• Philosopher19
    273
    There is no object called 'Infinity' in the sense you have been using it.

    Here is a way to say what you want to say:

    In mathematics, there are sets that are infinite but that have different cardinality from one another.

    Better yet:

    If x is infinite then there is a y that is infinite and y has greater cardinality than x.
    TonesInDeepFreeze

    But all of the above is exactly what I'm saying is contradictory. And my use of infinity which (if I've understood you correctly) you say is not the one that they use in maths, is the reason that I say all of the above is contradictory.

    There is no object called 'Infinity' in the sense you have been using it.TonesInDeepFreeze

    So what semantic are mathematicians using when they use the world/label "infinite"?

    There is no x such that for all y, y is a member of x iff y is not a member of y. Proof:TonesInDeepFreeze
    the axiom schema of separationTonesInDeepFreeze

    Something cannot be both a member of itself and a member of other than itself at the same time. For example, take z to be any set that is not the set of all sets, and take v to be any set. The z of all zs is a member of itself as a z (as in in the z of all zs it is a member of itself). But it is not a member of itself in the v of all vs, precisely because in the v of all vs it is a member of the v of all vs as opposed to a member of itself. If we view the z of all zs as a z, it is a member of itself. If we view the z of all zs as a v, it is a member of the set of all sets. You can't view it as both a member of the z of all zs and a member of the v of all vs at the same time. That will lead to contradictions. In other words, we can't treat two different references as one (as in are we focused on the context of vs or the context of zs?)

    Note that the above shows the impossibility of a set that contains all sets that are members of themselves where all equals more than one.

    For the fully fleshed out version of this, see my post on Russell's paradox which I posted the link to in this discussion and in the other one.

    'There exists a z such that for all y, y is a member of z' contradicts this instance of the axiom schema of separation: For all z, there is a x such that for all y, y is a member of x iff (y is a member of z and yis not a member of y).TonesInDeepFreeze

    If some theory suggests that you can view the z of all zs as both a member of the z of all zs and a member of the v of all vs at the same time, then that theory is contradictory. The z of all zs is either to be treated like a z or a v. If it is to be treated like a z, it is a member of itself. If it is to be treated like a v, it is not a member of itself (precisely because it is a member of the v of all vs)

    We do NOT claim that from "after each natural number there is a next number" and "there is no greatest natural number" that we can infer that there is a set of all the natural numbers. Indeed such an inference IS a non sequitur. And every mathematician and logician knows it is a non sequitur. So, we recognize that to have a set with all the natural numbers we need an AXIOM for that, which is NOT an inference.TonesInDeepFreeze

    I'm not sure what you mean by "So, we recognize that to have a set with all the natural numbers we need an AXIOM for that, which is NOT an inference."
  • TonesInDeepFreeze
    2.3k
    We go in a circles, as it is with cranks. The crank makes false claims and terrible misunderstandings. Then the crank is corrected and their error is explained. Then the crank ignores all the corrections and just posts the false claims and misunderstanding again as if the corrections and explanations never existed.

    So it is again regarding 'contradiction'.

    Go back and read what I wrote about "contradiction".

    Something cannot be both a member of itself and a member of other than itself at the same time.Philosopher19

    In certain alternative set theories, there are sets that both members of themselves and of other sets.

    In ordinary set theory, no set is a member of itself.

    By the way, we don't need to use temporal phrases such as "at the same time". Set theory does not mention temporality.

    Then the rest of your z's and v's is irrelevant if it is supposed to refute the proofs I gave. Moreover, if you knew anything about this subject or even mathematical discourse you'd see that your prose about it is ungrounded, impenetrable double-talk.

    To refute a purported proof, you need to show a step in the proof that is not permitted by the inference rules (which in this case are those of ordinary predicate logic).

    And you separately quoted me saying "The axiom schema of separation". What was the point of that? Did you look up what the axiom schema of separation is and you think your remarks relate to it in some way?

    I'm not sure what you mean by "So, we recognize that to have a set with all the natural numbers we need an AXIOM for that, which is NOT an inference."Philosopher19

    It is clear. You don't know what it means, because you are virtually completely ignorant of the subject matter.

    I'll spell it out even more:

    In set theory, to prove there exists a set having a certain property, we must do so from the axioms and rules of inference alone.

