• sandman
    20

    When JFK was assassinated, the general population could not accept that an ordinary individual could remove a popular public figure, so some thought it must be a conspiracy. I was never an advocate for that. Tragedies don't discriminate.
    I said 'many', not 'all'. SR is a great theory, and has so much experimental support, why is it still questioned.
  • sandman
    20

    Here is a paper that questions the 'diagonal argument'.
    https://app.box.com/s/vdop6iqhi8azgoc2upd76ifu8zacq8e4
  • Zuhair
    93
    As explained, this is absolutely falseMetaphysician Undercover

    Well you seem to refuse tribe as an object, well this is a deep point. Anyhow to me a tribe, a herd, a bunch, etc.. all of those are objects, and they are well specified objects as long as each individual member of them is a well specified entity. Anyhow I don't think I can discuss refusal of such clear kinds of objects.
  • fishfry
    837
    Here is a paper that questions the 'diagonal argument'.
    https://app.box.com/s/vdop6iqhi8azgoc2upd76ifu8zacq8e4
    sandman

    Cranky.

    Wittgenstein also criticized the diagonal argument. He was wrong but his objections were at least coherent and interesting.
  • Eee
    145
    Here is a paper that questions the 'diagonal argument'.sandman

    It's not a good paper. And Cantor was definitely sophisticated enough to see what he is supposed to have missed. The author is instead failing to see.

    When JFK was assassinated, the general population could not accept that an ordinary individual could remove a popular public figure, so some thought it must be a conspiracy. I was never an advocate for that. Tragedies don't discriminate.sandman

    What you still refuse to see is the proof is solid in the same way that a game of chess is legal. All of the moves are according to the rules. It does not prove something metaphysical about reality. Or at least the rules are agnostic about their 'real world' meaning.

    Speaking more personally (indulging in a real-world interpretation), I experience its intuitive content this way. If someone claimed to have a way/algorithm to list all infinite sequences of bits, I'd know they were wrong. I would just flip bits along the diagonal and have a sequence they didn't include on their list.
  • Metaphysician Undercover
    6.2k
    Well you seem to refuse tribe as an object, well this is a deep point. Anyhow to me a tribe, a herd, a bunch, etc.. all of those are objects, and they are well specified objects as long as each individual member of them is a well specified entity. Anyhow I don't think I can discuss refusal of such clear kinds of objects.Zuhair

    Correct, under your definitions, I refuse a 'tribe" as an object. If we are to consider a "tribe" as an object, the relations between the members are essential to the existence of that object. Remove those relations and you have no object.

    You seem to think that you can randomly point to a bunch of objects, say "abracadabra", and suddenly there is another object, which exists as the unity of those objects you have pointed to. Sorry, but that's not reality.
  • Eee
    145
    Yes, those metaphysical beliefs clearly do play a role in mathematical proofs because they are entrenched in the axioms, as foundational support for those axioms. And Cantor is a good example. What is at issue here is how we conceive of an "object".Metaphysician Undercover

    I agree that a community's embrace of a set of axioms manifests among other things something like metaphysical preferences or basic intuitions. But this is obvious. And where are the mathematicians that deny it? You and @sandman might as well complain about the rules of chess for not conforming to your metaphysical preferences.

    Anti-Cantor cranks are fencing with their own shadows. To be a mathematician it suffices to prove things using 'the rules.' One can think of it as a game with symbols. One can also, to be sure, think that one is doing the True Metaphysics. One can, as I do, think of it as working within a system that strives imperfectly to articulate and accord with intuitions of space, quantity, and algorithm. Imperfectly! I like non-mainstream versions of mathematics. They are fascinating. No need for dogmatism or a fixed position. And that's also how I enjoy philosophy.

    So to me the idea that mathematicians are true believers is in general ridiculous, and, in my experience, most online anti-Cantorism is purveyed by those who seemingly can't even play the game agnostically. That would take work, serious interest, and not just self-inflating online conspiracy theory.
  • Metaphysician Undercover
    6.2k
    You and sandman might as well complain about the rules of chess for not conforming to your metaphysical preferences.Eee

    I don't see how that's an adept analogy. Metaphysical principles are based in how one apprehends the nature of reality. Chess is a game which we can choose not to play if we don't like the rules, how does that relate to reality? Not even by killing oneself can one choose not to partake in reality.

    To be a mathematician it suffices to prove things using 'the rules.'Eee

    If the rules do not conform to reality, then I'd have to ask you what are these proofs sufficient for? I'd say they're sufficient to produce unsound conclusions. What exactly would the mathemagician be proving, if one uses poorly formed rules? I'd say that the mathemagician would be proving that confusion follows from the use of poorly formed rules.

