification?

Anyway, let's go back to the point which raised this issue. You said the following, which i said was contradictory:

First, the elements of a set need not be "the same" in any meaningful way. The only thing they have in common is that they're elements of a given set. — fishfry

Do you now see, and agree, that since a set must be specified in some way, then the elements must be "the same" in some way, according to that specification, therefore it's really not true to say that "the elements of a set need not be "the same" in any meaningful way." So we can get rid of that appearance of contradiction by stating the truth, that the elements of a set must be the same in some meaningful way. To randomly name objects is not to list the members of a set, because a set requires a specification.

What I am trying to get at, is the nature of a "set" You say that there is no definition of "set", but it has meaning given by usage. Now I see inconsistency in your usage, so I want to find out what you really think a set is. Consider the following.

The elements of a set need have no relation to one another nor belong to any articulable category or class of thought, OTHER THAN being gathered into a set. — fishfry

Since we now see that a set must have a specification, do you see how the above quote is inconsistent with that principle? Since a set must have a specification, a set is itself an "articulable category or class of thought". And, it is not the "being gathered into a set" which constitutes the relations they have with one another, it is the specification itself, which constitutes the relations. So if you specify a set containing the number five, the tuna sandwich you had for lunch, and the Mormon tabernacle choir, this specification constitutes relations between these things. That's what putting them into a set does, it constructs such relations.

Now here's the difficult part. Do you agree that there are two distinct types of sets, one type in which the specification is based in real, observed similarities, a set which is based on description, and another type of set which is based in imaginary specifications, a set produced as a creative act? Do you acknowledge that these two types of sets are fundamentally different?

I'm perfectly happy to stipulate so for purposes of discussion. After all, there are no infinite sets in physics, at least at the present time. So, what of it? The knight doesn't "really" move that way. Everybody knows that knights rescue damsels in distress, a decidedly sexist notion in our modern viewpoint. Therefore chess is misleading and unrepresentative nonsense. Nevertheless, millions of people enjoy playing the game. And millions more enjoy NOT playing the game. What I don't understand is standing on a soapbox railing against the game. If math is nonsense, do something else. Nobody's forcing you to do math, unless you're in school. And then your complaints are not really about math itself, but rather about math pedagogy. And I agree with you on that. When I'm in charge, a lot of state math curriculum boards are going straight to Gitmo. — fishfry

So are you agreeing that mathematical infinity has neither philosophical nor scientific relevance and that everyone knows this, or am i right to stand on a soap box and point out the idiocies and misunderstandings that ZFC seems to encourage?

I might point out, though, that assuming the negation of the axiom of choice has consequences every bit as counterintuitive as assuming choice. Without choice you have a vector space that has no basis. An infinite set that changes cardinality if you remove a single element. An infinite set that's Dedekind-finite. You lose the Hahn-Banach theorem, of vital interest in functional analysis, which is the mathematical framework for quantum mechanics. The axiom of choice is even involved in political science via the Arrow impossibility theorem. — fishfry

Obviously, a denial of AC doesn't amount to an assertion of ~AC, given that things are generally undecidable, but i see no counter-intuitive examples in what you present. In fact, many examples you raise should be constructively intuitive if we recall that construction can proceed either bottom-up from the assumptions of elements into equivalence classes, or vice versa, so an inability to locate a basis in a vector space using top-down construction seems reasonable.

As for the sciences, AC is meaningless and inapplicable when it comes to the propositional content. At best, AC serves a crude notation for referring to undefined sets of unbounded size, but ZFC is a terribly crude means of doing this, because it only recognises completely defined sets and completely undefined sets without any shade of grey in the middle as is required to represent potential infinity.

QM has also been reinterpreted in toposes and monoidal categories in which all non-constructive physics propositions have been removed, which demonstrates that non-constructive analysis is dying and going to be rapidly replaced by constructive analysis, to the consternation of inappropriately trained mathematicians who resent not knowing constructive analysis.

Besides, if you have a nation made up of states, can't you always choose a legislature? A legislature is a representative from each state. If there were infinitely many states, couldn't each state still choose a representative? The US Senate is formed by two applications of the axiom of choice. The House of Representatives is a choice set on the 435 Congressional districts. The axiom of choice is perfectly true intuitively. If you deny the axiom of choice, you are asserting that there's a political entity subdivided into states such that it's impossible to form a legislature. How would you justify that? It's patently false. If nothing else, each state could choose a representative by lot. — fishfry

Obviously, the axiom of choice isn't used in the finite case. In the infinite case, the sets of states needs to be declared as being Kuratowski infinite in order to say that the elements of the set are never completely defined, and so a forteriori the size of the set cannot be defined in terms of it's finite subsets.

Secondly, the set should be declared as Dedekind finite, in order to say that the set is an observable collection of elements and not a function (because only functions can be dedekind-infinite).

So, yes, you can choose as many representatives as you wish without implying a nonsensical completed collection of legislatures that are a proper subset of themselves, but formalisation of these sets isn't possible in ZFC, because AC and it's weaker cousin, the axiom of countable choice, forces equivalence of Kuratowski finiteness and Dedekind finiteness.

I might point out, though, that assuming the negation of the axiom of choice has consequences every bit as counterintuitive as assuming choice. Without choice you have a vector space that has no basis. An infinite set that changes cardinality if you remove a single element. An infinite set that's Dedekind-finite. You lose the Hahn-Banach theorem, of vital interest in functional analysis, which is the mathematical framework for quantum mechanics. The axiom of choice is even involved in political science via the Arrow impossibility theorem. — fishfry


Obviously, a denial of AC doesn't amount to an assertion of ~AC, given that things are generally undecidable, but i see no counter-intuitive examples in what you present. In fact, many examples you raise should be constructively intuitive if we recall that construction can proceed either bottom-up from the assumptions of elements into equivalence classes, or vice versa, so an inability to locate a basis in a vector space using top-down construction seems reasonable. — sime

If one doesn't wander into transfinite math, Choice is not necessarily required. A finite dimensional vector space doesn't require it for a basis. And if one works with a normed linear space that is separable, the Hahn-Banach theorem doesn't require it either. However, much modern math goes beyond these restrictions and requires transfinite results.

. . . which demonstrates that non-constructive analysis is dying and going to be rapidly replaced by constructive analysis, to the consternation of inappropriately trained mathematicians who resent not knowing constructive analysis — sime

When I write mathematical analysis programs for my computer I am certainly involved in constructive math, but pen on paper, not necessarily. I suspect constructive analysis will not overwhelm the math community, unless you know something I don't - which is possible. :cool:

So are you agreeing that mathematical infinity has neither philosophical nor scientific relevance — sime

Not entirely, but I'm not disagreeing either. Infinitary set theory has been used in the 20th century as the foundation of math, and math is the language of physics. So if we say that infinitary math is unrealistic or fiction or nonsense or false (any one of which I'd agree with for sake of discussion), we still have to explain the "unreasonable effectiveness" of math. So infinitary math is of great philosophical importance. And surely there are a whole lot of professional philosophers of math who do discuss and care about infinitary math. So infinitary math is both philosophically and scientifically relevant, even if we can plausibly argue that it's fiction. After all, as I'm fond of pointing out, Moby Dick is a work of fiction, but it still teaches us to avoid following our obsessions to our doom. Even fiction can be useful i the real world.

and that everyone knows this, — sime

I don't think everyone sees this issue the same way. There are a lot of different opinions, even different learned opinions.

or am i right to stand on a soap box — sime

That's always a personal choice. I may dislike the designated hitter rule in baseball, but I don't go out and rant about it in public. One chooses one's battles. There are those who dislike contemporary mask mandates, but wear their mask as required by local laws. There are others who go into grocery stores and confront the hapless clerks. Each of us has many opinions, but we still have to choose which hills to die on and which soapboxes to stand on.

and point out the idiocies and misunderstandings that ZFC seems to encourage? — sime

I disagree with you about this. The way the knight moves in chess is a fiction, it's an arbitrary rule of a formal game. Does it produce idiocies and misunderstandings? No, it's just a rule of the game. I've seen prominent set theorists admit that they don't know if set theory is true or meaningful. Set theory is the study of certain formal structures. People do it because they find it interesting. Others are Platonists and believe they're seeking some higher truth.

I'm not sure what idiocies and misunderstandings you mean. If you don't like the way the knight moves, don't play the game. Or play some alternate variant of chess. There are many alternate variants of set theory. And billions of people live perfectly happy lives without ever knowing or caring about set theory.

So you're not "wrong," pe se, in disliking or objecting to infinitary math. But if in addition to that you have a strong emotional aversion to it, that's ... well, it's a personal issue. You might introspect as to why. Maybe you had a screechy math teacher in third grade. A lot of math anger started that way.

Obviously, a denial of AC doesn't amount to an assertion of ~AC, given that things are generally undecidable, — sime

Ok. Still, absent choice, infinite sets are badly behaved. There's a vector space without a basis, a surjection without a right inverse (or section), a commutative ring with unity with no maximal ideal. These things are very inconvenient in math.

