I found the comments about Cohen's Filter in the article I linked fascinating. Like most math people I knew of his breakthrough results, but was unfamiliar with the actual math. I'd be interested to hear opinions from the set theorists on the forum about this. — jgill
I didn't know about structuralism in math! That the number one is an idea, a true idea, seems to me to be the basis of all that follows though, kinda that unity before the plurality. But structuralism in all forms is a really interesting idea! — Gregory
a set is something that contains and not something in its own right — Gregory
That said, can you say what "this" refers to? Cohen's invention of forcing in general? — fishfry
If you replace numbers with sets, do you keep the same ontology? — Gregory
What of structuralism? — Gregory
Set theory does provide the structural relations we expect. Even though the objects have "extra-structural" properties (e.g. that 0 is the empty set and 1 is the set {0}), the structural relations are captured (e.g. that 0 < 1). — TonesInDeepFreeze
In set theory everything is a set. — fishfry
Sets whose elements are sets whose elements are sets, drilling all the way down to the empty set. — fishfry
No, not at all. 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
The concept of "set" itself has no definition, as I've pointed out to you in the past. — fishfry
There is no set of ordinals, this is the famous Burali-Forti paradox. — fishfry
There is no general definition of number. — fishfry
You see you're at best a part-time Platonist yourself. — fishfry
If I put on my Platonist hat, I'll admit that the number 5 existed even before there were humans, before the first fish crawled onto land, before the earth formed, before the universe exploded into existence, if in fact it ever did any such thing. — fishfry
I must say, though, that I am surprised to find you suddenly advocating for mathematical Platonism, after so many posts in which you have denied the existence of mathematical objects. Have you changed your mind without realizing it? — fishfry
But Meta, really, you are a mathematical Platonist? I had no idea. — fishfry
I agree with the points you're raising. I don't know if 5 existed before there were humans to invent math. I truly don't know if the transfinite cardinals were out there waiting to be discovered by Cantor, and formalized by von Neumann. After all, set theory is an exercise in formal logic. We write down axioms and prove things, but the axioms are not "true" in any meaningful sense. Perhaps we're back to the Frege-Hilbert controversy again. — fishfry
That said, can you say what "this" refers to? Cohen's invention of forcing in general?
— fishfry
That would be good. I had heard the expression but had no idea what it was. The article came as a revelation to me. And here I thought the reals consisted of rationals and irrationals. — jgill
I reject "the empty set" for a reason similar to the reason why I rejected a set with no inherent order. it's a fiction which has no purpose other than to hide the shortcomings of the theory. There are very good reasons why "0" ought to represent something in a class distinct from numbers. There are even reasons why "1" ought to be in a distinct class. — Metaphysician Undercover
This is the best analogy for forcing. — fishfry
She puts the "pop" in pop science — fishfry
I'm not a set theorist, but I have some thoughts. — TonesInDeepFreeze
My initial guess was that a set is something that contains and not something in its own right. — Gregory
So zero remains a nothingness of anything in that case. — Gregory
Very abstract ideas. Couldn't structure just be that which contains a process and thus, like sets which compose it, it is nothing in itself. — Gregory
This would certainly make mathematics a system of process and divorce it from the notion that anything rests and stays permanent within it — Gregory
Since no two things are identical in spatiotemporal reality, do you also reject the number 2? — Luke
Sets can contain other sets. In fact a set is "something" in addition to its constituent elements. It's a "something" that allows us to treat the elements as a single whole. If I have the numbers 1, 2, and 3, that's three things. The set {1,2,3} is one thing. It's a very subtle and profound difference. A set is a thing in and of itself. — fishfry
The number 2 is an unnecessary intermediary between the symbol, and what the symbol represents, or means, in use. — Metaphysician Undercover
You write very well. That must be why I like to engage with you, not that I want to troll you. — Metaphysician Undercover
In set theory everything is a set.