    One can think of it as a game with symbols.Eee

    I don't like playing games, I'd rather be engaged in something meaningful.

    So to me the idea that mathematicians are true believers is in general ridiculous, and, in my experience, most online anti-Cantorism is purveyed by those who seemingly can't even play the game agnostically. That would take work, serious interest, and not just self-inflating online conspiracy theory.Eee

    That's the point, why put serious effort, work, toward something which is just a game? I know that athletes do it. Sorry, but I'm not interested.
  • Zuhair
    93
    You seem to think that you can randomly point to a bunch of objects, say "abracadabra", and suddenly there is another object, which exists as the unity of those objects you have pointed to. Sorry, but that's not reality.Metaphysician Undercover

    So in your sense if I bought two applies today, then I only have two objects, that is the apples themselves, there is no other object that is the totality of these two applies, i.e. the sum material of these two apples, i.e. an object such that each of these two apples is a part of it, and that doesn't have a part of it that is disjoint of these two apples. To my naive understanding, I see it obvious that there is that object.
  • Metaphysician Undercover
    6.2k

    I see each apple as an individual object. The claim that an apple is an object is justified by sense perception of its existence separate from other things. If you show me two apples and assert that the two are one, then you need to supply a principle to justify this claim.

    Let's say that the two apples are "the same", in the sense that both are apples. So we place them both in that category, the set of apples. Notice that "the same" here is not being used in a way which is consistent with the law of identity. The apples are not really "the same" in that sense, as they remain two distinct objects. Now, what is one is the category, or set we have created called "apples". The apples are not really unified to be one, they are judged as being members of one set, according to the principle (the Idea) whereby we class them both as apples. This is explained by Plato in his famous theory of participation. The distinct objects partake in the Idea. It is well explained in The Symposium.

    But it is important to note the deficiency of the theory of participation which is developed in Plato's Parmenides, which some people argue leads to the refutation of Pythagorean/Platonic idealism by Aristotle. Let's say that the Idea, which is "the set of apples", provides the unity whereby the two apples are judged as one. It is the Idea itself, which is one object, not the two apples. The problem is that no matter how many apples partake in this Idea (the set of apples), the Idea as one object does not change. Apples may come and go from the set, as time passes, but despite this activity the set itself, as an object never changes. We see that the sense of "the same" here means that the Idea is unchanging over time, whereas "the same" in the law of identity allows that the same object may be changing as time passes.

    This makes the "Idea", or "set", as an object, passive and unchanging, and therefore independent from, separate and distinct from, the objects identified through the law of identity, which are the members of the set. Through generation and corruption, identified objects which are members of a set, like apples, come into being and cease being, while the idea, or set itself is supposed to be unchanging.

    So when you show me two apples and say that they are one object, you might as well show me three or four apples, a thousand, or a million apples, and each time you are showing me the exact same object, the set, or Idea of apples. That object is the Idea, the principle whereby you unify them as one. But that seems like nonsense, that showing me two apples, or four apples, or a million apples, is showing me the exact same object. Therefore it appears like we cannot properly refer to these "Ideas", or "sets", as objects, because we do not allow them to change when change is warranted by what is exhibited. Showing me two apples cannot be showing me the same object as showing me a hundred apples. Therefore either the two apples do not properly make an object, or we do not understand the way that this object, being the Idea or set, changes when members are added or subtracted from it.
  • Zuhair
    93
    So when you show me two apples and say that they are one object, you might as well show me three or four apples, a thousand, or a million apples, and each time you are showing me the exact same objectMetaphysician Undercover

    No that is not correct. If you show me three or more apples, the totality object would be some OTHER object. I just showed you two particular apples (those that I've bought today), and I asked you simply if by today when I bought them, do I have an object that is the whole of both of them, and I explained this object in terms of Part-whole relationship, an object such that each apple (that I bought today) is a part of it, and such that it doesn't have a part of it that is disjoint (doesn't share a common part) of both these apples. This object is the smallest object that has both of these apples as parts of. It is simply this object that I've asked you to tell me whether it exists or not. I didn't speak about a Naming category (like the one you've spoken about) nor did I pose the problem of changing material of a set. I said at the moment when I bought both of these apples is there the object that I've defined or not? To me this 'whole' or 'totality' object of these two particular apples, to me I say, is as concrete as the existence of each apple, and it is an object as each apple is an object. Of course this object can be ruined with time, as each apple can be, and actually only when one of the apples constituting it would start to ruin. But this is another question. My simple question is whether such an object exists in the first place.
  • Metaphysician Undercover
    6.2k
    No that is not correct. If you show me three or more apples, the totality object would be some OTHER object. I just showed you two particular apples (those that I've bought today), and I asked you simply if by today when I bought them, do I have an object that is the whole of both of them, and I explained this object in terms of Part-whole relationship, an object such that each apple (that I bought today) is a part of it, and such that it doesn't have a part of it that is disjoint (doesn't share a common part) of both these apples.Zuhair

    By showing me the apples you are not showing me the "totality object". This you would show to me in your explanation of "part-whole relationship", etc.. That is why number as an object, if it is an object, is something other than the objects which you use to demonstrate its rationality. The proper demonstration is an explanation, and the apples are just props.

