which is a conclusion that Cantor accepted because it resonated with his theology. — sime

Are you suggesting that is a reason for rejecting his conclusions? Either way, I would suggest that we leave Cantor's theology as a matter between Cantor and his God.

Why do you think the proposition that the natural numbers is countable does not contradict the proposition that the natural numbers are infinite, in the way I explained? — Metaphysician Undercover

It depends, as I explained earlier, how you define "countable". I don't say that it's just all just a matter of definitions, but it's probably a good idea to get those agreed so that we can be sure we are talking about the real issues. As it is, we don't agree and so we never get to identify and discuss the real issues.

That's exactly right. To say that 2 is between 1 and 3 is to say that it serves as a medium. However, in the true conception and use of numbers, 1, 2, 3, is conceived as a unified, continuous idea. This unity is what allows for the simple succession representation which you like to bring up. No number is between any other number, they are conceived as a continuous succession. To say that 2 comes between 1 and 3 is a statement of division, rather than the true representation of 1, 2, 3, as a unity, in the way that the unified numbering system is conceived and applied. — Metaphysician Undercover

I'm not sure what you mean by "serves as a medium". I accept you are right to observe that the numbers are defined as a succession. (I don't know why you call the successor function a representation of something, but let that pass...) But the point of a succession is that every step (apart, perhaps, from 0) has a predecessor and a successor. That is what it means to say that n is between n-1 and n+1. It is not wrong to say that 2 unites 1 and 3 and it is not wrong to say that 2 divides 1 and 3. But it is wrong not to say both.

"Infinite" means limitless, boundless. The natural numbers are defined as infinite, endless. limitless. All measurement is base on boundaries. To say a specific parameter is infinite, means that it cannot be measured. Counting is a form of measurement. Therefore the natural numbers cannot be counted. To propose that they are countable, is contradictory, because to count them requires a boundary which is lacking, by definition. — Metaphysician Undercover

This just turns on your definition of what it is to count something.
Using a ruler to measure a (limited) distance means counting the units. Obviously, we need enough numbers to count any distance we measure. So having an infinite number of numbers is not a bug, but a feature. It guarantees that we can measure (or count) anything we want to measure or count.
I maintain that if you can start to count some things, they are countable. You maintain that things are countable only if you can finish counting them., It's a rather trivial disagreement about definitions. But I do wonder how it is possible to start counting if I can only start if I can finish.

- I've been meaning to return to this for a while now, but just haven't had time. You're already juggling multiple interlocutors; hopefully this won't be interpreted as "piling on".

For example, imagine that there is forty chairs in a room somewhere. There is simply an existing bijection between the chairs and the integers, so that the count is already made without having to be counted. It's just a brute fact that there is forty chairs there, without anyone counting them. This is a form of realism known as Platonic realism. The numbers simply exist, and have those relations, which we would put them into through our methods, but it is not required that we put them into those relations for the relations to exist. — Metaphysician Undercover

I see what you are saying here. I was coming at this from a slightly different angle.

I take it that you are aware that there are several different axiomatizations of set theory. Some examples are: ZFC, ZF, Z, CZF, IZF and various Finitistic and even Ultrafinitistic systems.

The argument about measurement that you provided in your reply is interesting, and I can see how it is relevant to question of whether (or in what sense) a countably infinite set can be said to "exist". But the word "exists" can have different meanings depending on the context. Within the context of ZFC set theory, to say that a countably infinite set "exists" doesn't imply that it exists in some Platonic heaven. That's not to say that you couldn't interpret it in a Platonic way, just that nothing in ZFC itself forces this interpretation.

Now, I see that a few others on the thread have raised a similar point and that you have not been convinced. That's fair. I doubt that I will be able to convince you either, but I will try to explain how I see it and then you can let me know what you think.

The way I (and many others) interpret the word "exists" with respect to ZFC set theory is something like "there is a derivation from the axioms of ZFC using the inference rules of classical first-order logic, of the formula ∃x P(x)". Or, more compactly, ZFC ⊢ ∃x P(x).

So to say that "a countably infinite set exists" is just to say "ZFC ⊢ ∃x CountablyInfinite(x)". The actual derivation follows very simply from the axiom of infinity in combination with the definition of "countably infinite".

In my view, accepting this does not mean that you have to believe that countably infinite sets "exist" in any other sense, whether that be in a Platonic heaven, in the mind of God, or as an actual collection of objects somewhere within the physical universe.

What are your thoughts on this?

(The finitude of an object's exact position in position space, becomes infinite when described in momentum space, and vice versa. Zeno's paradox is dissolved by giving up the assumption that either position space or momentum space is primal) — sime

i don't consider this to be a solution, because the result is the uncertainty principle. What you indicate is two distinct concepts of space which are incompatible, "position space", and "momentum space".

It depends, as I explained earlier, how you define "countable". I don't say that it's just all just a matter of definitions, but it's probably a good idea to get those agreed so that we can be sure we are talking about the real issues. As it is, we don't agree and so we never get to identify and discuss the real issues. — Ludwig V

We went through the common definition of "countable" provided by jgill, and the contradiction remained. So I really don't know what type of definition of "countable" you might be thinking of.

I'm not sure what you mean by "serves as a medium". — Ludwig V

"Medium is commonly defined as "something in a middle position". If something is between two things, it is distinct from each of the two as in the middle.