And that's the biggest reason for adopting AC versus rejecting it. Convenience. The reasons for adopting or rejecting axioms are pragmatic. We are not asserting any kind of absolute truth. We're only choosing the axioms that make it convenient to do math. Maddy explains all this in her classic articles, Believing the Axioms, I and II. We want expansive rather than restrictive axioms, and so forth.

but i see no counter-intuitive examples in what you present. — sime

If you don't find a vector space without a basis, a surjection without a right inverse, an infinite set that changes cardinality when you remove one element, etc., counterintuitive, then we see that differently. In the end, AC makes infinite sets well-behaved. For example without AC there are the Alephs, and then there are many infinite cardinalities that aren't Alephs. With AC, all the cardinals are Alephs. There's no "absolute truth" in that, just convenience.

I think if you regard AC as a pragmatic choice, it's easier to understand. It relieves you of needing to stand on a soapbox. Mathematicians are choosing convenience and a more orderly and expansive set-theoretic universe. What need is there to stand on a soapbox against someone's pragmatic choices?

In fact, many examples you raise should be constructively intuitive if we recall that construction can proceed either bottom-up from the assumptions of elements into equivalence classes, or vice versa, so an inability to locate a basis in a vector space using top-down construction seems reasonable.p/quote]

If you're making a constructivist argument, I can't argue with you. Many people agree with your point of view.
— sime

As for the sciences, AC is meaningless and inapplicable when it comes to the propositional content. At best, AC serves a crude notation for referring to undefined sets of unbounded size, but ZFC is a terribly crude means of doing this, because it only recognises completely defined sets and completely undefined sets without any shade of grey in the middle as is required to represent potential infinity. — sime

I suppose I can concede your point that AC is not strictly necessary for the sciences. Nor is the knight move necessary to cook a lasagna.

QM has also been reinterpreted in toposes and monoidal categories in which all non-constructive physics propositions have been removed, which demonstrates that non-constructive analysis is dying and going to be rapidly replaced by constructive analysis, to the consternation of inappropriately trained mathematicians who resent not knowing constructive analysis. — sime

ZFC is not even a 100 years old. In its present form it dates from Zermelo's 1922 axiomatization. And Mrs. Zermelo was pro choice. (/joke). There's no telling how these matters will be seen in another hundred years. As Max Planck said, science proceeds one funeral at a time. Meaning that the old guard die off and a new generation grows up accepting the new ideas.

Obviously, the axiom of choice isn't used in the finite case. In the infinite case, the sets of states needs to be declared as being Kuratowski infinite in order to say that the elements of the set are never completely defined, and so a forteriori the size of the set cannot be defined in terms of it's finite subsets. — sime

Why can't a nation in some alternate universe have infinitely many states, and choose a legislature? My argument here is that although AC is independent of ZF, it's nevertheless intuitively true. Even the ld joke admits that. AC is definitely true, the well-ordering theorem is definitely false, and Zorn's lemma, who knows! The joke being that they're all logically equivalent.

Secondly, the set should be declared as Dedekind finite, in order to say that the set is an observable collection of elements and not a function (because only functions can be dedekind-infinite). — sime

You lost me there. Absent AC there is a set that is infinite (not bijective with any natural number) yet Dedekind-finite (no proper subset is bijective with the entire set). I don't know what you mean here.

So, yes, you can choose as many representatives as you wish without implying a nonsensical completed collection of legislatures that are a proper subset of themselves, but formalisation of these sets isn't possible in ZFC, because AC and it's weaker cousin, the axiom of countable choice, forces equivalence of Kuratowski finiteness and Dedekind finiteness. — sime

I don't think you're engaging with my point. Why can't the uncountably many provinces of the planet Zork choose themselves a legislature? Why on earth can't we form a set consistent of one element from each of a collection of nonempty sets? It's intuitively true that we can, even if AC is not provable from ZF.

Besides, Gödel showed that AC is true in his constructible universe in which, as I understand it, all ordinary mathematics takes place anyway. So there is a perfectly good model of math in which AC is true. You can't stand on a soapbox and deny that.

But if you prefer to do math without AC, or if you are a constructivist, or a finitist or ultrafinitist, you're in good company. I can't argue you out of your preferences. I can only question your soapbox emotions. After all most serious constructive mathematicians still use some version of AC, because without it it's more difficult to do math.

That's right, to specify that they are real numbers is to specify, just like to specify that the guests at the hotel are human beings is to specify. The fact that a specification is vague, incomplete, or imperfect does not negate the fact that it is a specification. — Metaphysician Undercover

Only if you change what a specification is. In set theory, a specification is a predicate, a statement that can be true or false of a given item. The items for which the predicate is true, go into some set.

By saying, "Oh, they're all real numbers" when the set in question only contains SOME of the real numbers, does not specify the set.

But your definition of specification is not even valid by the everyday English meaning of the word. If you go into a store to buy a computer and you ask the salesman for the specifications of a particular model you have in mind, and they say, "Oh, it's a material object made of atoms," you'd walk out of the store. He has told you the truth, but has not given the specifications of the computer.

Likewise if you are a manufacturer and you subcontract out a particular part, asking to have it made to a particular specification, and they send you back a lump of metal totally unlike what you asked for, saying, "Well it's a physical object made of atoms, what more to you want," you'd sue them for breach of contract.

I told you how so. You've specified that the set contains real numbers. You are the one who explained to me, that 'set" is logically prior to "number", and that not all sets have numbers as elements. This means that "set" is the more general term. How can you now deny that to indicate that a particular set consists of some real numbers, is not an act of specifying? — Metaphysician Undercover

You have not specified which real numbers are in the set and which aren't. And if you really believed what you are saying, you would have told me long ago that every set is either empty or contains other sets; and THAT is a specification. Is that what you claim? Every set is fully specified by saying its elements are other sets? That's nonsense. That's not what a set specification is.

A set specification consists of two things: One, an existing set; and two, a predicate saying which elements of the given set are members of your specified set. You haven't done that. You've made a trivial and sophistic point, saying that every set of real numbers is "specified" because it contains real numbers. That's nonsense. It's childish.

Good, you now accept that every set has a specification. — Metaphysician Undercover

Only by completely changing both the mathematical AND the everyday meaning of specification.

Do you also agree now that this type of specification, which "doesn't tell me how to distinguish members of a set from non members", is simply a bad form of specification? — Metaphysician Undercover

Abraham Lincoln used to ask, If you call a tail a leg, how many legs does a dog have? And he answered: Four. Calling a tail a leg doesn't make it a leg.

Likewise, calling your vague and sophistic characterization of a set a specification does not make it a specification. Of course it could be argued that logically IF I call a tail a leg, then a dog has five legs. In THAT sense, you have made your point. Which is to say: You haven't made your point.

Do you now see, and agree, that since a set must be specified in some way, then the elements must be "the same" in some way, according to that specification, therefore it's really not true to say that "the elements of a set need not be "the same" in any meaningful way." — Metaphysician Undercover

Not in the least. Of course in pure set theory (ie set theory without urelements), the elements of every nonempty set are other sets. So it's true that the elements of every set are sets, and they have that in common. But that is not a property that distinguishes any given set from all other sets, so it's not a specification. If you want to call it a specification that's your right, just as you can call a tail a leg and say that a dog has five legs. If you can get anyone to take you seriously.

So we can get rid of that appearance of contradiction by stating the truth, that the elements of a set must be the same in some meaningful way. To randomly name objects is not to list the members of a set, because a set requires a specification. — Metaphysician Undercover

You haven't specified a set by pointing out that all its elements are sets. Because that doesn't distinguish that set from any other set. But like I say, you can hang on to your childish sophistry or you can hang on to your credibility. You might as well hang on to the former, having long ago lost the latter.

What I am trying to get at, is the nature of a "set" You say that there is no definition of "set", but it has meaning given by usage. Now I see inconsistency in your usage, so I want to find out what you really think a set is. Consider the following. — Metaphysician Undercover

Again -- AGAIN -- you are equivocating between the definition of a set in set theory, of which there is none -- just check the axioms please -- and the fact that some sets are given by specifications, where a specification consists of an already-existing set and a predicate. See the axiom schema of specification for an explication of this point. It's called a schema because it actually consists of infinitely many axioms, one for each predicate.

Since we now see that a set must have a specification, — Metaphysician Undercover

The Vitali set has no specification. It's true that all its elements are real numbers; and in fact all its elements are also sets, since in set theory, real numbers are modeled as sets. But that's as I said a childish and sophistic point, which you can hold to only at the loss off your own credibility.

do you see how the above quote is inconsistent with that principle? — Metaphysician Undercover

No, I see you making an astonishingly childish and sophistic point, calling a tail a leg and saying a dog has five legs.

Since a set must have a specification, a set is itself an "articulable category or class of thought". — Metaphysician Undercover

Most sets have no specifications. You've lost all credibility with me at this point. Well to be fair, you lost all credibility last week when you denied that pi is a particular real number. I have no idea what you might be thinking, but whatever it is, it's wrong.

And, it is not the "being gathered into a set" which constitutes the relations they have with one another, it is the specification itself, which constitutes the relations. — Metaphysician Undercover

Sure, trivially. But not meaningfully, since it's circular. How do I tell which real numbers are in the Vitali set? Well, they're in the Vitali set. Not helpful.

So if you specify a set containing the number five, the tuna sandwich you had for lunch, and the Mormon tabernacle choir, this specification constitutes relations between these things. That's what putting them into a set does, it constructs such relations. — Metaphysician Undercover

If it makes you happy to hold to this line of argument, I would not take that away from you.