    This object is the smallest object that has both of these apples as parts of.Zuhair

    So by the time you are talking about a number of apples, "number" is completely abstract, and you are applying that abstract idea back, onto the apples. The claimed object which is signified by "2" does not have any apples as a part of it, it is abstract. And you cannot demonstrate that object to me by showing me two apples, because such a demonstration can only be done through an explanation of what it means to be 2.

    It is simply this object that I've asked you to tell me whether it exists or not.Zuhair

    That claimed object, the "totality object", is what I see no reason to believe exists. You made some claim about "part-whole" relationship, but I see no reason why two apples, ten apples, or a million apples is a "whole" anything. Therefore there is no "whole" to this object which you are describing, and your descriptive terms are misleading. You are assuming that two apples is a whole, without any reasons why two apples may be a whole, and why it doesn't take three, four, five, or the proper totality of all apples, to make a whole. Surely there are more apples than two, so by what principle do you apprehend two as a whole?

    This is what Plato's theory of participation demonstrates to us. The "whole" is an Idea, the parts participate in the Idea. Now the existence of the Idea, as the whole which unifies the participants producing the object, must be supported or else the whole theory of participation falls apart. "One" is fundamentally a whole, that is its essence, a unified entity, but "one" is fundamentally different from "two" by the difference between a singularity and a multiplicity. So what principle will you introduce to support your assertion that "two" is a whole, just like "one"?
  • Zuhair
    93
    You are assuming that two apples is a whole, without any reasons why two apples may be a whole, and why it doesn't take three, four, five, or the proper totality of all apples, to make a whole. Surely there are more apples than two, so by what principle do you apprehend two as a whole?Metaphysician Undercover

    All of those have their wholes, For any predicate that hold of apples there is a totality of all apples fulfilling that predicate. And those totalities would be different totalities if the apples constituting them are different. But I've just presented to you a particular case. There is nothing special about two here or three or any number.

    I like to present matters in a Mereo-topological manner. Now a unit is an object that is not a whole of two separate (not in contact) parts, and at the same time it is separate from any other object. A totality of units is a collection. The smallest collection is a unit. An element of a collection is a unit part of that collection. So the unit collection is the sole element of itself. Multipleton collections are those that are constituted of many units. So they are not the elements of themselves. Now for the sake of simplicity let's assume the ideal condition of all units being unbreakable and actually in-changeable over time, and they won't be in contact with other objects at other moments of time. So no unit object can be split into two separate objects at some other moment of time, nor it would be a part of another unit object at other moment of time. Of course this is an ideal condition. Under that assumption we can have stable totalities and thus I can extend any predicate in the object world as far as that predicate only hold of unchangeable unit objects. If the units are breakable (as it is the case with the real object world) or can come in contact with other units to form bigger units (as it is the case with the real object world) then this method fail, or at least becomes very extremely complex.

    Set theory can be explained as an imaginary try to REPRESENT stable collections of units, by stable units. So any two stable collections (i.e. their units are unchangeable over time) would have distinct representative units (whether those representative units are part of those collections or external to them) as long as they are not the same, and each collection is only represented by one unit. This theory of representation of collections by units, is the essence of Set theory. Of course the representative units are ideal, i.e. unchangeable over time. Now while element-hood of collections are being unit parts of those collections, yet "membership" in a set is another matter. Membership in sets can be defined in two ways, l personally like the definition of them being elements of collections represented by the unit, i.e. every set is actually a unit object that represents a collection of units, now those units of the represented collection are the members of that set. Let me put it formally:

    x is member of y if and only if there exists a collection z such that y is the representative of z and x is an element of z.

    We start with the non representative unit, i.e. a unit object that do not represent any collection of units, this would stand for the empty set. then go upwards in a hierarchy. What I call as the representational hierarchy, where collections are represented by sets (units) and sets themselves are collected into collections, which are represented by sets, etc..... This step-wise hierarchical approach enables a gradual transition from the less complex to the next more complex to the next, and so on... So a nice way would be to start with the empty object (the non representing unit), then to the collection of all sets (i.e. units) representing parts of that empty object, then to the collection of all sets (i.e. units) representing parts of the resulting objects, etc... According to this view a set is always a unit, and that unit act to represent a collection of units.