But the point of a succession is that every step (apart, perhaps, from 0) has a predecessor and a successor. That is what it means to say that n is between n-1 and n+1. It is not wrong to say that 2 unites 1 and 3 and it is not wrong to say that 2 divides 1 and 3. But it is wrong not to say both. — Ludwig V

Yes, every step has a successor, but the succession is described as a continuous process. No individual step can serve as a division between the prior step and the posterior, as each is continuity, not a division. To say that one step is a division would produce two distinct successions, one prior one posterior. then the one which served as the divisor would have no place in either of the two successions.

So I dont't understand what you are saying here, especially what you mean by "2 divides 1 and 3". One divided by two produces a half, and three divided two produces one and a half. But it doesn't make sense to say that two acts as a division between one and three in the way that you propose.

This just turns on your definition of what it is to count something.
Using a ruler to measure a (limited) distance means counting the units. Obviously, we need enough numbers to count any distance we measure. So having an infinite number of numbers is not a bug, but a feature. It guarantees that we can measure (or count) anything we want to measure or count.
I maintain that if you can start to count some things, they are countable. You maintain that things are countable only if you can finish counting them., It's a rather trivial disagreement about definitions. But I do wonder how it is possible to start counting if I can only start if I can finish. — Ludwig V

So it looks like you disagree with my premise that counting is a form of measurement. Since you claim that starting to count something is sufficient to claim that it is countable, then if we maintain consistency for other forms of measurement, puling out the tape measure would be sufficient to claim that the item is measurable. Since this is obviously not true, it seems you are claiming that counting is not a form of measurement at all. How would you define "countable"?

Are you suggesting that is a reason for rejecting his conclusions? Either way, I would suggest that we leave Cantor's theology as a matter between Cantor and his God. — Ludwig V

I'm saying that in the presence of an inconsistency between ZFC and computable notions of mathematics, coupled with the obvious uselessness of of non-constructive cardinal analysis, the theological origin of ZFC becomes conspicuous.

i don't consider this to be a solution, because the result is the uncertainty principle. What you indicate is two distinct concepts of space which are incompatible, "position space", and "momentum space". — Metaphysician Undercover

yes, there are two incompatible bases for describing dynamics, and in line with your proposal, what looks like a hypertask iwhen measured in one basis, is a mere task in the other (where measurement is understood as destructive interference).

I presently suspect that the structure of the uncertainty principle, that concerns non-commutative measurements, is a logical principle derivable from Zeno's arguments, without needing to appeal to Physics.

So I dont't understand what you are saying here, especially what you mean by "2 divides 1 and 3". One divided by two produces a half, and three divided two produces one and a half. But it doesn't make sense to say that two acts as a division between one and three in the way that you propose. — Metaphysician Undercover

I'm sorry. I should have said "separates", not "divides".

So it looks like you disagree with my premise that counting is a form of measurement. — Metaphysician Undercover

Can you think of a form of measurement that is not counting - apart from guessing or "judging"?

Since you claim that starting to count something is sufficient to claim that it is countable, then if we maintain consistency for other forms of measurement, puling out the tape measure would be sufficient to claim that the item is measurable. — Metaphysician Undercover

I disagree. Since this is not an argument, it seems inappropriate to reply.

I'm saying that in the presence of an inconsistency between ZFC and computable notions of mathematics, coupled with the obvious uselessness of of non-constructive cardinal analysis, the theological origin of ZFC becomes conspicuous. — sime

OK. Obviously I'm not in a position to comment.

Again, there seems to me to be a bunch of errors in what you have said here. The core one seems to be equating P(N) with the decidable sets.

The statement “We can construct an injection P(N)→N via Turing machine encoding of decidable sets”
would mean every subset of N can be uniquely encoded by a natural number. But that is equivalent to saying ∣P(N)∣≤∣N∣, which directly contradicts Cantor’s theorem. So if the statement were true, Cantor’s theorem would already be false.

There are undecidable subsets of N. We cannot construct an injection P(N)→N via Turing machine encoding of decidable sets

I'll stop there. I can't see that your account works.

, Given Meta's rejection of quantification, and now of numbers being ordered, it's about as clear as it could be that for Meta there is very little left of mathematics.

There is a point at which one's interlocutor's commitments collapse the subject matter under discussion.

That's where we are at with Meta.

As Frank points out,

It really comes down to which view best accommodates what we do with math. — frank

And Meta's view undermines most of mathematics, despite what we do with it.

Meta treats the ∃ of quantification, a logical move within the game of maths that understands there is a symbol n in the domain of discourse that satisfies P according to the rules of the theory, as if it implies n exists as an abstract object independent of language, symbols, or human conventions. That's just a muddle. At the core he perhaps does not understand the difference between syntax, semantics and ontology.

Given that Meta asserts that 2 is not between 1 and 3, I think I'm done here. I don't see any gain in showing further absurdities in his position.

And Meta's view undermines most of mathematics, despite what we do with it. — Banno

A nominalist would provide an argument for why we can use math without committing to abstract objects. I guess Meta is a math skeptic.

've been meaning to return to this for a while now, but just haven't had time. — Esse Quam Videri

I'm glad you're back.