Now here's the difficult part. — Metaphysician Undercover

Oh this should be good.

Do you agree that there are two distinct types of sets, one type in which the specification is based in real, observed similarities, a set which is based on description, and another type of set which is based in imaginary specifications, a set produced as a creative act? — Metaphysician Undercover

Only to the extent that your first category is empty. You ask if there's a set based on "observed" similarities. I have never observed a set. I've been to math grad school and never observed a set. Set's are entirely abstract objects and are not observable in the sense of physics nor in the sense of everyday English. You cannot observe a set.

And "real?" What on earth do you mean by that?

FWIW I will grant that you might have meant to ask: Are there some sets given by the axiom of specification; and others that are not, and that are essentially nonconstructive? In which case yes, the set of even natural numbers is an example of the former, and the Vitali set the latter. But neither set is particular "real" or "observable." I have never observed an even number. I've seen four apples. And after they downgraded poor old Pluto, I then knew about the eight planets. But four? Or eight? I've never observed them. They're both abstract entities.

Do you acknowledge that these two types of sets are fundamentally different? — Metaphysician Undercover

Sure. All the sets that there are, are in the latter category; and none are in the former. No sets are based on anything "real" or "observed," and all sets are "based in imaginary specifications, a set produced as a creative act." Those are the only types of sets there are.

Just as there are two types of elephants: those that fly, and those that don't fly. That is a true statement. It's just that all the elephants are in the latter category and none in the former.

But being charitable and assuming you meant to ask about constructive and nonconstructive sets, sure. Constructivist mathematicians make the distinction all the time. Just ask @sime, who rejects the axiom of choice. He does not believe in the Vitali set (I assume, though we've never directly discussed it), and believes all sets are the output of some algorithm or deterministic process. It's a fair distinction.

if one works with a normed linear space that is separable, the Hahn-Banach theorem doesn't require it either. — jgill

Nice to have an actual mathematician around here!

OED: specify, "to name or mention". Clearly the set you called "V" is not unspecified, and it's you who wants to change the meaning "specify" to suit your (undisclosed) purpose. Sorry fishfry, but you appear to be just making stuff up now, to avoid the issues.

AC, ZL and WO are not logically equivalent. But they are equivalent in Z set theory.

OED: specify, "to name or mention". Clearly the set you called "V" is not unspecified, and it's you who wants to change the meaning "specify" to suit your (undisclosed) purpose. Sorry fishfry, but you appear to be just making stuff up now, to avoid the issues. — Metaphysician Undercover

Whatever, man.

"suit your (undisclosed) purpose" -- what does that mean? I'm using specification as in the axiom schema of specification. We are talking about set theory after all. If we're talking about baseball, a "fly" is a ball that's hit by the batter and remains in the air without touching the ground. It's not a winged insect.

What "undisclosed purpose" would I have? The corruption of the youth? What are you talking about?

Nevermind, I don't want to know.

If as you agree, all sets in standard set theory are composed of nothing but other sets; and that therefore every nonempty set whatsoever can be said to have elements that are sets; then isn't the fact that the elements of any set have in common the fact that they are sets, a rather trivial point? Can you see that this is not a helpful criterion to specify which elements are in the set and which are not? It's the example I gave of the doorman at a highly exclusive club who's told to just let everyone in. To be a valid specification, you have to tell the doorman which people to let in, and which to not let in.

But like I said, I am perfectly willing to accept your personal definition of the word; at the cost of no longer being able to take you even slightly seriously; since you're making such an unserious point.

AC, ZL and WO are not logically equivalent. But they are equivalent in Z set theory. — TonesInDeepFreeze

Not sure I follow. You mention Z a lot but that's a pretty obscure system unless one is a specialist. AC, ZL, and WO are surely equivalent in ZF. In what way would you say they're not equivalent?

if one works with a normed linear space that is separable, the Hahn-Banach theorem doesn't require it either. — jgill

Nice to have an actual mathematician around here! — fishfry

But one left on the ground while math, like Buzz Lightyear, has gone "To infinity and beyond ! "

They are equivalent in Z, so, a fortiori, they are equivalent in ZF. But they are not logically equivalent.

Z |- AC <-> ZL & ZL <-> WO & AC <-> WO

But it is not the case that

|- AC <-> ZL & ZL <-> WO & AC <-> WO

They are equivalent in Z, so, a fortiori, they are equivalent in ZF. But they are not logically equivalent. — TonesInDeepFreeze

Is Z supposed to be Zermelo set theory? In Wiki they call it $Z^-$, is that the same thing?

https://en.wikipedia.org/wiki/Zermelo_set_theory

Why do you work in Z so much, why do you care, why exactly should I care? I'm not familiar with it. Perhaps you can supply some context please.

Z |- AC <-> ZL & ZL <-> WO & AC <-> WO

But it is not the case that

|- AC <-> ZL & ZL <-> WO & AC <-> WO — TonesInDeepFreeze

Why is that last line not true? What exactly do you mean? Are you just saying it's not a tautology? Can you give a model in which it's false? In my experience, the claim that "AC, WO, and ZL are equivalent" is totally noncontroversial and nobody (but you) would look twice at it. Why do you take exception?

What I'm looking for is context and understanding of your thought process.

Z is axiomatized by:

extensionality
schema of separation
pairing
union
power
infinity
regularity

It is uncontroversial that AC, ZL, and WO are equivalent in the sense that in Z (perforce ZF) we can prove one from the other. But it is not the case that they are logically equivalent. To be logically equivalent they would have to be provable one from the other using only the pure predicate calculus.

Absent AC there is a set that is infinite (not bijective with any natural number) yet Dedekind-finite (no proper subset is bijective with the entire set). — fishfry

Absent AC, it is undecided whether there is such a set.

Absent AC, it is undecided whether there is such a set. — TonesInDeepFreeze

Yes ok I meant assuming the negation. But if you don't specify one or the other, you're right.

About the schema of separation, if we say there is one axiom for each predicate, we need to be careful what 'predicate' means. There is one axiom for each formula. We could say that each formula permits definition of a predicate symbol, so, in that sense there is one axiom for each defined predicate symbol. But we should be clear to say we don't mean 'predicate' in the sense of 'property'.

Not sure, but offhand, I suspect it is not the case that ~AC implies there is Tarski infinite but Dedekind finite set. The converse holds though.

If as you agree, all sets in standard set theory are composed of nothing but other sets; and that therefore every nonempty set whatsoever can be said to have elements that are sets; then isn't the fact that the elements of any set have in common the fact that they are sets, a rather trivial point? — fishfry

I don't see that as a trivial point, because not only is "set" undefined, but also "element" is undefined. So we have a vicious circle which makes it impossible to understand what type of thing a set is supposed to be, and what type of thing an element is supposed to be. What is a set? It's something composed of elements. What is an element? It's a set.

Under this description, a particular set is identified by its elements, not by a specification, definition, or description. Do you see what I mean? Under your description, any particular set cannot be identified by the predicates which are assigned to the elements, because it is not required that there be any assigned predicates.. But there still might be such an identified set. So a set must be identified by reference to its members. This is why, under this description of sets, the empty set is logically incoherent. A proposed empty set has no members, and therefore cannot be identified.

If, on the other hand, a set is identified by it's specification, definition, or description, (which you deny that it is), then there could be a definition, specification, or predication which nothing matches, and therefore an empty set.

Hopefully you can see that the two, identifying a set by its elements, and identifying a set by its predications, are incompatible, because one allows for an empty set, and the other does not. So as much as "set" may have no formal definition, we cannot confuse or conflate these two distinct ways of using "set" without the probability of creating logical incoherency.

By saying that "set" has no definition, we might be saying that there is nothing logically prior to "set", that we cannot place the thing referred to by the word into a category. But if you make a designation like "there is an empty set", then this use places sets into a particular category. And if you say that a set might have no specification, this use places sets into an opposing category. If you use both, you have logical incoherency.

Therefore it is quite clear to me, that the question of whether a set is identified by reference to its elements, or identified by reference to its specification, is a non-trivial matter because we cannot use "set" to refer to both these types of things without logical incoherency.

I don't see that as a trivial point, — Metaphysician Undercover

What's trivial is saying that the Vitali set is "specified" because all its elements are real numbers. That's like saying the guests at a particular hotel this weekend are specified because they're all human. It's perfectly true, but it tells you nothing about the guests at the hotel. That's why your point is trivial.

because not only is "set" undefined, but also "element" is undefined. — Metaphysician Undercover

If $x \in y$ is true, then If $x$ is an element of If $y$. That's perfectly well defined in terms of If $\in$, which is an undefined primitive. I've referred you many times to the axioms of Zermelo-Fraenkel set theory, or ZF, which if you'd read the page, might answer many of your questions.

It's also sometimes called ZFC in honor of Mrs. Zermelo, who was pro choice.

So we have a vicious circle which makes it impossible to understand what type of thing a set is supposed to be, — Metaphysician Undercover

On the one hand you're the only one who doesn't understand this. On the other hand, Skolem argued that the notion of set is too vague to be useful as a foundation for mathematics, and he was one of the greatest of the early set theorists. So you're not wrong. It would be better if you had more sophisticated arguments, because it would then be more interesting and fun to engage with you.

and what type of thing an element is supposed to be. — Metaphysician Undercover

If $x \in y$ then the thing on the left is an element of the set on the right. In pure set theory the thing on the left is also a set; and in set theories with urelements, the thing on the left might not be. Likewise in applications the thing on the left might be something else such as a voter in social choice theory, or a rational actor in an economic theory, etc.