    We can extend the representational hierarchy as long as we don't have a clear inconsistency with it. This way we can encode almost all of mathematical objects in that hierarchy.

    Sets not only can represent finished collections, it can also represent unfinished collection, as long as the process of producing the elements of that collection is well defined, like the process of making the naturals by succession from prior naturals and so on.. we can have a set that would represent the process of that natural production. And those are the infinite sets of naturals.

    So set theory of mathematics like in ZFC are just a theory about representation of actually finished collections and of potentially non-finishing collections.

    I just wanted to put you in the picture, that sets (as used in mathematics) are different from the collections I've spoken about. While the genre of collections is the same as the genre of their elements, sets on the other hand can be totally external to the collections they represent and can indeed be of a different nature. There is a lot of confusion between collections and sets, even in standard text-books of mathematics, and especially there is the confusion between element-hood of collections and membership in sets, that many mathematical textbooks on set theory introduce sets in terms of collections and set membership in terms of element-hood of collections, and this is a great confusion. Sets do not function as collections, no they function as unit representatives of collections, thereby enabling us to speak of a hierarchy of multiplicities within multiplicities and so on... So the set concept is a stronger concept than the collective concept. The former is representational and latter is mereological.
  • Metaphysician Undercover
    6.2k
    All of those have their wholes, For any predicate that hold of apples there is a totality of all apples fulfilling that predicate. And those totalities would be different totalities if the apples constituting them are different. But I've just presented to you a particular case. There is nothing special about two here or three or any number.Zuhair

    Right, that's why I say you're a magician. You claim that you can point to any random objects, and say those objects make another object, a whole. Your claim here amounts to an assertion that you can state "the predicate" which pertains to these two apples, or three apples, or any other number of apples, but no other apples. In reality there is no such predicate, it is a hollow claim. Therefore you have no principles for what constitutes an object, only "if I say it's a whole, then it's a whole". A principled "whole" has a defined completion, not the hollow (and sometimes impossible) claim that it's possible to define the completion.

    I like to present matters in a Mereo-topological manner. Now a unit is an object that is not a whole of two separate (not in contact) parts, and at the same time it is separate from any other object. A totality of unites is a collection. The smallest collection is a unit. An element of a collection is a unit part of that collection. So the unit collection is the sole element of itself. Multipleton collections are those that are constituted of many unites. So they are not the elements of themselves.Zuhair

    See, you are using "totality" as if any collection of things is a totality of things. Now "totality" is meaningless and redundant because any collection of things is, by your usage, automatically a totality of things. But to have a proper totality, which gives "totality" meaning, it is required that we name a type of thing, and sum the complete number of those named things. To say that any random number of things is a "totality" is nonsense, because it's not the total of anything, it's just a random collection.

    Now you claim that any random collection of elements is a "unit", but you have no principle of unity to substantiate that assertion, only the hollow claim that there is such a predicate which unifies them while excluding others. In reality, the elements cannot be a unit unless they are united by something which produces a unity. To say that any random collection of elements is a unity is to utter nonsense. By what means are those elements united?

    Do you not respect the fact, that "a collection" must be defined? And, to complete the specified collection, to produce a totality, or "whole", it is required that the entire collection be summed. To insist that my collection is 'definable', and demonstrably the totality or whole of that 'definable' collection, does not justify the claim that it is a totality or whole. The definition must be produced, and it must be demonstrated that there are no other elements which would fulfill the conditions of that definition in order to justify the claim that the collection is whole, a totality, complete.

    Set theory can be explained as an imaginary try to REPRESENT stable collections of unites, by stable unites. So any two stable collections (i.e. their unites are unchangeable over time) would have distinct representative unites (whether those representative units are part of those collections or external to them) as long as they are not the same, and each collection is only represented by one unit. This theory of representation of collections by units, is the essence of Set theory. Of course the representative unites are ideal, i.e. unchangeable over time. Now while element-hood of collections are being unit parts of those collections, yet "membership" in a set is another matter. Membership in sets can be defined in two ways, l personally like the definition of them being elements of collections represented by the unit, i.e. every set is actually a unit object that represents a collection of unites, now those unites of the represented collection are the members of that set.Zuhair

    If I correctly understand what you are saying here, a "set" is a collection. As a "unit" the set is complete, a totality, or whole. The completion is determined, fixed, perhaps even "caused", by the definition, the predicate. The definition makes the unit definite, and that is complete.