The argument about measurement that you provided in your reply is interesting, and I can see how it is relevant to question of whether (or in what sense) a countably infinite set can be said to "exist". — Esse Quam Videri

The issue of platonism is more about the existence of any bijection in general, and the question of whether a measurement exists without requiring someone to measure it. It's a form of naive realism, which in our conventions and educational habits, we tend to take for granted. We look at an object, a tree, a mountain, etc., and we assume that it has a corresponding measurement, without requiring that someone measures it. Then, when someone goes to measure it, the correct measurement is assumed to be the one which presumably corresponds with the supposed independent measurement. This type of realism requires platonism because there must be independent numbers and measuring standards which exist independently from any mind, in order that the object has a measurement before being measured.

But if we understand that numbering conceptions, and measuring conceptions are products of the human mind, then it's impossible that an independent object could have a measurement prior to the measurement being made by a human being. This rules out the possibility that the natural numbers could have a measurement, or be countable because we know that human beings could not count them all.

The argument about measurement that you provided in your reply is interesting, and I can see how it is relevant to question of whether (or in what sense) a countably infinite set can be said to "exist". But the word "exists" can have different meanings depending on the context. Within the context of ZFC set theory, to say that a countably infinite set "exists" doesn't imply that it exists in some Platonic heaven. That's not to say that you couldn't interpret it in a Platonic way, just that nothing in ZFC itself forces this interpretation. — Esse Quam Videri

So we could say, that numbers "exist" in a way other than platonic realism, but we must consider what would be meant by this. We need to ask, what is the criteria for existence. Consider the difference between the following two statements. 1. "the set of natural numbers between 0 and 5". 2. "{1,2,3,4.}". We might say at first glance that they both say the very same thing, and they both necessitate the existence of those four integers, but this would not be correct. That is because the first is a formula, and the existence of the named integers requires that the formula be carried out correctly. So we need to respect this difference, the existence of the integers in the first example is conditional, or contingent, on "correctness", and in the existence is necessary. And when we say 1, 2, 3, ..., or use the successor function, the existence of those numbers is conditional, contingent on correctness. Now we have the problem mentioned above, the natural numbers cannot have a measurement, because the procedure cannot be carried out to the end, "correctly".

The inclination might be to deny that distinction between necessary existence and conditional existence, which I provided. But we cannot do this because we need to account for the reality of human error. A formula does not necessitate the existence of numbers because error may arise in a number of different ways. The formula might be carried out incorrectly, or it might be a mistaken formula in the first place. So we might add, "the designated numbers exist if the the formula is properly formulated, and if it is carried out correctly. But that makes it conditional.

So to say that "a countably infinite set exists" is just to say "ZFC ⊢ ∃x CountablyInfinite(x)". The actual derivation follows very simply from the axiom of infinity in combination with the definition of "countably infinite". — Esse Quam Videri

So you are talking about a conditional existence. The supposed existence of the natural numbers, is dependent on the correct procedure. The issue with the definition of "countably infinite" is that the procedure cannot be carried out. The formula states something which is impossible to correctly finish, therefore the numbers cannot exist.

Furthermore, platonism doesn't solve the problem because the infinite is defined as being impossible, so the numbers cannot even exist in a platonic realm. That would be like saying that the full extension of pi exists in a platonic realm, when this has been demonstrated to be impossible.

I presently suspect that the structure of the uncertainty principle, that concerns non-commutative measurements, is a logical principle derivable from Zeno's arguments, without needing to appeal to Physics. — sime

I agree. Many people conclude that calculus solved Zeno's paradoxes. I've argued elsewhere, that all calculus provided was a workaround, which was sufficient for a while, until the problem reappeared with the Fourier transform.

I'm sorry. I should have said "separates", not "divides". — Ludwig V

I don't think this makes any difference.

Can you think of a form of measurement that is not counting - apart from guessing or "judging"? — Ludwig V

Sure, I believe measuring is fundamentally a form of ordering. So most comparisons which are intended to produce an order are instances of measuring. Get a bunch of people, compare their heights, and order them accordingly. That's a form of measuring.

As Frank points out,
It really comes down to which view best accommodates what we do with math.
— frank
And Meta's view undermines most of mathematics, despite what we do with it. — Banno

You mention "what we do with math", but are neglecting something very important, "what we can't do with math". This is the limitations, like the uncertainty principle mention above. We do a lot with math, sure, but there is a lot more we would be able to do if we work out some of the bugs. Then there's the even worse problem of the many things that people believe we do with mathematics, which we really don't. Some people think that calculus has solved Zeno's paradoxes. It has not. Some people think that mathematics allows us to determine the velocity of an object at a single instant in time. It doesn't. Some people think that mathematics has provided a way to make infinite numbers countable. It has not. That's what I'm talking about. To have the attitude that math is perfect, ideal, therefore it is wrong to subject it to philosophical skepticism is the real problem.

I guess Meta is a math skeptic. — frank

I like to apply a healthy dose of skepticism to any so-called knowledge. Nothing escapes the skeptic's doubt.

I guess Meta is a math skeptic.
— frank

I like to apply a healthy dose of skepticism to any so-called knowledge. Nothing escapes the skeptic's doubt. — Metaphysician Undercover

I suppose so, but the GPS in your phone was designed using math invented by Descartes. It's so weird that your GPS works even though math does not exist. :confused:

Again, there seems to me to be a bunch of errors in what you have said here. The core one seems to be equating P(N) with the decidable sets.