What is a set? — Metaphysician Undercover

The axioms don't tell us. A set is characterized by its behavior under the axioms. Even you've agreed to that previously.

It's something composed of elements. What is an element? It's a set. — Metaphysician Undercover

So what? You want an explicit definition, but in set theory there are no such definitions. Read the axioms. You're just recapitulating Frege's complaints to Hilbert, but I can't argue the point because there's no right or wrong to the matter. You don't like it, that's your right.

Under this description, a particular set is identified by its elements, not by a specification, definition, or description. Do you see what I mean? — Metaphysician Undercover

LOL. Yes, that's the axiom of extensionality, which I've been explaining to you for at least two years. Glad you finally got it. A set is entirely characterized by its elements.

Under your description, any particular set cannot be identified by the predicates which are assigned to the elements, because it is not required that there be any assigned predicates.. — Metaphysician Undercover

Some sets are specified by predicates, such as the set of all natural numbers that are prime. Some sets aren't specified by predicates, such as the Vitali set.

And as I've noted, constructivist mathematicians, neo-intuitionists, finitists, and ultrafinitists only believe in sets that can be constructed by an algorithm or explicit procedure. So this is a fair debate in the philosophy of math.

But there still might be such an identified set. — Metaphysician Undercover

Like the set of natural numbers that are prime. That's a set given by the axiom schema of specification.

So a set must be identified by reference to its members. T — Metaphysician Undercover

"By reference?" No. The Vitali set is characterized by its members, but I can't explicitly refer to them because I don't know what they are. It's a little like knowing that there are a billion people in China, even though I don't know them all by name.

his is why, under this description of sets, the empty set is logically incoherent. A proposed empty set has no members, and therefore cannot be identified. — Metaphysician Undercover

On the contrary. Since everything is equal to itself, the empty set is defined as $\{x : x \neq x\}$. I rather thought you'd appreciate that, since you like the law of identity. The empty set is in fact the extension of a particular predicate.

If, on the other hand, a set is identified by it's specification, definition, or description, (which you deny that it is), then there could be a definition, specification, or predication which nothing matches, and therefore an empty set. — Metaphysician Undercover

Exactly. The empty set is the extension of the predicate $x \neq x$. Or if you like, it's the extension of the predicate "x is a purple flying elephant." Amounts to the same thing.

Hopefully you can see that the two, identifying a set by its elements, and identifying a set by its predications, are incompatible, because one allows for an empty set, and the other does not. — Metaphysician Undercover

Since the empty set is the extension of a particular predicate, your point is incoorect.

So as much as "set" may have no formal definition, we cannot confuse or conflate these two distinct ways of using "set" without the probability of creating logical incoherency. — Metaphysician Undercover

You're wrong, since the empty set is the extension of a predicate.

By saying that "set" has no definition, we might be saying that there is nothing logically prior to "set", that we cannot place the thing referred to by the word into a category. — Metaphysician Undercover

In set theory that's true. Although in the formal theory, the axioms and the inference rules of first-order predicate logic are taken as logically prior.

But if you make a designation like "there is an empty set", then this use places sets into a particular category. — Metaphysician Undercover

Well it's in the category of sets. Which is formally true in the category of sets, and is also true in the sense of "category" that you're using. A set is a set.

And if you say that a set might have no specification, this use places sets into an opposing category. If you use both, you have logical incoherency. — Metaphysician Undercover

I don't see why. Every set is fully characterized by its elements. Specification is one particular way of identifying or constructing or proving the existence of particular sets. Other ways are union, intersection, and so forth. It's all given in the axioms, which I've given you the link to many times and which I'm sure you've never even bothered to glance at. You know the most credible way to make an argument is to take the trouble to become familiar with the thing you're arguing against. You won't do that. So you make very trivial and sophistic arguments.

Therefore it is quite clear to me, that the question of whether a set is identified by reference to its elements, or identified by reference to its specification, is a non-trivial matter because we cannot use "set" to refer to both these types of things without logical incoherency. — Metaphysician Undercover

I don't see why. Now do I agree that constructive mathematicians do make this sharp distinction between sets that can be constructed by an algorithm or mechanical procedure (there are many sub-flavors of this idea) versus mathematicians who believe in nonconstructive sets such as the Vitali set. @sime, for example, is one who makes this distinction. If you wish to make an argument along these lines, that would be an interesting conversation. But I don't think that's what you're doing. I don't know what you're doing. i don't know what your point is.

tl;dr: There are many sophisticated arguments against set theory. Therefore you're not shocking me by questioning set theory. But your arguments are from ignorance rather than knowledge. So your arguments are not interesting.

ps -- Am I being to harsh to your ideas? I can't really follow your logic. The axioms are very clear. There's no circularity or infinite regress involved. There's no formal definition of set in the axioms; but some sets are given by specifications and others not. They're not "defined" by the specifications, rather their existence is given by the the axiom of specification. Some other sets are given by various other axioms. Replacement is one of the more subtle ones. Perhaps it's helpful to think of the axioms as a toolkit for determining which sets can exist. In that respect, specification isn't all that special. As to what it all means, perhaps it means nothing at all. That's not a problem, neither does chess.

What's trivial is saying that the Vitali set is "specified" because all its elements are real numbers. That's like saying the guests at a particular hotel this weekend are specified because they're all human. It's perfectly true, but it tells you nothing about the guests at the hotel. That's why your point is trivial. — fishfry

It's not trivial, because it's a demonstration of what "specified" means. If you specify that the guests are all human, then clearly that is a specification. If you do not appreciate that specification because it does not provide you with the information you desire, then the specification is faulty in your eyes. But it's false to say that just because you think the specification is faulty, then there is no specification. There is a specification, but it is just not adequate for you. That is simply the nature of specification, it comes in all different degrees of adequacy, depending on what is required for the purpose. But an inadequate specification, for a particular purpose, is in no way a total lack of specification.

A set is entirely characterized by its elements. — fishfry

Do you see then, that if "A set is entirely characterized by its elements", then a so-called empty set is not possible? If there are no elements, under that condition, then there is no set. A set is characterized by its elements. There are no elements. Therefore there is no set. If we adhere to this premise, "the set is entirely characterized by its elements", then when there is no elements there is no set.

Some sets are specified by predicates, such as the set of all natural numbers that are prime. — fishfry

This is logically inconsistent with "a set is entirely characterized by its elements", as I explained in the last post. Either a set is characterized by its elements, or it is characterized by its specified predicates, but to allow both creates the incoherency which I referred to. One allows for an empty set, the other does not.

"By reference?" No. The Vitali set is characterized by its members, but I can't explicitly refer to them because I don't know what they are. It's a little like knowing that there are a billion people in China, even though I don't know them all by name. — fishfry

We've been through this already. You clearly have referred to the members of the Vitali set. You've said that they are all real numbers. Why do you believe that this is not a reference to the members of the set? You can say "all the people in China", and you are clearly referring to the people in China, but to refer to a group does not require that you specify each one individually.

This seems to be where you and I are having our little problem of misunderstanding between us. It involves the difference between referring to a group, and referring to individual. I believe that when you specify a group, "all the guests at the hotel" for example, you make this specification without the need of reference to any particular individuals. You simply reference the group, and there is no necessity to reference any particular individuals. In fact, there might not be any individuals in the group (empty set). You seem to think that to specify a group, requires identifying each individual in that group.

This is the two distinct, and logically inconsistent ways of using "set" which I'm telling you about. We can use "set" to refer to a group of individuals, each one identified, and named as a member of that set (John, Jim, and Jack are the members of this set), or we can use "set" to refer simply to an identified group, "all the people in China". Do you see the logical inconsistency between these two uses, which I am pointing out to you? In the first case, if there are no identified, and named individuals, there is no set. Therefore in this usage there cannot be an empty set. But in the second case, we could name the group something like "all the people on the moon", and this might be an empty set.

On the contrary. Since everything is equal to itself, the empty set is defined as {x:x≠x}{x:x≠x}. I rather thought you'd appreciate that, since you like the law of identity. The empty set is in fact the extension of a particular predicate. — fishfry

I must say, I really do not understand your notation of the empty set. Could you explain?

The empty set is the extension of the predicate x≠xx≠x. Or if you like, it's the extension of the predicate "x is a purple flying elephant." Amounts to the same thing. — fishfry

This doesn't help me.

Since the empty set is the extension of a particular predicate, your point is incoorect. — fishfry

Actually you don't seem to be getting my point. The point is that if a set is characterized by its predicates, then an empty set is possible, so I have no problem with "the empty set is the extension of a particular predicate". Where I have a problem is if you now turn around and say that a set is characterized by its elements, because this would be an inconsistency in your use of "set", as explained above. A set characterized by its elements cannot be an empty set, because if there is no elements there is no set. Do you apprehend the difference between "empty set" and "no set"?

I don't know what you're doing. i don't know what your point is.

...

I can't really follow your logic. — fishfry

Perhaps it's a bit clearer now?