    Here is the issue which I see. In reality, it is only possible that the definition may produce a complete collection, in some cases it may be the case that a complete collection is impossible according to some definitions. Completion of the collection is really dependent on the quality of the definition. A good definition may produce a complete set, while a bad definition will produce a set which is impossible to complete. Notice your opening sentence in the paragraph above: "Set theory can be explained as an imaginary try to REPRESENT stable collections...". The operative word here being "try". So if we take a poorly defined set, and try to produce a completion, or whole, or if we assert that a poorly defined set has produced a complete whole, this is a mistake.

    We start with the non representative unit, i.e. a unit object that do not represent any collection of unites, this would stand for the empty set.Zuhair

    I see this as a self-refuting, contradictory start. The empty set would be complete, whole, with no members. It requires a definition which nothing could fulfill, a definition of nothing. But it is the definition itself which produces the entity, the unit, so the empty set is at the same time a unit, and also nothing. In common terms, to try and produce the empty set is to try to produce something out of nothing. There is no such thing as the empty set, it is an impossibility, by way of contradiction.

    Therefore I propose as the true start, the definition of a unit, what it means to be an object, complete, whole, a totality. We cannot start with an empty symbol because that creates infinite random possibilities for nonsense. We need to start with what the symbol is really supposed to represent, a set, which is an object, a unit, or whole. In this way we may restrict the use of the symbol, to eliminate vain attempts to produce an impossible whole.

    Remember, the possibility of completion is directly dependent on the quality of the definition. Therefore we ought to restrict poor quality definitions which are not conducive to the possibility of completion. Such definitions may produce the illusion of completion when no such thing is possible, allowing for misleading, or deception. This can be done with a proper definition of what it means to be a unity, a whole, complete or totality.

    So the empty set, as a starting point ought to be replaced with the set of one, a whole, an object, complete, whole and total in itself, by its very nature; so that the principal or primary set is consistent with itself, and not self-contradictory as the empty set is.

    According to this view a set is always a unit, and that unit act to represent a collection of units.Zuhair

    One problem, the empty set cannot be a unit, as described above, that's self-contradictory. It's an object composed of nothing.

    I just wanted to put you in the picture, that sets (as used in mathematics) are different from the collections I've spoken about. While the genre of collections is the same as the genre of their elements, sets on the other hand can be totally external to the collections they represent and can indeed be of a different nature. There is a lot of confusion between collections and sets, even in standard text-books of mathematics, and especially there is the confusion between element-hood of collections and membership in sets, that many mathematical textbooks on set theory introduce sets in terms of collections and set membership in terms of element-hood of collections, and this is a great confusion. Sets do not function as collections, no they function as unit representatives of collections, thereby enabling us to speak of a hierarchy of multiplicities within multiplicities and so on... So the set concept is a stronger concept than the collective concept. The former is representational and latter is mereological.Zuhair

    I would say that this confusion is actually produced by the convention of allowing for empty sets. By allowing for a set with no members we produce a separation between the set, and the members of the set. But no such separation is warranted. The set itself is the object, the elements comprise that object, and the object is created by the definition. The quality of the definition determines the definiteness or definitiveness of the object. A vague definition produces a vague object. So the set, and the elements or members, must be one and the same, in order that the set be the object composed of those elements, while the definition is what has separate existence. Therefore judgement lies in conformity between the definition and the elements, such that a poorly defined set makes a vague object, or in some (impossible or contradictory) cases not an object at all.
  • Zuhair
    93
    Now you claim that any random collection of elements is a "unit"Metaphysician Undercover

    I never said that, nor did I claim it. Actually what I said refutes that!

    If I correctly understand what you are saying here, a "set" is a collection. As a "unit" the set is complete, a totality, or wholeMetaphysician Undercover

    You didn't correctly understand what I was saying!

    to say that any random collection of elements is a unity is to utter nonsense.Metaphysician Undercover

    I cannot agree more! Of course, and that's what I was saying. But you totally misread what I was writing. I think because of you "apparently" not having experience with the topic of Mereo-topology.

    What I'm saying is a little bit complicated. Seeing your comments, I realize that you completely mis-understood me. But I do concede that what I wrote was too compact.