The statement “We can construct an injection P(N)→N via Turing machine encoding of decidable sets”
would mean every subset of N can be uniquely encoded by a natural number. But that is equivalent to saying ∣P(N)∣≤∣N∣, which directly contradicts Cantor’s theorem. So if the statement were true, Cantor’s theorem would already be false.

There are undecidable subsets of N. We cannot construct an injection P(N)→N via Turing machine encoding of decidable sets

I'll stop there. I can't see that your account works. — Banno

I'm not saying that the powerset of N is defined as only referring to the decidable sets (apologies if that is how it looked). Rather, I was overloading the notation of P(N) to refer exclusively to the decidable subsets of the natural numbers (i.e. to what is sometimes written Pdec(N))), in order to inspect what CSB implies in that special case, because the results are illuminating.

A decidable set A is a set whose members can be enumerated, and whose complement ~A can also be enumerated. Applying diagonalization to A, as per Cantor's theorem, must produce a novel decidable set, at least if we assume that an injection N --> Pdec(N) represents an effective procedure (a point that I ought to have stressed earlier). Thus to improve upon the above, diagonalization shows that:

if N --> Pdec(N) is a computable injection, then N --> Pdec(N) cannot be a computable surjection. Hence in this case, diagonalisation is a proof of the undecidability of the halting problem, rather than a proof of "more numbers".

Furthemore, Pdec(N) --> N exists as a computable injection. Hence according to CSB, N --> Pdec(N) must necessarily exist as a surjection, which is false if by surjection we mean a computable surjection.

The biggest failure of CSB in this context, is its insistence that if A --> B is a surjection, then a surjection B --> A must also exist. This is constructively false as discussed above, and the reason for why cardinal arithmetic is pointless, misleading, and false from a computational perspective.


Formally, it is might be said that CSB isn't refuted by the above, due to an "apples versus oranges" argument: Classical set theory makes no reference to decidability, meaning that N --> Pdec(N) is allowed to exist in ZFC 'rent free' as a non-computable surjection in the strictly syntactical sense of a quantiified predicate that cannot be converted into data.

In the constructive setting however, CSB is usually said to be false rather than inapplicable.

Personally, I am of the opinion that CSB along with infinite cardinal analysis, should be called "correct" in relation to the language of ZFC, but false when no background set theory is specified, due to the fact 1) that exclusively classical theorems aren't relatable to reality, and 2) they are often appealed to without anyone remembering that their correctness is relative to a classical axiomatization of set theory.

The issue with the definition of "countably infinite" is that the procedure cannot be carried out. The formula states something which is impossible to correctly finish, therefore the numbers cannot exist. — Metaphysician Undercover

This seems to be the crux of the issue for you, and I can appreciate the tension that you are raising, but personally I don't see this as an issue. I see the logical proof of the bijection as adequate to accept its existence, but scoped only to within the "game" of ZFC set theory.I'll try to explain my reasoning as clearly as I can.

For many on this thread, to say "the bijection exists", is literally to say nothing more than:

(1) the bijection is formally derivable from the axioms of ZFC in combination with the inference rules of classical first-order logic.

That's it. So when we say "the bijection exists" we are saying something more like "the bijection exists within ZFC".

For many of us on the thread, (1) is straight-forwardly true. So when someone denies "the bijection exists", we hear it as a denial of (1), since that is all we mean, whereas I think the people making the denial are (perhaps?) not intending it in this way. Hence all of the confusion.

What are your thoughts on this?

It's not even an issue of does math, or does math not exist. We are talking about whether we have an accurate description of it. To assign a name "exists" without any principles for understand what that word means, does nothing toward the purpose of describing.

If set theory starts with the premise that numbers are countable objects, and this is false, then it is a theory based on a false premise. We can ignore the question, and say that a number is "an object" in a different sense of the word, from how we commonly use "object", but then we have to ask, in what sense is it "countable" then. Counting is how we measure ordinary objects, and now we are not dealing with ordinary objects. It's incoherent to say that a number is countable in the sense of an ordinary object, but the number is not an object in the sense of an ordinary object. That's mixing apples and oranges. So this type of object, the mathematical object, must be "countable" in a different way from the way that ordinary objects are.

And the matter of infinity confirms this. If we assume the real possibility of an infinity of ordinary objects, they would not be countable. But then we say that an infinity of the type of object, which numbers are supposed to be, is countable. This confirms the problem. We count ordinary things with a bijection. Numbers are not counted in the same way as ordinary things. How are they counted? Or is the act of counting just a pretense? It appears to me to be the latter. Numbers are not counted at all, there is a stipulated formula for determining cardinality.

The matter of how does math work if it does not exist, is a completely different issue. That math works, implies that it is the means to an end. As such it is a technique, a "way of acting". Even though a "way of acting" cannot be said to exist, it can reliably bring about the desired end. A way of acting is perhaps best described as a sort of conditional. If the conditions are X, then the response is Y. So it might be described as a prescriptive rule. It's a way of applying a general principle to the particular situation.