It's not trivial, because it's a demonstration of what "specified" means. If you specify that the guests are all human, then clearly that is a specification. If you do not appreciate that specification because it does not provide you with the information you desire, then the specification is faulty in your eyes. But it's false to say that just because you think the specification is faulty, then there is no specification. There is a specification, but it is just not adequate for you. That is simply the nature of specification, it comes in all different degrees of adequacy, depending on what is required for the purpose. But an inadequate specification, for a particular purpose, is in no way a total lack of specification. — Metaphysician Undercover

I've already agreed numerous times that if you insist on your own definition, you're right. You can't convince me that your way of looking at it isn't trivial. I originally understood you to be claiming that the members of every set had something in common that distinguished them of all the non-members. That's an interesting statement, and you'd find many constructivists who agree with you (to the extent that I understand the mindset of the constructivists).

I was surprised -- shocked, in fact -- to discover that you didn't mean that at all, but only meant that you could find some superset that contained the potential candidates for our set. That's a much weaker criterion, I hope you at least agree to that.

So yes, all the guests are human, but that hardly helps the security guard to know who to let in and who to keep out. And every element of the set of primes is a natural number, but that doesn't tell me what's a prime and what's not. It's an incredibly weak criterion. I'm happy to let you have it, but it's trivial because it's such a weak criterion as to be utterly useless in determining which elements are in a given set.

Do you see then, that if "A set is entirely characterized by its elements", then a so-called empty set is not possible? If there are no elements, under that condition, then there is no set. A set is characterized by its elements. There are no elements. Therefore there is no set. If we adhere to this premise, "the set is entirely characterized by its elements", then when there is no elements there is no set. — Metaphysician Undercover

First, the empty set is the unique set that has no elements at all. It's characterized by not having any elements.

But for a more precise answer, we have to look at the actual, exact formal statement of the axiom of extensionality. The natural language version, "A set is entirely characterized by its elements," is just an approximation to what the axiom actually says. The problem is that you don't relate to symbolic reasoning at all. That said, and for the record, I'll walk you through what the axiom of extensionality actually says; and for reference, you can see the Wiki link.

Axiom of extenionality: $\forall A \forall B (X \in A \iff X \in B) \implies A = B$

We unpack this as follows. It says that

For all sets $A$ and $B$:

If it happens to be the case that for all sets $X$, $X \in A$ if and only if $X \in B$;

Then $A = B$.

This says in effect that if two sets have exactly the same elements, they're the same set. But the way it's written, it also includes the case of a set with no elements at all. If you have two sets such that they have no elements, they're the same set; namely the empty set.

Now I know this isn't your cup of tea. And that's ok. All you need to know about this is that when you drill down into the technical details of what the axiom of extensionality actually says, the case of the empty set is included.

A more general point is that you want to criticize set theory based on vague natural language descriptions rather than grappling with the actual formal symbology. But in the end, we all have to roll up our sleeves and grapple with the symbology.

Another point is that everyone has trouble with vacuous arguments and empty set arguments. If 2 + 2 = 5 then I am the Pope. Students have a hard time seeing that that's true. The empty set is the set of all purple flying elephants. A set is entirely characterized by its elements; and likewise the empty set is characterized by having no elements. John von Neumann reportedly said, "You don't understand math. You just get used to it." The empty set is just one of those things. You can't use your common sense to wrestle with it, that way lies frustration.

This is logically inconsistent with "a set is entirely characterized by its elements", as I explained in the last post. Either a set is characterized by its elements, or it is characterized by its specified predicates, but to allow both creates the incoherency which I referred to. One allows for an empty set, the other does not. — Metaphysician Undercover

You keep equivocating specification by predicates, on the one hand; and the axiom of extensionality, on the other. Every set is entirely characterized by its elements. Secondly, quite separately from that fact, are various ways of showing the existence of sets. Given a collection of sets we can take their union; or their intersection. Given a set we can take its powerset. Given a set and a predicate we can use the axiom of specification to obtain a subset of the original set consisting of exactly those of its members satisfying the predicate. There's also the axiom of replacement, and the axiom of choice. So first, a set is entirely characterized by its elements. And secondly, we have a toolbox for showing the existence of various sets: union, intersection, powerset, specification, replacement, choice. It's sort of like knowing what a house is, then learning how to build one. There's no ambiguity. Those are two separate things.

We've been through this already. You clearly have referred to the members of the Vitali set. You've said that they are all real numbers. Why do you believe that this is not a reference to the members of the set? You can say "all the people in China", and you are clearly referring to the people in China, but to refer to a group does not require that you specify each one individually. — Metaphysician Undercover

Of course all the members of the Vitali set are real numbers. As are all the elements that are NOT members of the Vitali set. So you can have your point, but it's rather pointless. It doesn't do you any good.

This seems to be where you and I are having our little problem of misunderstanding between us. It involves the difference between referring to a group, and referring to individual. I believe that when you specify a group, "all the guests at the hotel" for example, you make this specification without the need of reference to any particular individuals. You simply reference the group, and there is no necessity to reference any particular individuals. In fact, there might not be any individuals in the group (empty set). You seem to think that to specify a group, requires identifying each individual in that group. — Metaphysician Undercover

Well that's fine, then you believe in nonconstructive sets. You are willing to take the Vitali set and its members at face value. That's great. But you can see that it's very different than the set of prime numbers. With the set of prime numbers, we can look at a given individual number and say, "Yes you're in the club," or "No you're not in the club." With the Vitali set, there is no way to do that.

This is the two distinct, and logically inconsistent ways of using "set" which I'm telling you about. — Metaphysician Undercover

They're not logically inconsistent, they're different ways of building sets; just as using brick or using wood are two different ways of building houses.

We can use "set" to refer to a group of individuals, each one identified, and named as a member of that set (John, Jim, and Jack are the members of this set), or we can use "set" to refer simply to an identified group, "all the people in China". — Metaphysician Undercover

Yes. We can do one or the other. We can talk about the set of prime numbers, in which we can talk about the entire set AND determine exactly which natural numbers are allowed into the set; and we can talk about the Vitali set, where we can NOT determine for any particular real number whether it belongs in the set or not.

Two different ways of obtaining or showing the existence of sets. There are lots of different ways of building houses and lots of different ways of building sets. I don't know why you think this is a problem.

Do you see the logical inconsistency between these two uses, which I am pointing out to you? — Metaphysician Undercover

No, I only see various ways of building sets. Unions, intersections, powersets, specification, replacement, and choice. I believe those are all the set-building or set-existence tools. Like styles of houses or perhaps construction techniques or different choices of materials.

In the first case, if there are no identified, and named individuals, there is no set. Therefore in this usage there cannot be an empty set. — Metaphysician Undercover

The axiom of extensionality provides for the empty set. It's the set with no elements.

But in the second case, we could name the group something like "all the people on the moon", and this might be an empty set. — Metaphysician Undercover

Aha! You're on the verge of getting it. The set of pink flying elephants is an empty set. The set of people on the moon is an empty set. And the axiom of extensionality says that these must be exactly the same set. Because an object is an element of one if and only if it's an element of the other. I hope you'll take a moment to work through the logic. There is only one empty set, because the axiom of extensionality says that if for every object, it's a person on the moon if and only if it's a pink flying elephant, that the two sets must be the same.

I must say, I really do not understand your notation of the empty set. Could you explain? — Metaphysician Undercover

We know from the law of identity that everything is equal to itself. So what is the set of all things that are not equal to themselves? It's the empty set. And by the axiom of extensionality, it's exactly the same as the set of pink flying elephants and the people on the moon.

We notate this as $\{x : x \neq x\}$. It's read: "The set of all x such that x is not equal to x." As an expert on the law of identity you will agree that there are no such x that satisfy that condition. So we have specified the empty set.

This doesn't help me. — Metaphysician Undercover

I hope my more detailed explanation was better.

Actually you don't seem to be getting my point. The point is that if a set is characterized by its predicates, then an empty set is possible, so I have no problem with "the empty set is the extension of a particular predicate". — Metaphysician Undercover

Ok.

Where I have a problem is if you now turn around and say that a set is characterized by its elements, — Metaphysician Undercover

You're confusing two different things. The axiom of extensionality tells you when two sets are identical: Namely, when they have exactly the same elements.

The axiom of specification, the powerset axiom, the axiom of choice, the axiom schema of replacement, and the axioms of union, intersection, and pairing (I forgot to mention that one) are all axioms that tell us which particular sets exist.

So we have an axiom that tells us when two sets are equal. That's extensionality, or "a set is entirely characterized by its elements." And we have a toolbox of ways to show that various sets exist. The're not in conflict with each other, any more than saying what a house is, is in conflict with the various construction techniques and styles of houses.

because this would be an inconsistency in your use of "set", as explained above. — Metaphysician Undercover

No. The axiom of extensionality tells us when two sets are the same; namely, when they have exactly the same elements. The other axioms are a toolbox for knowing which sets exist. Specification is one of the tools as are union, intersection, etc.

A set characterized by its elements cannot be an empty set, because if there is no elements there is no set. — Metaphysician Undercover

The empty set satisfies the symbolic expression I discussed earlier, whether or not it satisfies the English-language version. I can only refer you to the Wiki page on the axiom of extension, which I keep pointing you to and you keep not reading.

Do you apprehend the difference between "empty set" and "no set"? — Metaphysician Undercover

Most definitely. The empty set is a set. No set is no set.