— fishfry
I didn't know that, but it makes the problems which I've apprehended much more understandable. — Metaphysician Undercover
If everything is a set, in set theory, then infinite regress is unavoidable. — Metaphysician Undercover
A logical circle is sometimes employed, like the one mentioned here ↪jgill to disguise the infinite regress, but such a circle is really a vicious circle. — Metaphysician Undercover
I reject "the empty set" for a reason similar to the reason why I rejected a set with no inherent order. it's a fiction which has no purpose other than to hide the shortcomings of the theory. — Metaphysician Undercover
There are very good reasons why "0" ought to represent something in a class distinct from numbers. There are even reasons why "1" ought to be in a distinct class. — Metaphysician Undercover
No, not at all. 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
This may be the case, but you ought to recognize that being elements of the same set makes them "the same" in a meaningful way. — Metaphysician Undercover
Otherwise, a set would be a meaningless thing. So when you said for instance, that {0,1,2,3,} is a set, there must be a reason why you composed your set of those four elements. — Metaphysician Undercover
That reason constitutes some criteria or criterion which is fulfilled by each member constituting a similarity. — Metaphysician Undercover
This is a simple feature of common language use. A word may receive its meaning through usage rather than through an explicit definition. — Metaphysician Undercover
That the word has no definition does not mean that it has no meaning, its meaning is demonstrated by its use, as is the case with an ostensive definition. — Metaphysician Undercover
Allowing that a word, within a logical system, has no explicit definition, allows the users of the system an unbounded freedom to manipulate that symbol, (exemplified by TonesInDeepFreeze's claim with "least"), but the downfall is that ambiguity is inevitable. — Metaphysician Undercover
This is an example of the uncertainty which content brings into the formal system, that I mentioned in the other thread. — Metaphysician Undercover
There is no set of ordinals, this is the famous Burali-Forti paradox.
— fishfry
This I would say is a good representation of the philosophical concept of "infinite". Note that the philosophical conception is quite different from the mathematical conception. If every ordinal is a set composed of other ordinals, and there is no limit to the "amount" of ordinals which one may construct, then it ought to be very obvious that we cannot have an ordinal which contains all the ordinals, because we are always allowed to construct a greater ordinal which would contain that one as lesser. — Metaphysician Undercover
So we might just keep getting a greater and greater ordinal, infinitely, and it's impossible to have a greatest ordinal. — Metaphysician Undercover
I think there is a way around this though, similar to the way that set theory allows for the set of all natural numbers, which is infinite. As you say, "set" has no official definition. And, you might notice that "set" is logically prior to "cardinal number". So all that is required is a different type of set, one which is other than an ordinal number, which could contain all the ordinals. It would require different axioms. — Metaphysician Undercover
There is no general definition of number.
— fishfry
This is not really true now, if we accept set theory. — Metaphysician Undercover
If "set" is logically prior to "number", then "set"
is a defining principle of "number". — Metaphysician Undercover
That is why you and I agreed that each ordinal is itself a set. We have a defining principle, an ordinal is a type of set, and a cardinal is a type of ordinal. — Metaphysician Undercover
Correction, at my worst I am a part-time Platonist. At my best I am a fulltime Neo-Platonist. — Metaphysician Undercover
We do not have to go the full fledged Platonic realism route here, to maintain a realism. This is what I tried to explain at one point in another thread. — Metaphysician Undercover
We only need to assume the symbol "5", and what the symbol represents, or means. There is no need to assume that the symbol represents "the number 5", as some type of medium between the symbol, and what the symbol means in each particular instance of use. So when I say that a thing exists, and has a measurement, regardless of whether it has been measured, what I mean is that it has the capacity to be measured, and there is also the possibility that the measurement might be true. — Metaphysician Undercover
If you think that I was advocating for mathematical Platonism, then you misunderstood. I was advocating for realism. — Metaphysician Undercover
A mathematical Platonist thinks of ideas as objects. I recognize the reality of ideas, and furthermore I accept the priority of ideas, so I am idealist. But I do not think of ideas as objects, as mathematical Platonists do, I think of them as forms, so I'm more appropriately called Neo-Platonist. — Metaphysician Undercover