The issue of whether a general principle, a Form "exists" is difficult. When you ask a bunch of philosophers what type of existence a rule has, you get as many different answers as people. But now we go to a further extent, and question the relationship between the general rule and the particular, in application, as a way of acting. It's like asking about the existence of a habit. It's best to avoid "exists" altogether in this context. Plato exposed this difficulty with "the good", as something which does not fit into the realm of existence. So we just judge habits as good or bad, without regard to whether they have "existence".

This seems to be the crux of the issue for you, and I can appreciate the tension that you are raising, but personally I don't see this as an issue. — Esse Quam Videri

That's fair, it appears like most people do not see it as an issue.

I'll try to explain my reasoning as clearly as I can. For me, to say "the bijection exists", is literally to say nothing more than:

(1) the bijection is formally derivable from the axioms of ZFC in combination with the inference rules of classical first-order logic. — Esse Quam Videri

How I interpret this, is that you believe it exists by stipulation. if something is stipulated to exist, then it does. I have a problem with this, because it circumvents the judgement of truth, allowing you to employ premises (axioms) without the requirement of truth. Ultimately the conclusions are unsound.

But I've discussed this with others, and it appears like "pure mathematics" likes to provide itself with the ultimate freedom of being unfettered by empirical judgements. This is advantageous to the mathematician for a number of reasons, but significantly it allows judgement based on results, rather than prior constraints. Therefore "unsound" is not necessarily bad. Forging ahead with unsound principles often produces what is good.

So in the reply to frank above, I described mathematics as "a way of acting". The way of acting is judged relative to the end, the consequences, rather judging the premises. This is pragmaticism, if it produces good results then the way of acting is itself good. We don't need to address the truth or falsity of the premises (axioms), and if you analyze the numerous different ways of acting you'll find that often the basic rules being followed cannot even be identified. At the base level, we have habits, and when a person acts by habit one cannot say that there is a rule which is being followed. So the pragmatist perspective renders the exact nature of the rules, and principles which are followed in a procedure, as fundamentally unimportant, because success is what is desired and therefore the focus.

This means that my strategy of attacking the premises, axioms, is rather pointless, as you and others will say "personally I don't see this as an issue". Even if there are blatant contradictions, it wouldn't be an issue, because success is what is important, and that's what frank pointed to. Contradiction at the base level is unimportant if the system reliably produces success. This means that the true test, the real judgement requires a focus soley on "success". In ethics this is consequentialism. However, it requires that we clearly define the intended goal, the end, in order to determine whether there really is success or not.

As a philosopher, I look beyond all the worldly goods, GPS mentioned by frank, computers, internet, AI, all these things, to say that the ultimate goal is to understand the true nature of reality. If this is the case, then as sime pointed out, mathematics delivers us a problem known as the uncertainty principle. This is a roadblock which stands in the way of success, under that description. Since this problem has been revealed, and is demonstrably the result of the mathematics, it is incumbent on us philosophers to analyze the axioms and premise of the mathematics, to determine where we are misguided. Since success is lacking, we need to take issue with the rules.

How I interpret this, is that you believe it exists by stipulation. if something is stipulated to exist, then it does. I have a problem with this, because it circumvents the judgement of truth, allowing you to employ premises (axioms) without the requirement of truth. Ultimately the conclusions are unsound. — Metaphysician Undercover

I think we are on the same page now. I personally don't think that the axioms of ZFC are "true" in any metaphysical, transcendental or empirical sense. However, I accept that existence claims derived from those axioms are nonetheless valid within the formal system. This is formal/heuristic truth, rather than metaphysical or empirical truth.

Is this a form of pragmatism? Yes, I think it probably is. I am not adopting the axioms because they are "true" in any robust sense, but because they enable so much interesting, beautiful and indispensably useful mathematics.

That said, I don't deny that your critique may have real bite against those who would take the axioms to be true in a more robust sense.

For many of us on the thread, (1) is straight-forwardly true. So when someone denies "the bijection exists", we hear it as a denial of (1), since that is all we mean, whereas I think the people making the denial are (perhaps?) not intending it in this way. Hence all of the confusion.
What are your thoughts on this? — Esse Quam Videri

It is possible, and has been the case at times in this discussion, that both sides of a debate are thinking in terms of "straightforwardly true". But there is a case for saying that, in this instance, "straightforward truth" just isn't available. To put it one way, there is truth in the orthodox account of infinity, and the "Aristotelian" and nominalist accounts. Much of this debate has circled round this, without producing much in the way of mutual understanding. Classic philosophy.

Wittgenstein's approach seems to me much more likely to be fruitful - which is not to say that everything that he argues for is equally convincing - ref; "infinity". The aim of dismantling philosophical theories by showing that some term or other had not been effectively defined is an excellent test. But nonetheless, the comparison of a philosophical view with an interpretation of a picture suggest that a more laid-back approach is more likely to enable us to understand the issue and its difficulties, at least. Then the option of admitting that the problem is insoluble or has more than one solution may be open to us or that the issue

I suppose so, but the GPS in your phone was designed using math invented by Descartes. — frank

Indeed it was. And it is important to see and to notice that mathematical ideas apply to the physical universe, at least sometimes. Whether this is unreasonable or not, is not obvious to me. But reasonable or not, it is so. Pragmatic approaches often get short shrift around here and I don't equate "works" with "true". But where some idea or technique does work, that seems an important fact about it - just as whether it is verifiable or not is not the whole story, but is an important part of it.