    In this instance, the property in question is "has as members all the natural numbers"

    Without the axiom of infinity, we cannot prove that there is a set with the property "has as members all the natural numbers". But with the axiom of infinity we can prove that there is a set with the property "has as members all the natural numbers".

    You argued that from "after each natural number there is a next natural number and there is no greatest natural number" we cannot infer "there is a set of all the natural numbers". And you are correct about that!

    So I pointed out that indeed set theory does not make that unjustified inference, but rather, set theory has an axiom from which we CAN infer that there is a set of all the natural numbers. And THAT inference, from the axiom, does NOT use the unjustified inference from "after each natural number there is a next natural number and there is no greatest natural number" to "there is a set of all the natural numbers".

    /

    You know virtually nothing about set theory. You should present whatever concept of infinity you like, but you shouldn't be presenting it as a refutation of a subject you are ignorant about.
  • TonesInDeepFreeze
    2.3k
    You know, its a funny thing, but when I don't know much about a subject, I pay attention to people who do know something about it. And especially I don't slather the Internet with stubbornly false and confused claims about it.
  • TonesInDeepFreeze
    2.3k
    So what semantic are mathematicians using when they use the world/label "infinite"?Philosopher19

    We don't say "using semantic".

    Rather, we just state the definitions.

    I stated the definitions in my first post in this thread:

    https://thephilosophyforum.com/discussion/comment/878326
  • jgill
    3.5k
    So what semantic are mathematicians using when they use the world/label "infinite"?Philosopher19

    Mathematicians, like myself, may get a little sloppy about using the word, infinity, at times. For example, for those of us in complex variable theory The point at infinity has a specific reality as the north pole of the Riemann sphere. There is a technical way of saying this.
  • TonesInDeepFreeze
    2.3k
    I have an ability to understand concepts without even knowing of themVaskane

    Please forgive the cliche, but it is especially apt: Above is Dunning-Kruger on steroids.
  • Corvus
    2.4k
    Here is a finite definition of an infinite set: "A given set S is infinite iff there exists a bijective function between S and a proper subset of S." Furthermore, such a bijective function can be stated finitely.

    Here is an example. Take the set of natural numbers ℕ = { 0, 1, ··· }. Now take a proper subset of ℕ containing only even the numbers, ℙ = { 0 , 2 , ··· }. These two are equinumerous because there is a bijective function f : ℕ → ℙ, given by f(n) = 2n.

    The proof that "f" is bijective is finite. So is the proof that ℙ is a proper subset of ℕ.
    DanCoimbra
    Great post, thanks. How do you prove then N is different size to P?
  • Mark Nyquist
    729

    Did you give Philosopher19 the finger or is there real math behind the north pole of the riemann sphere? It would be cool if you meant it both ways.
  • Philosopher19
    273
    In certain alternative set theories, there are sets that both members of themselves and of other sets.TonesInDeepFreeze

    We go in a circles, as it is with cranks. The crank makes false claims and terrible misunderstandings. Then the crank is corrected and their error is explained. Then the crank ignores all the corrections and just posts the false claims and misunderstanding again as if the corrections and explanations never existed.TonesInDeepFreeze

    Evidently, there's no point in continuing this discussion. If you believe your mathematics is free from contradictions or paradoxes, then in my opinion, you are not blameworthy for upholding them or sticking to them (unless of course someone presents a better or more complete thing to you and you reject greater for lesser), but if you see paradoxes and contradictions or incompleteness and you treat them as other than paradoxes/contradictions/incompletions...
    I see no paradoxes or contradictions or foundational incompleteness in the beliefs that I uphold (mathematical or otherwise).

    Peace
  • Philosopher19
    273
    I will just say this. That a set cannot be both a member of itself and a member of other than itself is the equivalent of saying that a shape cannot be both a square and a triangle (I have taken out the "at the same time" and the effect is still the same).

    The above point I felt was worth adding to this discussion, but I will probably stop posting here as I don't think there's anything left to add to this discussion.
  • Mark Nyquist
    729

    This isn't really the place to come to get people to agree with you. I think the math boys really did give you a good amount of feedback that would be hard to get anywhere else. So if you want to run something past us we'll tell you what we think and you can react accordingly. Most of what you say really irks a formally trained mathematician.

    To me it seems like arguing about mental fantasies but for someone who has studied it there would be something to defend.