    Lets come to what I meant by "UNIT", I mean by that an individual. For example an apple is a unit, while the collection of two separate apples is not a unit. Now I envision a unit as an object that is not the whole of two separate objects, that is at the same time separate form other objects. This has something to do with separateness and contact. So a single apple has any two parts of it connected by a part of the apple, so it is in continuity, there is no breach to its material. While the collection of some two separate apples is not like that, you have one apple being a part of that collection and the other apple also being a part of that collection but you have a breach of material between them, i.e. the two apples are separate, i.e. not in contact with each other and no part of that collection is in contact with these two parts, such collections are NOT units, they are collections of separate units. The only collection that is a unit is a collection that have one individual, like the collection of one apple, like the collection of one bird, etc.. those are unit collections. You need some experience with Mereology (Part-whole formal study) and connectedness (Separate-contact) formal study, joining both fields you have what is known as Mereo-topology. You need to be familiar with the axiomatization of Mereo-topology, in order to get the grasp of what I'm writing here. These are particular concepts, they are not that philosophical, but of course they can be realized on philosophical grounds.

    I define "collection" as a totality of units, of course that totality itself may be a unit (in the case the collection has only one unit part of it), or might not be a unit (like a collection of multiple units: like of two apples, 10 cats, etc...). I need to stress here that "being a unit" or not, has nothing to do with the collection being definable or not, even if it is definable after some predicate still the collection if it contains many units, still it is NOT a unit. Being a unit depends on the continuity of the material in the collection, and not on definability issues or the alike. The only collection that is at the same time a unit, is the singular collection, i.e. the collection having one element, i.e. has one unit part. Otherwise collections having multiple elements whether definable or not, are always not units.

    A set (as that term is used in set theory) is a unit object that represent a collection of units, like in how a lawyer represent a collection of many accused persons. Each accused person is a unit object (because its material is in continuity, and it itself is separate form other material) and the lawyer is also a unit object, so here you have an example of some Representation relation where a collection of unit objects (that is itself (i.e. the collection) not a unit since there are many accused person in that collection of our example) that is represented by a unit object (the lawyer). That was an example of EXTERNAL REPRESENTATION. On the other hand there is INTERNAL REPRESENTATION where a single unit in the collection can stand to represent the total collection, like for example when the HEAD of some tribe represents the whole of its tribe in some meeting of head of tribes. The head of a tribe is a unit part of that tribe, and yet it can represent the whole tribe. Any group (collection) of people can always chose one among them that can stand to represent the whole group. This is internal representation.

    The usual set theory with well founded sets is a theory of external representation of collections of representatives of collections of representatives of..... It is about tiers of representation of collections.

    The empty set can be ANY non-representing individual object. For example take any particular apple. This can serve as the empty set, since apples are not representatives of collections of representatives..

    Now take some unit object that serves to represent the chosen apple above (the one we called the empty set). This must be different form that apple, because the apple is not a representative of anything, while that object is representing that apple itself. This latter object would act as the singleton set of the empty set, denoted by {{}}. Now you can take a third object that act as a representative of the collection of the apple (the empty set) and the object that represents that apple (the singleton of the empty set), now this representative object would be the set of the empty set and the singleton of the empty set, denoted by { {}, {{}} }. And so on....

    So one need to discriminate sets (which are unit collections that act as representatives of collections) from collections (which are totalities of unit objects). If one manage to fathom that discrimination, then one can of course understand the difference between being an element of a collection, which is being an individual (i.e. a unit) part of that collection, and between being a "member" of a set, here a set is a representative of a collection, and with well founded sets, they are always external representatives of collections (like in the lawyer, accused example), now being a member of a set is actually to be an element (i.e. a unit part) of the collection represented by that set. Membership of sets is a representational act, it is a kind of a singular representational act. Discrimination between the concepts of Collections and their elements, from Sets and their members, is vital for a proper understanding of the subject of sets and classes, and it is something often misunderstood, and misrepresented even at official text-books unfortunately.

    Actually If I was to rename matters, I'd call collections as sets, and what is termed as "sets" in set theory I'll call as representatives, and epsilon membership, I'll re-name as "representation step". Anyhow

    One needs to be careful! Not every collection has a representative! Even some well definable collections might not have representatives. Although this largely depends on what is meant by "well definable".

    I hope you can re-read my prior posting with this clarification.

    As about the question of random collections and defined ones, this is another matter, I didn't allude to those yet. I want to define the basic terms, and then if we have some agreement over those, we can go to those issues. But basically I do agree with the sentiment that ALL collections are aught to be definable!
  • ssu
    1.7k
    Here is a paper that questions the 'diagonal argument'.
    https://app.box.com/s/vdop6iqhi8azgoc2upd76ifu8zacq8e4
    sandman

    Cranky.fishfry

    The author giving just one reference and that being the Wikipedia page of the diagonal argument is telling by itself.

    And seems like the author is simply confused about infinite sets. And one really has to understand how different the reals are.
  • Metaphysician Undercover
    6.2k
    I cannot agree more! Of course, and that's what I was saying. But you totally misread what I was writing. I think because of you "apparently" not having experience with the topic of Mereo-topology.