Is this a form of pragmatism? Yes, I think it probably is. I am not adopting the axioms because they are "true" in any robust sense, but because they enable so much interesting, beautiful and indispensably useful mathematics. — Esse Quam Videri

Yes. That's why I prefer the classical approach here. But I don't rule out that perhaps there is something about the alternative approaches that has not yet come to light. There is a connection to theology, which might explain why those approaches survive, though I confess it would not recommend them to me.

So this is where Cantor specifically went wrong: he should have interpreted diagonalization as showing that a surjection cannot always exist between countable sets. But instead, Cantor started with the premise that a surjection A --> B must always exist when A and B are countable, which forces the conclusion that diagonalisation implies "even bigger" uncountable sets, which is a conclusion that Cantor accepted because it resonated with his theology. — sime

OK, this might be difficult to understand as I don't have a clear way to say this, but I'll try to be as clear and as simple as possible:

Cantor's diagonalization, just as the example with Turing's machine, is basically about negative self reference and this negative self-reference and it's effects are the key issue. With Cantor's diagonal argument, one way to say is that it's about showing that there's a real number which isn't on the list of real numbers where every real number ought to be. With Turing, it's about another Turing Machine that does the opposite of the first Turing Machine, where the first Turing Machine should be capable of computing and giving an answer on everything. Both cases, the limitation is, de facto, negative self reference. Negative self refence simply means that it's not possible for me or anybody to do something, that I or they, don't do.

I myself have used many times the following example of how negative self reference, how easily this diagonalization works:

Forecast what number, 1 or 2, I will write in my next response and make the forecast before I respond (in a day, at least). I will follow exactly these lines: If you say I will write 1, I will write 2. If you say I will write 2, I will write 1. If you don't answer anything, just copy this or answer something else or disregard this, I will write 1.

In game theory, the answer is obvious:
_You say_/_I say_
__1__/__2__
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Is it easy for anyone else than you to forecast correctly what I will say. Yes, very likely it's going to be 1, because usually people won't even bother to participate in this simple forecasting game of mine. But you cannot write the correct forecast because the correct forecast will be what you don't write. Hence the negative self-reference. Is there a correct number to be forecasted? Definitely, yes, but it depends on your action.

Let's put this back into the context of what we have been talking about:

Is Cantor-Schröder-Bernstein theorem correct? Yes, it is.

Just as Gödel's completeness theorem goes perfectly with his incompleteness theorems, the real question is what changes in the more complex systems?

You argue it this way:

So let's take P(N) to be the decidable subsets of the natural numbers. Then is CSB true or false?

1. We know that we can construct an injection P(N) --> N via Turing machine encoding of decidable sets. (|P(N)| <= N)

2. We can build 'any old' injection f : N -> P(N) to show that |N| <= |P(N)|.

3. Hence according to CSB, the set of decidable sets of P(N) has the same size as N.

And yet f cannot be a surjection: For diagonalising over f must produce a new member of P(N), but this isn't possible if f is surjective. Hence f cannot be a surjection, and this is the reason why diagonalization can produce new members of P(N), without P(N) ever being greater than N. — sime

OK, perhaps I don't get 100% of this, but I assume your on the correct track.

Yet does diagonalization really "produce a new member"? I don't think so. Diagonalization shows us the limits of computability or in the case of Cantor the futility of trying of treating an uncomputable set as a computable set. Remember that the diagonal proof is a Reductio ad absurdum proof. A list of all real numbers cannot simply done. That it cannot be done means that it's uncomputable. Yet is there the set of real numbers? Yes. And obviously there's a bijection from the set of real numbers to the set of real numbers.

I would argue that the limits of computability and provability just show where the line between objectivity and subjectivity go. The part of mathematics that is uncomputable tells actually a lot about subjectivity and being part of the universe, which creates problems for objectivity.

N --> Pdec(N) — sime

P(N) and Dec(N) are different sets. Pdec(N) is an odd notation; I presume you mean it as the decidable subsets of P(N). I'll use Dec(N) there, to avoid any ambiguity.

We can inject Dec(N) into P(N) but not P(N) into Dec(N).

That is, there are undecidable subsets of N.

This is true computationally as well as classically.

We know that we can construct an injection P(N) --> N via Turing machine encoding of decidable sets. (|P(N)| <= N) — sime

No, we don't. Encoding Turing machines only enumerates decidable subsets of N, not all of P(N).

An addition:

The Cantor–Schröder–Bernstein theorem states:
If there exists an injection f:A→B and an injection g:B→A, then there exists a bijection h:A↔B.

That's all.

CSB does not claim:

• that a surjection A→B implies a surjection B→A;
• that injections can be replaced by surjections;
• that such functions must be computable;
• that the bijection is constructively obtainable.

I think we are on the same page now. I personally don't think that the axioms of ZFC are "true" in any metaphysical, transcendental or empirical sense. However, I accept that existence claims derived from those axioms are nonetheless valid within the formal system. This is formal/heuristic truth, rather than metaphysical or empirical truth. — Esse Quam Videri

We're still not on the same page, because I claim contradiction within the system, therefore I cannot conclude that the existence claims are valid. That would require that existing things could violate the law of noncontradiction. I say that there is no coherent way to make whatever is said to be infinite also countable, regardless of how you describe "existence" (except by violation of noncontradiction).