Perhaps it's a bit clearer now? — Metaphysician Undercover

Yes. You are confusing the axiom of extension, which tells us when two sets are the same, with the other axioms that give us various ways to build sets or prove that various sets exist.

You lost me there. Absent AC there is a set that is infinite (not bijective with any natural number) yet Dedekind-finite (no proper subset is bijective with the entire set). I don't know what you mean here. — fishfry

I'm talking about stream objects in computer science, or equivalently the isomorphism
S <----> 1 x S in the Set category, where S is a stream object, 1 is the terminal object, i.e the singular set, and x is the Cartesian product. Unfolding the definition:

S <----> 1 x S <----> 1 x (1 x S) <----> 1 x (1 x (1 x S)) ....

Since each of the arrows is invertible, S clearly has, by definition, a surjection onto any finite set, which is what i meant by Kuratowski infiniteness.

On the other hand, recall that in category theory every element of a set S is an arrow of the form
1 --> S. However, these arrows haven't been specified in my above definition of S, and therefore the number of elements of S is currently zero, i.e. S is the empty set, which is another way of saying that S is a completely undefined set until the first observation is made.

Every time an observation is made, an arrow of the form 1--> S is introduced into the above category, and we can denote the current state of the stream by shifting from left to right in the above diagram. But at every moment of time, the arrow S --> S that is implicit in the the product S ---> 1 x S is a bijection, meaning that is S is forever dedekind-finite.

S cannot exist as an internal set of ZF because it isn't a well-founded set, although it can exist in the sense of an "external set" that is to say, inside ZF as part of a non-standard interpretation of an internal well-founded set. And it it cannot exist in any capacity inside ZFC.

All of which is tantamount to saying that ZF has only partial relevance to modern mathematics in terms of being an axiomatization of well-foundedness, whilst ZFC is completely and utterly useless, failing to axiomatize the most rudimentary notions of finite sets as used in the modern world.

This says in effect that if two sets have exactly the same elements, they're the same set. But the way it's written, it also includes the case of a set with no elements at all. If you have two sets such that they have no elements, they're the same set; namely the empty set. — fishfry

You do not seem to be grasping the problem. If a set is characterized by its elements, there is no such thing as an empty set. No elements, no set. Do you understand this? That is the logical conclusion we can draw from " a set is characterized by its elements". If we have no elements, we have no set. If you do not agree with this, explain to me how there could be a set which is characterized by its elements, and it has no elements. It has no character? Isn't that the same as saying it isn't a set?

So we cannot proceed to even talk about an empty set because that's incoherent, unless we dismiss this idea that a set is characterized by its elements. Can we get rid of that idea? Then we could proceed to investigate your interpretation of the axiom of extensionality, which allows you to say "If you have two sets such that they have no elements, they're the same set; namely the empty set", because "empty set" would be a coherent concept. Until we get rid of that premise though, that a set is characterized by its elements there is no such thing as a set with no elements, because such a set would have no identity whatsoever, and we could not even call it a set.

Another point is that everyone has trouble with vacuous arguments and empty set arguments. If 2 + 2 = 5 then I am the Pope. Students have a hard time seeing that that's true. The empty set is the set of all purple flying elephants. A set is entirely characterized by its elements; and likewise the empty set is characterized by having no elements. John von Neumann reportedly said, "You don't understand math. You just get used to it." The empty set is just one of those things. You can't use your common sense to wrestle with it, that way lies frustration. — fishfry

You are not grasping the distinction between 'characterized by its elements', and 'characterized by its specification' which I'm trying to get though to you. When you say "the set of all purple flying elephants", this is a specification, and this set is characterized by that specification. There are no elements being named, or described, and referred to as comprising that set, there is only a specification which characterizes the set.

Every set is entirely characterized by its elements. — fishfry

Where do you get this idea from? Clearly your example "the set of all purple flying elephants" is not characterized by its elements. You have made no effort to take elements, and compose a set You have not even found any of those purple flying elephants. In composing your set, you have simply specified "purple flying elephants". Your example set is characterized by a specification, not by any elements. If you do not want to call this "specification", saving that term for some special use, that's fine, but it's clearly false to say that such a set is characterized by its elements.

This is what happens when we proceed deep into the workings of the imagination. We can take a symbol, a name like "purple flying elephants", or any absurdity, or logical incoherency, like "square circles", each of which we assume has no corresponding objects However, we can then claim something imaginary, a corresponding imaginary object, and we can proceed under the assumption that the name actually names something, a purple flying elephant in the imagination. You might then claim that this imaginary thing is an element which characterizes the set. But if you then say that the set is empty, you deny the reality of this imaginary thing, and you are right back at square one, a symbol with nothing corresponding. And so we cannot even call this a symbol any more, because it represents nothing.

The set of pink flying elephants is an empty set. The set of people on the moon is an empty set. And the axiom of extensionality says that these must be exactly the same set. — fishfry

Now you've hit the problem directly head on. To be able to have an empty set, a set must be characterized by it's specification, as I've described, e.g. "pink flying elephants". So. the set of pink flying elephants is one set, characterized by the specification "pink flying elephants", and the set of people on the moon is another set, characterized by the specification "people on the moon". To say that they are exactly the same set, because they have the same number of elements, zero, is nor only inconsistent, but it's also a ridiculous axiom. Would you say that two distinct sets, with two elements, are the exact same set just because they have the same number of elements? I think you'll agree with me that this is nonsense.

And to say that each of them has the very same elements because they don't have any, is clearly a falsity because "pink flying elephants" is a completely different type of element from "people on the moon". If at some point there is people on the moon, then the set is no longer empty. But the two sets have not changed, they are still the set of pink flying elephants, and the set of people on the moon, as specified, only membership has changed. Since the sets themselves have not changed only the elements have, then clearly they were never the same set in the first place.

Of course, you'll claim that a set is characterized by its elements, so it was never "the set of pink flying elephants in the first place, it was the empty set. But this is clearly an inconsistency because "pink flying elephants was specified first, then determined as empty. So that is not how you characterized these sets. You characterized them as "the set of pink flying elephants", and "the set of people on the moon".

If you had specified "the empty set", then obviously the empty set is the same set as the empty set, but "pink flying elephants", and "people on the moon" are clearly not both the same set, just because they both happen to have zero elements. The emptiness of these two sets is contingent, whereas the emptiness of "the empty set" is necessary, so there is a clear logical difference between them.

There is only one empty set, because the axiom of extensionality says that if for every object, it's a person on the moon if and only if it's a pink flying elephant, that the two sets must be the same. — fishfry

I don't know why you can't see this as a ridiculous axiom. You say that a "person on the moon" is a "pink flying elephant". That's ridiculous.

We know from the law of identity that everything is equal to itself. So what is the set of all things that are not equal to themselves? It's the empty set. And by the axiom of extensionality, it's exactly the same as the set of pink flying elephants and the people on the moon. — fishfry

See the consequences of that ridiculous axiom? Now you are saying that a pink flying elephant is a thing which is not equal to a pink flying elephant, and a person on the moon is not equal to a person on the moon. Face the facts, the axiom is nonsensical.

You are confusing the axiom of extension, which tells us when two sets are the same, with the other axioms that give us various ways to build sets or prove that various sets exist. — fishfry

Obviously, the axiom of extension is very bad because it fails to distinguish between necessity and contingency.

All of which is tantamount to saying that ZF has only partial relevance to modern mathematics in terms of being an axiomatization of well-foundedness, whilst ZFC is completely and utterly useless, failing to axiomatize the most rudimentary notions of finite sets as used in the modern world. — sime

This has murky and unclear relevance to what went before in the same post. I can't tell if you are trying to explain something to me or promoting an agenda. More clarity and less stridence would be helpful to me; but I'm not sure if being helpful is your intention.

What I mean is that one can look up the entry for the axiom of choice on nLab, without encountering a rant against ZFC. So I think you're the one adding that part, and not your fellow constructivists / category theorists / programmers or whatever direction you're coming from.

Indeed, nLab expresses choice as "every surjection splits," which they note means "every surjection has a right inverse," in set theory. This formulation is easily shown to be equivalent to the traditional statement of the axiom of choice. There is no distance between the category-theoretic and set-theoretic views of choice.

You do not seem to be grasping the problem. — Metaphysician Undercover

Rather than try to understand set theory on its own terms, you just want to fight with it. Why? I'm taking the trouble to explain it to you, on its own terms. I'm on record multiple times saying that I make no claims that it's "true" in any meaningful sense; or even meaningful in any meaningful sense. So why try to argue with me about the subject? The person you want to address your complaints to is [url=https://en.wikipedia.org/wiki/Ernst_ZermeloErnst Zermelo[/url], who more than anyone is responsible for the modern incarnation of the standard axioms of set theory. Cantor gets all the credit and Zermelo did the heavy lifting. Zermelo died in 1953 so he's not available for you to complain to; but I am not available for you to complain to either. I"m not defending the truth, meaning, sanity, or sense of set theory. I'm only describing to you how it is. You will have to take your complaints elsewhere.

Perhaps this is a fundamental confusion on your part, but I don't see why. I have explained my viewpoint many times. I'm describing set theory to you. I am not defending it, not advocating it, not promoting it. But I am, to the best of my ability, giving an accurate account of the basics of the subject as it is understood by mathematicians. So if you want to learn it, let's proceed. If you just want to argue with me about it, that makes for a one-sided and tedious interaction.