    What I'm saying is a little bit complicated. Seeing your comments, I realize that you completely mis-understood me. But I do concede that what I wrote was too compact.
    Zuhair

    Sorry for the misunderstanding, I'll try to stay on track.

    Lets come to what I meant by "UNIT", I mean by that an individual. For example an apple is a unit, while the collection of two separate apples is not a unit. Now I envision a unit as an object that is not the whole of two separate objects, that is at the same time separate form other objects. This has something to do with separateness and contact. So a single apple has any two parts of it connected by a part of the apple, so it is in continuity, there is no breach to its material.Zuhair

    I tend to think that this is not a very good representation of what an object, or "unit" really is. It isn't based on an accurate description of the relationship between parts and wholes. A "unit" for you is something with existence separate from other units, yet its parts do not have such separate existence. In reality though, there is vagueness in what constitutes "separate". Due to this vagueness, there may be discrepancy in judgements as to what is the unit, and what is a part of a unit, depending on one's perspective. For example, the apple is really a part of the tree. Its generation, existence and subsistence is dependent on the tree, such that as soon as it gains "separate" existence it starts to degenerate. Also, consider a "unit" like the earth. You might think of it as "separate" from the sun, but really it only exists as a part of the solar system. Then the solar system only exists as a part of the galaxy, and so on. And if we look the other way, we are faced with the question of why this composition of molecules which is "the apple" is properly "the unit", and not the molecules themselves. After the apple separates from the tree, the molecules of the apple separate from each other in the process of degeneration. That is why I think your determination of what constitutes an object or "unit" is rather arbitrary, and dependent on one's perspective.

    So a single apple has any two parts of it connected by a part of the apple, so it is in continuity, there is no breach to its material. While the collection of some two separate apples is not like that, you have one apple being a part of that collection and the other apple also being a part of that collection but you have a breach of material between them, i.e. the two apples are separate, i.e. not in contact with each other and no part of that collection is in contact with these two parts, such collections are NOT units, they are collections of separate units.Zuhair

    Now your description of a "collection" doesn't seem to provide principles to distinguish between an artificial collection and a natural collection. So for example you do not distinguish between a collection of apples in a bag, placed there artificially, and a collection of apples hanging on a tree. The tree might be an object, a unit, and the collection of apples exist as parts of that unit, but they could also be rearranged as parts of an artificial collection.

    So a single apple has any two parts of it connected by a part of the apple, so it is in continuity, there is no breach to its material.Zuhair

    So this appears to be the critical question, what constitutes a "breach to its material"? The apples hanging on the tree clearly have no breach of material and are therefore part of the tree, but that's a simple example. Is there a breach of material between the earth and the sun, when the two are connected by things like gravity and light?

    On the other hand, an artificial collection might very well be connected by something. Apples in a plastic bag are "connected". Apples of the same variety are "connected". Furthermore, when we manufacture things like cars for example, we connect parts together to produce a unit. So the distinction between artificial and natural, though it serves as an example, is not even a good distinction itself.

    The issue here seems to be what constitutes a "material" connection. You would say that having a material connection to something else negates the status of being a "unit", making the thing a "part" of a unit instead. If we switch to Aristotelian terms we'd replace "material" with "substance". In his "Categories", "substance" in the truest and primary sense, is defined as that which is neither predicable of, nor present in, a subject. Notice that this produces a more rigorous restriction than your "material" connection. Not only do we have "present in" as a restriction, but also "predicable of". So for example, if X is predicable of Y, X cannot be given the status of substance, and cannot therefore be a unit or object. This would extend your category of material connection to include predication as representing a material connection.

    I believe that the goal here is not to produce the artificial/natural distinction mentioned above, but to distinguish between substantial and non-substantial, or material and non-material collections. Consider my criticism of your last post accusing you of a "random collection of elements" which you flatly denied, accusing me of misunderstanding. If a collection is truly random, the so-called "parts" of that collection are actually units, there is nothing substantial connecting them, and the collection itself cannot be an object or unit. Therefore the "parts" of that collection are not properly "parts". But if the parts are connected for any valid reason, this must qualify as a substantial, or a material connection. Then the whole of the collection is a valid object or unit, and the parts cannot be understood as independent objects.