The problem is that "infinite" puts severe restrictions on what type of "existence" the supposed infinite thing can have. Some definitions of "existence" might not even allow for anything infinite. Therefore, if "exist" is to be a relevant term it needs to be defined in a way which would allow for an existing infinity. The specified "infinite" parameter must be limitless, boundless. This makes that parameter impossible to measure. Therefore if the infinite parameter is quantity, it is impossible to count.

So it's not a question of whether or not the said bijection "exists", it's a matter of whether it is possible for it to exist. I argue that the meaning of the terms, "infinite" and "bijection" render it impossible for that specified bijection to exist, no matter how you define "exist", unless you allow contradiction.

There is a connection to theology, which might explain why those approaches survive, though I confess it would not recommend them to me. — Ludwig V

The connection to theology which I get a glimpse of, is the ontological argument, which became unacceptable even in theology. The principal problem which I see with that argument is that it attempted to but a boundary, a limit to the limitless. The "infinite" God was described as "That than which nothing greater can be conceived". Notice how it is a deceptive way of describing what is supposed to be limitless, infinite, (God), as an actual limit, the limit to thought. (The greatest possible conception is a limit to thought.) So it's a sort of self-contradicting premise, because we can conceive of the limitless, "infinity", but then the premise presents this as a limit to thought, 'the greatest possible' conception.

So set theory clearly demonstrates how the ontological argument is faulty. If we represent the limitless, i.e. "infinite" as a limit to our thinking (as the ontological argument does), then our thinking will just think up a way to surpass that boundary. That is because the thinking mind will not allow itself to be limited by such self-deception. So when "infinite" is proposed as a limit, as is the case in calculus, then the mind which refuses such a constraint, will just say 'that's a countable infinity', and surpass it with a new, 'uncountable infinity'. Therefore, when the limitless is proposed, and accepted, as a limit to the thinking mind, the true reality of the thinking mind is that it will not allow itself to be limited in that way, so it simply surpasses that limit, and creates a new limitlessness for itself.

How I see this as a problem is that this method is complex and convoluted, so it needs to be simplified. And, the complexity obscures inherent contradiction, which I've argued for. This is because the way of set theory doesn't actually address the problem, it builds on top, working to cover up. The true problem is with the principles of calculus, where the limitless (infinite) is proposed as a limit. That's where the base of the contradiction lies. Then, instead of addressing this as fundamentally flawed, the mathematicians simply forge ahead, thinking that if the limitless has become a limit to us, we're just going to create a new limitlessness. That's the way the human mind works, if you constrain its freedom it will find itself a way around that constraint.

We're still not on the same page, because I claim contradiction within the system, therefore I cannot conclude that the existence claims are valid. — Metaphysician Undercover

I understand. I do not see a formal contradiction. It sounds like you do. I think this is where we must part ways.

I do not see a formal contradiction. — Esse Quam Videri

I wouldn't call the contradiction "formal", because it's not explicit as a statement of X and not-X. However, i see it as implicit, and unavoidable from the meaning given to those terms within the system. "Countable" implies capable of being put into a bijection, and "infinite" implies not capable of producing a bijection. The closest we can get to resolving this is with platonism, by saying that the bijection simply exists. But then "countable" would need to be replaced with "counted", and that implies that the limitless (infinite) is not limitless. So in my mind, the contradiction is unavoidable.

But there is a case for saying that, in this instance, "straightforward truth" just isn't available. — Ludwig V

I see what you are saying, but I would gently push back here. In my last post I distinguished between formal, empirical and metaphysical truth. I see the claim "there exists a bijection..." as being straight-forwardly true in a formal sense—given the axioms and the inference rules, it follows as a matter of course. As far as I am able to tell, this is precisely what is being denied by some others on the thread.

If we were talking at the level of metaphysical or empirical truth I would agree with you. But at the level of formal truth, either ZFC ⊢ ∃f (bijection) is true, or it isn't. I have a hard time making sense of the claim that it isn't.

However, i see it as implicit, and unavoidable from the meaning given to those terms within the system. — Metaphysician Undercover

I realize that you see the contradiction as implicit and unavoidable. But you are not recognizing the meaning given to the terms within the system.
"countable" within the system means only that some of them can be counted and we cannot find any numbers in the sequence that cannot be counted. Actually, since we had that discussion, I've come across the term "countably infinite" which I think is much less misleading.
And I think that you are not aware of how the term "limit" is used within the system. A limit, in this context, is a value that the series gets closer to, but never reaches. It is not a value derived from the function. It is not the last term in the series.
It does not constrain the series at all. So, in Zeno's paradox, Achilles gets closer and closer to the tortoise but never reaches it. (Forgive my inexpert account.)
From my perspective, the adjusted meaning of terms within the system is one of the biggest differences between us.

The connection to theology which I get a glimpse of, is the ontological argument, which became unacceptable even in theology. — Metaphysician Undercover

Thanks for that. I hadn't thought about it. But that wasn't what got Cantor into trouble.

Some Christian theologians saw Cantor's work as a challenge to the uniqueness of the absolute infinity in the nature of God. In particular, neo-Thomist thinkers saw the existence of an actual infinity that consisted of something other than God as jeopardizing "God's exclusive claim to supreme infinity". — Wikipedia - Georg Cantor

I think that a dissection of the ontological argument here might be thought off-topic. So I won't go any further into territory that I don't understand anyway.