    It's been one of the more lively threads here...seems to go on all day.

    As far as the math profession I do think you should show some respect because the world runs on the math they do and for some things only a few people per million or billion may be able to do it.
  • ssu
    7.9k
    This isn't really the place to come to get people to agree with you. I think the math boys really did give you a good amount of feedback that would be hard to get anywhere else. So if you want to run something past us we'll tell you what we think and you can react accordingly. Most of what you say really irks a formally trained mathematician.Mark Nyquist
    Here's actually some advice to all non-mathematicians (from a non-mathematician):

    If you really can ask an interesting foundational question that isn't illogical or doesn't lacks basic understanding, you actually won't get an answer... because it really is an interesting foundational question!

    Yet if the answer is, please start from reading "Elementary Set Theory" or something similar then yes, you do have faulty reasoning.
  • TonesInDeepFreeze
    2.3k
    How do you prove then N is different size to P?Corvus

    We don't. He proved that they are the same size.
  • TonesInDeepFreeze
    2.3k
    We go in a circles, as it is with cranks. The crank makes false claims and terrible misunderstandings. Then the crank is corrected and their error is explained. Then the crank ignores all the corrections and just posts the false claims and misunderstanding again as if the corrections and explanations never existed.
    — TonesInDeepFreeze

    Evidently, there's no point in continuing this discussion.
    Philosopher19

    As I said, the discussion will go in circles given that you skip answers given you and instead just repeat your refuted claims.

    If you believe your mathematics is free from contradictions or paradoxesPhilosopher19

    It's not my mathematics. I don't have allegiance to it. I find value in it, find wisdom in it, recognize that it axiomatizes reasoning used in the sciences, and enjoy it. But I don't claim that there might not be better approaches - philosophically, intuitively, and practically.

    I don't claim to perfect certainty that set theory is consistent. But it seems to me to be an extremely good bet that it is. (1) No contradiction has been found in it under incredibly intense and indefatigable scrutiny for about 125 years. (2) We can see specifically how it was devised to avoid Russell's paradox. (3) The concept of sets as a hierarchy itself suggests an intuitive approach that is consistent.

    Again, you use the word 'incompleteness', thus ignoring the information that was given you about incompleteness.

    I see no paradoxes or contradictions or foundational incompleteness in the beliefs that I uphold (mathematical or otherwise).Philosopher19

    You haven't proposed an alternative framework, let alone in axiomatic form. Articulate the principles by which you propose to derive mathematics adequate for the sciences, or, better yet, put it in axioms; then we can put it to the test.

    Set theory gets the job done of axiomatizing the mathematics for the sciences. By analogy: Set theory is an airplane that flies. If one thinks it's not a good airplane, then one is welcome to show us a better one.
  • Lionino
    849
    I see no paradoxes or contradictions or foundational incompleteness in the beliefs that I uphold (mathematical or otherwise).Philosopher19

    Which are?
  • TonesInDeepFreeze
    2.3k
    That a set cannot be both a member of itself and a member of other than itself is the equivalent of saying that a shape cannot be both a square and a trianglePhilosopher19

    He did it again! He completely skipped recognizing the refutation given him.
  • Corvus
    2.4k
    We don't. He proved that they are the same size.TonesInDeepFreeze
    From the point of the set N, it looks like it is. But from the point of the set P, it looks like it is only a half set to N. What's going on?
  • Philosopher19
    273


    If you look at the posts, I don't think I'm the one that has been showing the disrespect (if I have, it has been in response to disrespect). I wanted a discussion because I felt I had something to offer in response to something that I saw as contradictory. I don't think I entered the discussion closed-minded or dogmatic. And I think I tried to understand the other's point of view.

    If someone is an "expert" in the field of something, but that something is evidently paradoxical or foundationally incomplete, it's absurd to treat them like an expert of anything useful. Some people are unreasonable/absurd. They want to hold on to their paradoxical or contradictory theory or belief at the cost of sincerity to Truth/Goodness/Existence/God

    If people here witness that their beliefs or theories or axioms lead to no paradoxes or contradictions or foundational incompleteness, then I can't say to them they're misguided or lacking in knowledge.