    I define "collection" as a totality of units, of course that totality itself may be a unit (in the case the collection has only one unit part of it), or might not be a unit (like a collection of multiple units: like of two apples, 10 cats, etc...). I need to stress here that "being a unit" or not, has nothing to do with the collection being definable or not, even if it is definable after some predicate still the collection if it contains many units, still it is NOT a unit. Being a unit depends on the continuity of the material in the collection, and not on definability issues or the alike.Zuhair

    According to what I've explained above, I dismiss your criteria of "the continuity of the material", as being too vague, and replace it with the Aristotelian concept of substance. Therefore any valid "collection" is itself an object or unit, the parts having a substantial relation to one another, demonstrating the existence of a "whole". The parts are therefore not independent units. Having something in common for example, cannot be taken as merely coincidental, and must be understood as indicating that the parts are not independent objects, but parts of a whole.

    The only collection that is at the same time a unit, is the singular collection, i.e. the collection having one element, i.e. has one unit part. Otherwise collections having multiple elements whether definable or not, are always not units.Zuhair

    I foresee an issue with this concept of a "singular collection". I'm afraid it might be somewhat contradictory like the empty set, or simply purposeless. Let's say that every object is unique, as per the law of identity. Any individual thing which we come across could therefore be a singular collection. However, it is pointless to make such a collection, because the reason for making a collection is to acknowledge relationships between things. So we could only place a thing in the category of "singular collection" if and only if there could be no relations between that thing and anything else. Having a relation would make it a part of a collection negating the status of "singular collection". Perhaps we could keep the category of "singular collection", but it would most likely remain an empty category. It's not an empty set though, but an empty category, because under my categorization a valid set (reasonable relations) constitutes an object.

    A set (as that term is used in set theory) is a unit object that represent a collection of units, like in how a lawyer represent a collection of many accused persons. Each accused person is a unit object (because its material is in continuity, and it itself is separate form other material) and the lawyer is also a unit object, so here you have an example of some Representation relation where a collection of unit objects (that is itself (i.e. the collection) not a unit since there are many accused person in that collection of our example) that is represented by a unit object (the lawyer). That was an example of EXTERNAL REPRESENTATION. On the other hand there is INTERNAL REPRESENTATION where a single unit in the collection can stand to represent the total collection, like for example when the HEAD of some tribe represents the whole of its tribe in some meeting of head of tribes. The head of a tribe is a unit part of that tribe, and yet it can represent the whole tribe. Any group (collection) of people can always chose one among them that can stand to represent the whole group. This is internal representation.Zuhair

    So this is where your system gets very confused, and mine becomes much more practical. In your example, why not simply say that the collection of people represented by the lawyer is itself an object? They all have X in common, so they have that valid relationship to one another, and therefore exist together as that mentioned object. There is no need to assign to a member of the group the task of representation, such as the person who represents the group, and hand the group real existence through that representation, the group already has real existence through the real experience which they share, which constitutes a real relationship. Such real relations make real objects. And, in the other example, the tribe has real existence as an object, due to the relations between members, it does not require a "head" of the tribe, or representative of the tribe, to give it real existence as a collection or object. Requiring that the collection has a representative creates all sorts of problems, beginning with the representative's real capacity to adequately represent the collection. See, under this system, the collection, as an object, can only be apprehended as an object, to the extent provided by the representative. But the representative is not the true object, and we are better off to look directly at the object to understand its true existence.

    The usual set theory with well founded sets is a theory of external representation of collections of representatives of collections of representatives of..... It is about tiers of representation of collections.

    The empty set can be ANY individual object. For example take any particular apple. This can serve as the empty set, since apples are not representatives of collections of representatives..

    Now take some unit object that serves to represent the chosen apple above (the one we called the empty set). This must be different form that apple, because the apple is not a representative of anything, while that object is representing that apple itself. This latter object would act as the singleton set of the empty set, denoted by {{}}. Now you can take a third object that act as a representative of the collection of the apple (the empty set) and the object that represents that apple (the singleton of the empty set), now this representative object would be the set of the empty set and the singleton of the empty set, denoted by { {}, {{}} }. And so on....
    Zuhair

    Yes, this complexity is exactly why your system is bound to failure. As I described above, any individual object may be used to create a "singular collection". But that is to assume that the individual has no relations to anything else, and this produces an empty set. To make the set meaningful, another object must represent the singular collection, but that negates the status of singular collection. So your whole set system is based in something meaningless, or even contradictory, the empty set, which is represented by the singular collection. It's like you're building your sets bottom up, when they need to be produced top down to have any substance. The set must be principled on the real existence of parts to a whole, as an object, and not based on a part which is meant to represent a whole.

    One needs to be careful! Not every collection has a representative! Even some well definable collections might not have representatives. Although this largely depends on what is meant by "well definable".Zuhair

    This is evidence that your system is faulty. We need to recognize a collection as an entity itself, and not rely on a representative. A representative is often incapable of representing to us, the "thing" which is responsible for the real and valid existence of the collection. And this is proven by the fact that some valid collections have no representative.
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