If we were talking at the level of metaphysical or empirical truth I would agree with you. But at the level of formal truth, either ZFC ⊢ ∃f (bijection) is true, or it isn't. I have a hard time making sense of the claim that it isn't. — Esse Quam Videri

Yes. I was thinking in terms of a truth that could be recognized unconditionally, as it were. But then, "either ZFC ⊢ ∃f (bijection) is true, or it isn't" is just that. So I missed that truth. The difference is, I suppose, is that I doubt if one could defend similar claims beginning "In metaphysics..." or "Empirically..."

If there are an infinite number of natural numbers... — an-salad

The definition of "number" (in bold, in the quoted statement), is relevant.

The number of natural numbers is not a natural number. A natural number is a number that can be counted to. The same thing applies to every instance in your post where you refer to an "infinite number".

The only meaningful measure of infinities are transfinite numbers, and it's based on a mapping relation. The "number" of natural numbers (positive integers) equals the number of integers because a 1:1 mapping can be identified between the sets. Similarly, the (transfinite) number of real numbers between 1 and 2 is the same (transfinite) number of real numbers.

Transfinite numbers should not be confused with quantities, in the everyday sense. When one mistakenly treats them that way, it leads to paradox.

The "number" of natural numbers (positive integers) equals the number of integers because a 1:1 mapping can be identified between the sets. — Relativist

I think it is important to underline that the mapping between the sets is identified between the first few steps in the series (I may have the exact terms wrong). But the written identification runs out at some point and is conventionally followed by .... The conclusion that it applies right through the sequence follows from the fact that there is nothing to stop it continuing on the principle that every step in the sequence is structurally identical. (You would need to give a reason why things should be different at some point.)
I've found that this is not obvious to everyone.

I think it is important to underline that the mapping between the sets is identified between the first few steps in the series... — Ludwig V

So we picture {2, 4, 6...} and carry on in this way.

But we know that there are innumerable ways to continue this sequence.

The mapping is not based on "carry on in this way" but on the function

$f: \mathbb{N} \to \mathbb{E}, \quad f(n) = 2n$

And this is what shows the bijection. At every n∈N, f assigns exactly one even number, and every even number is assigned to exactly one n.

We know exactly how to carry on.

I realize that you see the contradiction as implicit and unavoidable. But you are not recognizing the meaning given to the terms within the system.
"countable" within the system means only that some of them can be counted and we cannot find any numbers in the sequence that cannot be counted. Actually, since we had that discussion, I've come across the term "countably infinite" which I think is much less misleading. — Ludwig V

I don't agree with that interpretation, because along with "countably infinite" there is "uncountably infinite". Furthermore, Banno produced the proof that the set of natural numbers can be in a bijection with itself. And there can be a bijection between the naturals and the even numbers, and so on, so that the cardinality of these types of sets is the same. To assign cardinality is not a case of "some of them can be counted".

And I think that you are not aware of how the term "limit" is used within the system. A limit, in this context, is a value that the series gets closer to, but never reaches. It is not a value derived from the function. It is not the last term in the series.
It does not constrain the series at all. So, in Zeno's paradox, Achilles gets closer and closer to the tortoise but never reaches it. (Forgive my inexpert account.)
From my perspective, the adjusted meaning of terms within the system is one of the biggest differences between us. — Ludwig V

That is exactly the problem. It is treated as, and called "the limit" but it is not the limit. Since it is not the true limit, you don't actually ever get closer to it. There is an infinite number of places between here and the designated "limit", and no matter how many places you proceed through, there is still always an infinite number of places. Therefore you are never actually getting any closer to the supposed limit through that means. It is a faulty means. It's supposed to be the limit, and it's supposed that the series gets closer to the limit, but it does not.

That is why this representation of "limit" is faulty. It is a false representation, because there is always a gap of infinity between where you are and the supposed limit. Therefore it's not the true limit, you're never closer to it, or further from it, by that method. So you don't actually keep getting closer, and it's nothing but an arbitrary, and false designation of "limit".

To say that you actually keep getting closer would be like saying as you count the natural numbers you keep getting closer to the end. Or, as you work out pi to more and more decimal places you keep getting closer to the end. That's false representation. You do not. Likewise, to say that the series gets closer to, but never reaches the limit is equally a false representation, because there is always an infinite gap between the series and the limit. You are not getting any closer. This shows it to be a faulty representation, and a faulty method. Zeno's paradox is unresolved.



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like saying as you count the natural numbers you keep getting closer to the end. Or, as you work out pi to more and more decimal places you keep getting closer to the end. That's false representation. — Metaphysician Undercover

No one ever says either of those things. You're arguing with someone in your head who knows no more about mathematics than you do.

*

Zeno's paradox comes down to this: the rational numbers in their natural order do not form a sequence, unlike the natural numbers.

As it happens, they can made to form a sequence; and as it happens, the real numbers cannot. But I don't think either of those things really matter.

Zeno quite reasonably approaches the problem of moving by attempting to break it into a sequence of "steps" as we call them, for obvious reasons. (A very powerful technique that underlies much of what we do.) But that sequence of actions cannot be mapped onto the rationals in their natural order because that's not a sequence.

Zeno's paradox is a convergent series, dude. It doesn't matter what order you sum it in.