Let me say this once and for all: I am not the lord-high defender of set theory. It's exactly like chess. I'm teaching you the rules. If you don't like the game, my response is for you to take up some other game more to your liking.

If a set is characterized by its elements, there is no such thing as an empty set. — Metaphysician Undercover

You're too hung up on the empty set. As I said in my previous post, students generally have a hard time getting accustomed to vacuous arguments. It's perfectly analogous to the subject of material implication in logic. Students don't get why "If 2 + 2 = 5 then I am the Pope" is a true statement. At some point most of them get it, and some never do.

No elements, no set. Do you understand this? — Metaphysician Undercover

No. It's not only that you're wrong, but it's a nothingburger of an issue. It's like a beginning logic student constantly arguing with the professor about material implication. The facts of the matter are never going to change. The student can only accept it, or drop the course and enroll in a different one more to his or her liking. One tactic is to just accept it ("it" being whatever is bothering the learner at the moment) on faith, keep working at the subject, and one they they'll wake up and realize that it's all perfectly obvious and they can't even remember a time when it wasn't. That's the tactic I recommend to you.

That is the logical conclusion we can draw from " a set is characterized by its elements". — Metaphysician Undercover

But I have already explained to you in my previous post, that "a set is characterized by its elements" is merely an English-language approximation to the axiom of extentionality, which actually says, $\forall A \forall B (\forall X (X \in A \iff X \in B) \implies A = B)$. That is the axiom that says that two sets are equal if they have exactly the same elements. And by a vacuous argument -- the same kind of argument that students have had trouble with since logic began -- two sets are the same if they each have no elements.

If you won't grapple with the symbology, you have to accept it on faith. You can't just fall back on the imprecise English-language version, now that I've shown you (twice) the formal version.

If we have no elements, we have no set. If you do not agree with this, explain to me how there could be a set which is characterized by its elements, and it has no elements. It has no character? Isn't that the same as saying it isn't a set? — Metaphysician Undercover

You're being tedious. First, the matter is trivial. The empty set is a thing in set theory. You can't allow your learning to be stuck on this one point. Accept it and move on, or go find something else to be interested in. Secondly, if you will put in the work to understand the symbology as described on the Wiki page for the axiom of extensionality, at some point you'll probably just get it.

So we cannot proceed to even talk about an empty set because that's incoherent, unless we dismiss this idea that a set is characterized by its elements. — Metaphysician Undercover

The formal symbolic expression of extensionality could not be more clear. The fact that you won't engage with it is not my problem.

Can we get rid of that idea? — Metaphysician Undercover

Perhaps. But again, you're trying to learn a subject, and every time you're shown one of the basic principles, you just want to argue. You make it difficult on yourself. Once you learn basic set theory, you can set about developing an alternative version if you like. Einstein changed physics, but before he did that he mastered classical physics. Right? Right.

Then we could proceed to investigate your interpretation of the axiom of extensionality, which allows you to say "If you have two sets such that they have no elements, they're the same set; namely the empty set", because "empty set" would be a coherent concept. Until we get rid of that premise though, that a set is characterized by its elements there is no such thing as a set with no elements, because such a set would have no identity whatsoever, and we could not even call it a set. — Metaphysician Undercover

You're just being tedious. The formal symbology is perfectly clear. And even if it isn't clear to you, you should just accept the point and move on, so that we can discuss more interesting things. You're causing yourself to get stuck on a relatively minor point. You have two ways out: One, grapple with the formal symbology here. Two, accept it and move on. Repeating the same tired and fallacious objections is no longer an option, at least with me. Maybe you can get someone else to play.

You are not grasping the distinction between 'characterized by its elements', and 'characterized by its specification' which I'm trying to get though to you. — Metaphysician Undercover

Here you have made up your own phrase, "characterized by its specification." I have not said that. The axiom of specification is one of the axioms that tell us when particular sets may be said to exist. I explained this to you in painful detail in my previous post, and you are just ignoring what I said. The axiom of extensionality tells us when two sets are the same. The axiom schema of specification tells us when certain sets exist. There are other set existence axioms: pairing, union, intersection, powerset, replacement, and choice. Frankly if you want to complain about axioms, it's replacement you should be concerned about. It's very murky. Zermelo's original formulation didn't even include it, as I've recently learned due to @TonesInDeepFreeze's repeated mention of Zermelo set theory, or Z. I went and looked it up and learned something new.

When you say "the set of all purple flying elephants", this is a specification, and this set is characterized by that specification. There are no elements being named, or described, and referred to as comprising that set, there is only a specification which characterizes the set. — Metaphysician Undercover

If by specification you mean predicate, then "E(x) = x is a flying elephant" is most definitely a specification. And in the axiom of specification, that's what's meant. The fact that a predicate may have an empty extension is not a bug, it's a feature.

Every set is entirely characterized by its elements.
— fishfry

Where do you get this idea from? — Metaphysician Undercover

From the axiom of extensionality. You know, I don't care if you believe in the empty set or not. But after my having directedyour attention to the axiom of extensionality so many times, I don't see how you can ask where I got the idea. It's one of the axioms.

Clearly your example "the set of all purple flying elephants" is not characterized by its elements. You have made no effort to take elements, and compose a set You have not even found any of those purple flying elephants. In composing your set, you have simply specified "purple flying elephants". Your example set is characterized by a specification, not by any elements. If you do not want to call this "specification", saving that term for some special use, that's fine, but it's clearly false to say that such a set is characterized by its elements. — Metaphysician Undercover

Accept it and let's move on; or don't accept it and quietly seethe while we move on. I can't engage with you on this point anymore. I've already explained it. The axiom of specification allows us to use a predicate to form a set. The predicate is not required to have a nonempty extension.

This is what happens when we proceed deep into the workings of the imagination. We can take a symbol, a name like "purple flying elephants", or any absurdity, or logical incoherency, like "square circles", each of which we assume has no corresponding objects — Metaphysician Undercover

As I'm always fond of pointing out, the unit circle in the taxicab metric is a square. There's a picture of a square circle at this link. How about married bachelor, that's a better example.

However, we can then claim something imaginary, a corresponding imaginary object, and we can proceed under the assumption that the name actually names something, a purple flying elephant in the imagination. You might then claim that this imaginary thing is an element which characterizes the set. But if you then say that the set is empty, you deny the reality of this imaginary thing, and you are right back at square one, a symbol with nothing corresponding. And so we cannot even call this a symbol any more, because it represents nothing. — Metaphysician Undercover

It's merely a predicate with an empty extension.

Now you've hit the problem directly head on. To be able to have an empty set, a set must be characterized by it's specification, as I've described, e.g. "pink flying elephants". So. the set of pink flying elephants is one set, characterized by the specification "pink flying elephants", and the set of people on the moon is another set, characterized by the specification "people on the moon". To say that they are exactly the same set, because they have the same number of elements, zero, is nor only inconsistent, but it's also a ridiculous axiom. — Metaphysician Undercover

What's true is that given any thing whatsoever, that thing is a pink flying elephant if and only if it's a person on the moon. So the axiom of extensionality is satisfied and the two sets are equal. If you challenged yourself to work through the symbology of the axiom of extentionality this would be perfectly clear to you.

Would you say that two distinct sets, with two elements, are the exact same set just because they have the same number of elements? — Metaphysician Undercover

Of course not. It's not a matter of cardinal equivalence. The elements themselves have to be respectively equal. {1,2} and {1,2} are the same set. {1,2} and {3,47} are not.

I think you'll agree with me that this is nonsense. — Metaphysician Undercover

It would be false. It would not be nonsense, that's a value judgment. And your value judgments regarding mathematics are not good.

And to say that each of them has the very same elements because they don't have any, is clearly a falsity because "pink flying elephants" is a completely different type of element from "people on the moon". If at some point there is people on the moon, then the set is no longer empty. — Metaphysician Undercover

You are right about that. But that's because we are making up examples from real life. Math doesn't have time or contingency in it. 5 is an element of the set of prime numbers today, tomorrow, and forever. The "people on the moon" example was yours, not mine. I could have and in retrospect should have objected to it at the time, because of course it is a temporally contingent proposition. I let it pass. So let me note for the record that there are no temporally contingent propositions in math.

But the two sets have not changed, they are still the set of pink flying elephants, and the set of people on the moon, as specified, only membership has changed. Since the sets themselves have not changed only the elements have, then clearly they were never the same set in the first place. — Metaphysician Undercover

It's a bad example because one of your propositions is temporally contingent. I noted that at the time you mentioned it but let it pass. I see that was a mistake. I have now rectified my error. There are no temporally contingent propositions in math.

Of course, you'll claim that a set is characterized by its elements, so it was never "the set of pink flying elephants in the first place, it was the empty set. But this is clearly an inconsistency because "pink flying elephants was specified first, then determined as empty. So that is not how you characterized these sets. You characterized them as "the set of pink flying elephants", and "the set of people on the moon". — Metaphysician Undercover

The two sets, assuming that we mean at the present moment, are the same, namely the empty set, because the condition in the axiom of extensionality is satisfied. A thing is in one of those sets if and only if it's in the other. Therefore the sets are the same. That's all there is to it.