    You have those who recognise/witness that their theories are incomplete and act as such (there is honesty to them), and then you have those who recognise/witness this, but act as though they are the knowledgeable ones whilst all others are ignorant (which to me is the very definition of a "bad guy"). I believe spending hours or years or decades on something that is foundationally corrupt, does not make you an expert in anything other than something that is useless. What good is an expert in multishapism geometry that deals with the study of shapes such as round triangles and circular pentagons?

    I don't feel like I have any contradictory or paradoxical theories or beliefs that I need to reconcile. I was trying to address what I saw as contradictory. If it's not contradictory, then it's not contradictory. But if it is contradictory/paradoxical and some are hardcore with regards to holding on to this, what can I say?
  • TonesInDeepFreeze
    2.3k


    The cardinality of N = the cardinality of P iff there is a bijection between N and P.

    There is a bijection between N and P.

    Therefore, the cardinality of N = the cardinality of P.

    Meanwhile, there is no apparent meaning in "from the point of view".

    Yes, P is a proper subset of N. Indeed the point is that it is a property of infinite sets that there are bijections between them and certain proper subsets of themselves.

    The fallacy is in saying "half" in this context. For infinite sets, there is no division operation such that there is 1/2 the cardinality of an infinite set.
  • Philosopher19
    273
    He did it again! He completely skipped recognizing the refutation given him.TonesInDeepFreeze

    Ok. Let me put it this way. I gave you a refutation with the z example. You started with insults, then you eventually said something like this:

    By the way, we don't need to use temporal phrases such as "at the same time". Set theory does not mention temporality.

    Then the rest of your z's and v's is irrelevant if it is supposed to refute the proofs I gave.
    TonesInDeepFreeze

    I decided discussing something with someone who seems to be emotional or biased is a waste of my time so I said I will stop, but I felt the need to add the following to the discussion:

    I will just say this. That a set cannot be both a member of itself and a member of other than itself is the equivalent of saying that a shape cannot be both a square and a triangle (I have taken out the "at the same time" and the effect is still the same).Philosopher19

    This dealt with your temporal phrases response.

    You have not yet answered:

    Is it logically possible for a set to be both a member of itself and a member of other than itself? If it is a member of other than itself, then it is not a member of itself, is it? And if it is a member of itself, it is not a member of other than itself is it?

    And don't say to me something like "some set theories allow for this or that". I'm asking a basic logical question that has a basic and straight forward answer. There is no need to dance around anything. Just deal with the main issue at hand.
  • Mark Nyquist
    729

    You are too pessimistic. You can have your view and they can have theirs.

    You can always declare victory, plant your flag and call it a day. Really, say what you like. I agree there are contradictions and what I brought up about parameters that can be anything your brain can dream up.

    Maybe there are real world applications to some of this as has been discussed by those who have actually done it. I assume they use what has proven to work. Math in practice has a precision component, not just theorizing.
  • TonesInDeepFreeze
    2.3k
    I don't think I'm the one that has been showing the disrespectPhilosopher19

    I don't care to say you are "disrespectful", but you are irrational and in bad faith when you skip refutations and explanations given you and instead just keep repeating your false and confused claims.

    I don't think I entered the discussion closed-minded or dogmatic.Philosopher19

    You are closed minded to the fact that you are close-minded and dogmatic. And you still won't face that your hyper-opinionating on a subject you know nothing about. If really were the fair minded person you claim to be, then you would get a book and find about the subject rather than posting misinformation and confusions about it.

    "expert" in the fieldPhilosopher19

    Just for the record, I don't claim to be an expert in anything other than jazz, and even in that field I'm deficient in important ways.

    They want to hold on to their paradoxical or contradictory theoryPhilosopher19

    There it is again! You say 'contradictory', again ignoring all the explanation given you about that.

    incompletePhilosopher19

    There is it is again! You say 'incomplete', again ignoring all the explanation given you about that.

    act as though they are the knowledgeable ones whilst all others are ignorantPhilosopher19

    I don't know anyone who has said that all others are ignorant. You are ignorant on the subject. That doesn't entail that others are ignorant on it. Indeed, there are people who critique classical set theory who are extremely knowledgeable about it. Critiques of set theory are quite fair game and bring profound insights into the subject. But those are knowledgeable, responsible and thoughtful critiques. And better yet, they are critiques that are followed up with actual mathematical alternatives to classical set theory.