This is that vague boundary, the grey area between fact and fiction which we might call "logical possibility". If we adhere to empirical principles, we see that there are individual objects in the world, with spatial separation between them. If we are realist, we say that these objects which are observed as distinct, really are distinct objects, and therefore can be counted as distinct objects. We might see three objects, and name that "3", but "3" is simply what we call that quantity. Being realist we think that there is the same quantity of objects regardless of whether they've been counted and called "3" or not. — Metaphysician Undercover
But if we give up on the realism, and the empirical principles, there is no need to conclude that what is being seen is actually a quantity of 3. There might be no real boundaries between things, and anything observed might be divisible an infinite number of times. Therefore whatever is observed could be any number of things. This is the world of fiction, which some might call "logical possibility", and you call pure mathematics. Empirical truths, like the fact that distinct objects can be counted as distinct objects, pi as the ratio between circumference and diameter of a circle, and the Pythagorean theorem, we say are discovered. Logical possibilities are dreamt up by the mind, and are in that sense fictions. — Metaphysician Undercover
I do not mean to argue that dreaming up logical possibilities is a worthless activity. — Metaphysician Undercover
What I think is that this is a primary stage in producing knowledge. We look at the empirical world for example and create a list of possibilities concerning the reality of it. The secondary stage is to eliminate those logical possibilities which are determined to be physically impossible through experimentation and empirical observation. — Metaphysician Undercover
So we proceed by subjecting logical possibilities, and axioms of pure mathematics, to a process of elimination. — Metaphysician Undercover
Sets can contain other sets. In fact a set is "something" in addition to its constituent elements. It's a "something" that allows us to treat the elements as a single whole. If I have the numbers 1, 2, and 3, that's three things. The set {1,2,3} is one thing. It's a very subtle and profound difference. A set is a thing in and of itself.
— fishfry
This is what I was asking about earlier, what allows for that unity if not some judgement of criteria, making the elements similar, or the same in some respect., a definition. — Metaphysician Undercover
This is a very important ontological question because we do not even understand what produces the unity observed in an empirical object. — Metaphysician Undercover
Suppose you arbitrarily name a number of items and designate it as a set. — Metaphysician Undercover
You have created "a thing" here, a set, which is some form of unity. But that unity is completely fictitious. You are just saying that these items compose a unity called "a set", without any justification for that supposed unity. — Metaphysician Undercover
In its simplest from, this is the issue of counting apples and oranges. We can count an apple and orange as two distinct objects, and call them 2 objects. But if we want to make them a set we assume that something unifies them. — Metaphysician Undercover
If we are allowed to arbitrarily designate unity in this way, without any criteria of similarity, then our concept of unity, which some philosophers (Neo-Platonist for example) consider as fundamental loses all its logical strength or significance. — Metaphysician Undercover
How can either the number 2 or the numeral "2" represent or mean anything in use if no two things are identical in spatiotemporal reality? Isn't the law of identity the basis of your mathematics? — Luke
No not at all. First, what's wrong with infinite regress? After all the integers go backwards endlessly: ..., -4, -3, -2, -1, 0, 1, 2, 3, 4, ... You can go back as far as you like. I'm fond of using this example in these endlessly tedious online convos about eternal regress in philosophy. Cosmological arguments and so forth. Why can't time be modeled like that? It goes back forever, it goes forward forever, and we're sitting here at the point 2021 in the Gregorian coordinate system. — fishfry
jgill was referring to the Riemann sphere, a way of viewing the complex numbers as a sphere. It's based on the simple idea of stereographic projection, a map making technique that allows you to project the points of a sphere onto a plane. There is nothing mystical or logically questionable about this. You should read the links I gave and then frankly you should retract your remark that the Riemann sphere is a "vicious circle." You're just making things up. Damn I feel awful saying that, now that you've said something nice about me. — fishfry
Jeez Louise man. I say: "The only thing they have in common is that they're elements of a given set." And then you say I "ought to recognize ..." that very thing. — fishfry
A very disingenuous point. 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
Ok, you are now agreeing with me on an issue over which you've strenuously disagreed in the past. You have insisted that "set" has an inherent meaning, that a set must have an inherent order, etc. I have told you many times that in set theory, "set" has no definition. Its meaning is inferred from the way it behaves under the axioms. — fishfry
And now you are making the same point, as if just a few days ago you weren't strenuously disagreeing with this point of view.