If you had specified "the empty set", then obviously the empty set is the same set as the empty set, but "pink flying elephants", and "people on the moon" are clearly not both the same set, just because they both happen to have zero elements. The emptiness of these two sets is contingent, whereas the emptiness of "the empty set" is necessary, so there is a clear logical difference between them. — Metaphysician Undercover

Only by virtue of the people on the moon being temporally contingent. So it's a bad example, which I should have pointed out when you first mentioned it.

I don't know why you can't see this as a ridiculous axiom. You say that a "person on the moon" is a "pink flying elephant". That's ridiculous. — Metaphysician Undercover

Nobody says that. What is true is that the axiom of extensionality is satisfied. Until you roll up your sleeves and put in the work to understand that, you'll spin yourself in circles.

See the consequences of that ridiculous axiom? — Metaphysician Undercover

You are being tedious. You wrote an entire post on the nonsense. Go understand what the axiom of extensionality says. You're running yourself in circles because you can't be bothered to challenge yourself to work through what the axiom says.

Now you are saying that a pink flying elephant is a thing which is not equal to a pink flying elephant, and a person on the moon is not equal to a person on the moon. Face the facts, the axiom is nonsensical. — Metaphysician Undercover

Nobody is saying a pink flying elephant is a thing. You're just a logic student having trouble with vacuous arguments. Put in the work to understand it, or just accept it and move on. Being endlessly tedious, writing an entire post about your own misconceptions, is pointless.

Obviously, the axiom of extension is very bad because it fails to distinguish between necessity and contingency. — Metaphysician Undercover

There are no temporally contingent propositions in math. The people on the moon example is a bad one for that reason. I was wrong rhetorically to let it pass without objection earlier, because now you just want to use it to make a sophistic point.

Bottom line is that the empty set is a purely formal object that satisfies some formal conditions. It's not "real" and it's not helpful to try to understand it in terms of common sense. The following SE thread, in particular the checked answer, may be helpful.

https://philosophy.stackexchange.com/questions/14823/why-do-we-have-a-problem-about-understanding-the-concept-of-the-empty-set

What I mean is that one can look up the entry for the axiom of choice on nLab, without encountering a rant against ZFC. So I think you're the one adding that part, and not your fellow constructivists / category theorists / programmers or whatever direction you're coming from.

Indeed, nLab expresses choice as "every surjection splits," which they note means "every surjection has a right inverse," in set theory. This formulation is easily shown to be equivalent to the traditional statement of the axiom of choice. There is no distance between the category-theoretic and set-theoretic views of choice. — fishfry

Category theory is a useful meta-language for understanding precisely where the mathematical foundation proposed in the early 20th century goes wrong, as well as being helpful for relating alternative theories. CT is itself philosophically neutral in the sense that it only assumes the presence of identity arrows in a category, but places no other constraints on either the presence or absence of arrows, provided the laws of arrow composition and association are obeyed. Therefore disputes between intuitionists, formalists and platonists carry over into the language.

Like a mathematics department, nlab as an encyclopedia is obviously going to disseminate mathematics in a politically neutral fashion. Or perhaps i should have said, unlike a mathematics department. But political neutrality doesn't amount to reasonableness regarding which mathematics should be prioritised.

. It's exactly like chess. I'm teaching you the rules. If you don't like the game, my response is for you to take up some other game more to your liking. — fishfry

I'm a philosopher, my game is to analyze and criticize the rules of other games. This is a matter of interpretation. If you do not like that, then why are you participating in a philosophy forum?

As much as you, as a mathematician are trying to teach me some rules of mathematics, I as a philosopher am trying to teach you some rules of interpretation. So the argument goes both ways, you are not progressing very well in developing your capacity for interpreting. But if you do not like the game of interpretation, then just do something else

But I have already explained to you in my previous post, that "a set is characterized by its elements" is merely an English-language approximation to the axiom of extentionality, which actually says, — fishfry

This is why the axiom of extensionality is not a good axiom. It states something about the thing referred to by "set", which is inconsistent with the mathematician's use of "set", as you've demonstrated to me.

That is the axiom that says that two sets are equal if they have exactly the same elements. And by a vacuous argument -- the same kind of argument that students have had trouble with since logic began -- two sets are the same if they each have no elements. — fishfry

We've already been through this problem, a multitude of times. That two things are equal does not mean that they are the same. That's why I concluded before, that it's not the axiom of extensionality which is so bad, but your interpretation of it is not very good. But I now see that the axiom of extensionality is itself bad.

The formal symbology is perfectly clear. And even if it isn't clear to you, you should just accept the point and move on, so that we can discuss more interesting things. — fishfry

In case you haven't noticed, what I am interested in is the interpretation of symbols. And obviously the symbology of the axiom is not perfectly clear. If you can interpret "=" as either equal to, or the same as, then there is ambiguity.

What's true is that given any thing whatsoever, that thing is a pink flying elephant if and only if it's a person on the moon. So the axiom of extensionality is satisfied and the two sets are equal. If you challenged yourself to work through the symbology of the axiom of extentionality this would be perfectly clear to you. — fishfry

Actually, I'm starting to see that this, what you claim in your vacuous argument, is not a product of the axiom of extensionality, but a product of your faulty interpretation. By the axiom of extensionality, a person on the moon is equal to a pink flying elephant, and you interpret this as "the same as". So the axiom is bad, in the first place, for the reasons I explained in the last post, and you make it even worse, with a bad interpretation.

The axiom of specification allows us to use a predicate to form a set. The predicate is not required to have a nonempty extension. — fishfry

You really do not seem to be getting it. If, we can "use a predicate to form a set" as the axiom of specification allows, then it is not true that a set is characterized by its elements. It's characterized by that predication. The two are mutually exclusive, inconsistent and incompatible. Specification allows for a nonempty set, I have no problem with this. But to say that this set is characterized by its elements is blatantly false. It has no elements, and it is characterized as having zero elements, an empty set. So it's not characterized by its elements, it's characterized by the number of elements which it has, none. .

Of course not. It's not a matter of cardinal equivalence. The elements themselves have to be respectively equal. — fishfry

Yes, this is the problem with the axiom of extension, in its portrayal of the empty set. It is saying that if two specified sets each have zero elements, then "the elements themselves" are equal. However, there are no such elements to allow one to judge the equality of them. So there is no judgement that "the elements themselves" are equal, because there are no elements to judge, and so the judgement of cardinal equivalence, that they have the same number of elements, zero, is presented as a judgement of the elements themselves.

You ought to recognize, that to present a judgement of cardinal equivalence, as a judgement of the elements themselves, is an act of misrepresentation, which is an act of deception. I know that you have no concern for truth or falsity in mathematical axioms, but you really ought to have concern for the presence of deception in axioms.

Of course not. It's not a matter of cardinal equivalence. The elements themselves have to be respectively equal. {1,2} and {1,2} are the same set. {1,2} and {3,47} are not. — fishfry

Now, do you agree, that when there are no elements, it makes no sense to say that the elements themselves are respectively equal? What is really being judged as equal is the cardinality. They both have zero elements.

The axiom of extensionality tells us when two sets are the same. — fishfry

No, the axiom of extensionality does not tell us when two sets are the same, that's the faulty interpretation I've pointed out to you numerous times already, and you just cannot learn. It tells us when two sets are equal.

That faulty interpretation is what enables the deception. Equality always indicates a judgement of predication, and in mathematics it's a judgement of equal quantity, which you call cardinal equivalence. When you replace the determination of the cardinality of two empty sets, "equal", with "the same", you transfer a predication of the set, its cardinality, to make a predication of its elements, "the same as each other". I believe that's known as a fallacy of division.

You are right about that. But that's because we are making up examples from real life. Math doesn't have time or contingency in it. 5 is an element of the set of prime numbers today, tomorrow, and forever. The "people on the moon" example was yours, not mine. I could have and in retrospect should have objected to it at the time, because of course it is a temporally contingent proposition. I let it pass. So let me note for the record that there are no temporally contingent propositions in math. — fishfry

Well, "pink flying elephants" was your example, and it's equally contingent. The issue of temporally contingent propositions raises a completely different problem. The only truly necessary empty set is the one specified as "the empty set". As your examples of square circles and married bachelors show, definitions and conceptual structures change over time, so your assertion that mathematics has no temporally contingent propositions is completely untrue. It may be the case that "the empty set" will always refer to the empty set, necessarily, but how we interpret "empty" and "set" is temporally contingent. So temporal contingency cannot be removed from mathematics as you claim. This is the problem of Platonic realism, the idea that mathematics consists of eternal, unchanging truths, when in reality the relations between symbols and meaning evolves.

You reject general relativity and set theory because of your ontology, and your ontology rejects emergence. There is the problem. An object has no weight on it's own, and neither does spacetime. But together they form the world of substance we experience as weight. In set theory Zero means "no thing" and Set is a collecting of something. But the concepts together form something that is useful in the practice of set theory

https://www.zacharyfruhling.com/philosophy-blog/strong-emergence-is-holism-not-magic

The "further reading" is good too

An objection was made that the axiom of extensionality does not "distinguish" between sufficiency and necessity.

The axiom is:

Axy(Az(zex <-> zey) -> x = y)

that is the sufficiency clause.

The necessity clause comes from identity theory:

Axy(x = y -> Az(zex <-> zey))

So together we have the theorem of sufficiency and necessity:

Axy(Az(zex <-> zey) <-> x = y)