    What good is an expert in multishapism geometry that deals with the study of shapes such as round triangles and circular pentagons?Philosopher19

    Nope. Set theory doesn't do that.
  • Philosopher19
    273


    I responded to you, you responded me with a refutation, I responded to your refutation with the following:

    Is it logically possible for a set to be both a member of itself and a member of other than itself? If it is a member of other than itself, then it is not a member of itself, is it? And if it is a member of itself, it is not a member of other than itself is it?Philosopher19

    Where is my response? Is it me who ignores you or you who ignores me?
  • Lionino
    849
    All of them are here:

    godisallthatmatters.com
    Philosopher19

    Ok, so you are saying that your beliefs are not incomplete or contradictory in any way. That is not philosophy, that is religion, aka delusion.
  • Philosopher19
    273


    Did you read anything from the link I gave you?
    I believe my beliefs are not foundationally incomplete or contradictory in any way from a rational/semantical point of view.
  • Philosopher19
    273
    I don't know anyone who has said that all others are ignorant. You are ignorant on the subject. That doesn't entail that others are ignorant on it. Indeed, there are people who critique classical set theory who are extremely knowledgeable about it. Critiques of set theory are quite fair game and bring profound insights into the subject. But those are knowledgeable, responsible and thoughtful critiques. And better yet, they are critiques that are followed up with actual mathematical alternatives to classical set theory.TonesInDeepFreeze

    I didn't say all others are ignorant. I just said there are people who are like this. I did not specify who.
  • TonesInDeepFreeze
    2.3k
    I gave you a refutation. You started with insultsPhilosopher19

    On some crucial points, you didn't even recognize them, let alone refute them. And when you did attempt to refute points, you failed, as your supposed refutations were false and confused.

    You started with insultsPhilosopher19

    That's a lie. I started with plain, cold information. And I did that for several posts. Eventually, it became clear that you are immune to rational discussion, and so I factually pointed out that you are confused, ignorant of the subject and in bad faith.

    emotional or biasedPhilosopher19

    As to bias, I have read a pretty good amount of the literature of this field with informed and responsible debates regarding classical mathematics. I am fascinated by and greatly enjoy informed and responsible critiques of classical mathematics. As to emotion, exasperation with cranks is natural.

    You have not yet answered:

    Is it logically possible for a set to be both a member of itself and a member of other than itself? If it is a member of other than itself, then it is not a member of itself, is it? And if it is a member of itself, it is not a member of other than itself is it?
    Philosopher19

    I did answer it. Specifically and exactly.

    And don't say to me something like "some set theories allow for this or that".Philosopher19

    The relative consistency of those theories indicates that it is not contradictory that a set is a member of itself and also a member of other sets.

    There is no need to dance around anything.Philosopher19

    There's nothing terpsichorean about my reply. I gave you an exact refutation. The fact that you are ignorant of the context of set theory and alternative set theories is not my fault.
  • TonesInDeepFreeze
    2.3k
    Is it logically possible for a set to be both a member of itself and a member of other than itself? If it is a member of other than itself, then it is not a member of itself, is it? And if it is a member of itself, it is not a member of other than itself is it?
    — Philosopher19

    Where is my response? Is it me who ignores you or you who ignores me?
    Philosopher19

    My response is right where it was when I gave it.

    And I also responded to your previous tu quoque, and you ignore that too.
  • Philosopher19
    273


    The relative consistency of those theories indicates that it is not contradictory that a set is a member of itself and also a member of other sets.TonesInDeepFreeze

    It is blatantly contradictory for x to be both x and not x. It is blatantly contradictory for a set to be both a member of itself and not a member of itself. Yet you want to persist by saying things like the above. Again, I asked:

    Is it logically possible for a set to be both a member of itself and a member of other than itself? If it is a member of other than itself, then it is not a member of itself, is it? And if it is a member of itself, it is not a member of other than itself is it?Philosopher19

    and I added:

    And don't say to me something like "some set theories allow for this or that". I'm asking a basic logical question that has a basic and straight forward answer. There is no need to dance around anything. Just deal with the main issue at hand.Philosopher19

    It seems that what I added was ignored and what I asked was not answered. Until I see a good enough response, I'm done putting any more time into this. Once again:

    It is blatantly contradictory for x to be both x and not x. It is blatantly contradictory for a set to be both a member of itself and not a member of itself.

    Who would reject this but the contradictory/unreasonable/irrational/absurd/insincere?
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