But in any event, welcome to my side of the issue. Set has no definition. Its meaning comes exclusively from its behavior as specified by the axioms. — fishfry
Not at all. Bricks are the constituents of buildings, but all the different architectural styles aren't inherent in bricks. There are plenty of sets that aren't numbers. Topological spaces aren't numbers. The set of prime numbers isn't a number. Groups aren't numbers. The powerset of the reals isn't a number. Just because numbers are made of sets in the formalism doesn't mean every set is a number. — fishfry
Meta I find you agreeing with my point of view in this post. — fishfry
So you would ban the teaching of Euclidean geometry now that the physicists have accepted general relativity? — fishfry
Would you ban Euclidean geometry from the high school curriculum because it turns out not to be strictly true? — fishfry
There is no criterion. In fact there are provably more sets than criteria. If by "criterion" you mean a finite-length string of symbols, there are only countably many of those, and uncountably many subsets of natural numbers. So most sets of natural numbers have no unifying criterion whatsoever, They're entirely random. — fishfry
I just proved that most sets of natural numbers are entirely random. There is no articulable criterion linking their members other than membership in the given set. There is no formal logical definition of the elements. There is no Turing machine or computer program that cranks out the elements. That's a fact. — fishfry
jgill was referring to the Riemann sphere, a way of viewing the complex numbers as a sphere . . .— fishfry
I beg to differ — Metaphysician Undercover
Obviously, "2" refers to two distinct and different things. If there was only one thing we'd have to use "1". — Metaphysician Undercover
Suppose you arbitrarily name a number of items and designate it as a set. You have created "a thing" here, a set, which is some form of unity. But that unity is completely fictitious. You are just saying that these items compose a unity called "a set", without any justification for that supposed unity. — Metaphysician Undercover
If we adhere to empirical principles, we see that there are individual objects in the world, with spatial separation between them. If we are realist, we say that these objects which are observed as distinct, really are distinct objects, and therefore can be counted as distinct objects. We might see three objects, and name that "3", but "3" is simply what we call that quantity. Being realist we think that there is the same quantity of objects regardless of whether they've been counted and called "3" or not.
But if we give up on the realism, and the empirical principles, there is no need to conclude that what is being seen is actually a quantity of 3. There might be no real boundaries between things, and anything observed might be divisible an infinite number of times. Therefore whatever is observed could be any number of things. — Metaphysician Undercover
I'm missing your point also. What's your gripe about the innocuous Riemann sphere? :chin: — jgill
"2" can also refer to two distinct but same things, such as "things" of the same type or category. — Luke
But all categories/classifications are equally as fictitious and man-made as the sets and orders you reject. — Luke
Scientists justified both the inclusion and exclusion of Pluto as a planet at different times. Like Pluto, many individual "things" are borderline cases in their classification. Moreover, nothing guarantees the perpetuity of any category/set, or of what defines ("justifies") the inclusion of its members. — Luke
Furthermore, if you base your mathematics on empiricism rather than on "abstraction" or "fiction", then you must also reject fractions, since a half cannot be exactly measured in reality. — Luke
If there are "no real boundaries between things", then acknowledging that "anything observed might be divisible an infinite number of times" is not to "give up on the realism", but to adhere to it. — Luke
"2" can also refer to two distinct but same things, such as "things" of the same type or category. — Luke
This is a different sense of "same", not consistent with the law of identity. — Metaphysician Undercover
But all categories/classifications are equally as fictitious and man-made as the sets and orders you reject. — Luke
When they are based in empirical observation they are not equally fictitious. — Metaphysician Undercover
Scientists justified both the inclusion and exclusion of Pluto as a planet at different times. Like Pluto, many individual "things" are borderline cases in their classification. Moreover, nothing guarantees the perpetuity of any category/set, or of what defines ("justifies") the inclusion of its members. — Luke
That a person later decides to have been wrong in an earlier judgement, is not relevant. — Metaphysician Undercover
I do reject fractions — Metaphysician Undercover
I believe that the principles employed are extremely faulty, allowing that a unit might be divided in any way that one wants. — Metaphysician Undercover
In reality, how a unit can be divided is dependent on the type of unit. — Metaphysician Undercover
That's the case if there are "no real boundaries between things". But I am arguing that empirical evidence demonstrates that there are real boundaries. — Metaphysician Undercover
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