# Continuity and uncountability

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26 minutes ago, uncool said:

The set of real numbers is not countable,

I know that this is the well accepted theory. But it would be great if the contrary could be proven. Yes, that is impossible. But one can always think otherwise.

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But it HAS been proven.  See Skolem's paradox.   Every countable axiomatisation of set theory in first-order logic, if it is consistent, has a model that is countable.

This result should only be surprising to those who literally believe in non-computable reals.

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7 hours ago, TheSim said:

But it HAS been proven.  See Skolem's paradox.   Every countable axiomatisation of set theory in first-order logic, if it is consistent, has a model that is countable.

This result should only be surprising to those who literally believe in non-computable reals.

Completeness is a second order property. You're reading too many Wiki pages and too little actual math. As I'm sure you would understand IF you understood, even in a countable model of the set theory, the reals are uncountable. If you don't understand why that is, you don't understand Skolem's result. Even in a countable model, there is still no bijection between the naturals and the reals.

You haven't troubled yourself to reply to my observation that the computable real line is full of holes and fails to satisfy the Intermediate value theorem, making it a poor representation of anyone's idea of a continuum.

Edited by wtf

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Second order logic has certainly been used to create a sentence that has traditionally been called "completeness of the reals",  but to 'prove' completeness of the reals in second-order logic requires assuming the very existence of uncountable and non-computable sets of real-numbers that are in question.   At best all you have is an empty axiom of syntax called "completeness of the reals" that cannot be meaningfully interpreted.  And if  non-constructable sets are dropped from second-order logic, it collapses into first order logic.

Cantor's so-called  "proof" of the existence of uncountable sets  is merely a conditional statement saying that if the power-set of natural numbers exists then it cannot be enumerated.   But we have no way of constructing this power-set,  since only a countable number of subsets of N are recursively enumerable, that is to say, can be generated by an algorithm.

Sure, we can define the phase "Uncountable sets exist" to refer to Cantor's proof, but this result is merely an empty and circular parlour game of syntactical gibberish without physical or practical significance.   It might be a fun parlour game,  but i don't see how it leads to philosophical clarity.

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15 minutes ago, TheSim said:

Cantor's so-called  "proof" of the existence of uncountable sets  is merely a conditional statement saying that if the power-set of natural numbers exists then it cannot be enumerated.   But we have no way of constructing this power-set,  since only a countable number of subsets of N are recursively enumerable, that is to say, can be generated by an algorithm.

Sure, we can define the phase "Uncountable sets exist" to refer to Cantor's proof, but this result is merely an empty and circular parlour game of syntactical gibberish without physical or practical significance.   It might be a fun parlour game,  but i don't see how it leads to philosophical clarity.

You're wrong about this. Cantor's theorem is a valid theorem of first-order ZF. It's true in any model of ZF, even a countable model. In set theory we are not required to "construct" anything, as I'm sure you know.

Point being (I do hope you understand this subtle point) that even in a countable universe, there is no bijection between a set and its powerset. And the simple and beautiful proof can be appreciated by a high school student.

But if you are using Skolem's theorem to resolve pengkuan's issues, this is surely far off the mark. Perhaps a separate thread on Skolem's paradox would be interesting, but in the context of the present thread I can't see how it sheds any light.

Edited by wtf

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Sure there is no bijection between a set and it's powerset in the case of finite sets since this result can be directly constructed.

But Cantor's theorem is a so-called 'proof' by way of contradiction that begs the law of excluded middle without right in order to lead to his desired conclusion.

He  seems to argue that since   N -> P(N)  cannot be a surjective function without leading to a contradiction, that it must therefore be the case  that N->P(N) isn't a surjection, thereby concluding that |P(N)| > |N|.    But  he has no right to do this  because he hasn't shown that P(N) is a set.

Obviously Cantor's theorem is expressible as a theorem of ZF since the rules of classical logic are permitted there.  But as Skolem's Paradox demonstrates this leads to a contradiction regarding its intended interpretation.    This is why it is probably better to abandon ZF for a constructive set theory.

Edited by TheSim

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16 minutes ago, TheSim said:

This is why it is probably better to abandon ZF for a more constructive set theory.

You're entitled to that opinion. You'll have to convince the entire worldwide community of mathematicians, along with all of the physical scientists (biologists, physicists, chemists, ...) whose work is grounded in infinitary math. Biology? Yes. How are you going to replace the importance of differential equations in the life sciences? Are you really going to recast everything in the world that depends on diffEq with the theory of finite differences? A huge intellectual project with ZERO practical payoff. Akin to recasting astronomy by taking the earth to be the center of the universe. It COULD be done with great difficulty, at the cost of making everything incredibly convoluted. But why?

I don't disagree that neo-intuitionism is making a comeback via automated proof checking software, but it's a long way from that to overthrowing LEM in mainstream math and logic.

But as I noted, you are raising interesting points that would be sensible in their own thread. Denial of LEM, advocacy of constructive math, denial of noncomputable reals (what do you make of Chaitin's constant then? Even in computability theory they prove the existence of noncomputable problems), etc., are all interesting. But you are hijacking this thread to grind your constructivist axe. Nothing you've said bears on the thread topic. Start a new thread titled, "What do you think about constructive math?" or "Down with ZF and the evil Cantorians," or "The hell with Mrs. Zermelo and her pro-Choice views." If you did that we could have an interesting discussion. But in this thread? Just a thread jack by someone pushing an agenda.

Edited by wtf

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2 hours ago, TheSim said:

he hasn't shown that P(N) is a set.

You never have to show this. It is an axiom that if X is a any set, then P(X) is a set.

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3 hours ago, TheSim said:

Sure there is no bijection between a set and it's powerset in the case of finite sets since this result can be directly constructed.

But Cantor's theorem is a so-called 'proof' by way of contradiction that begs the law of excluded middle without right in order to lead to his desired conclusion.

He  seems to argue that since   N -> P(N)  cannot be a surjective function without leading to a contradiction, that it must therefore be the case  that N->P(N) isn't a surjection, thereby concluding that |P(N)| > |N|.    But  he has no right to do this  because he hasn't shown that P(N) is a set.

Obviously Cantor's theorem is expressible as a theorem of ZF since the rules of classical logic are permitted there.  But as Skolem's Paradox demonstrates this leads to a contradiction regarding its intended interpretation.    This is why it is probably better to abandon ZF for a constructive set theory.

I agree with what you say. Is it a school of thought in mathematics? Are there many people who think like you ?

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15 minutes ago, pengkuan said:

Is it a school of thought in mathematics?

If you want to study this stuff, read this preface to

Computability and logic by

Boolos and Jeffrey

It will give you a roadmap or guide of what to look for.

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1 hour ago, studiot said:

Computability and logic by

Boolos and Jeffrey

Boolosian logic?

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8 minutes ago, Strange said:

Boolosian logic?

Sorry this one does not come out so well due to teh coloured background.

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2 hours ago, pengkuan said:

I agree with what you say. Is it a school of thought in mathematics? Are there many people who think like you ?

1) There is a school/philosophy of mathematics that TheSim is referencing, called constructivism. It is a relatively rare view.

2) To be honest, you don't have the experience, knowledge base, or understanding to deal with the differences between constructivism and the standard view. I would highly recommend that you study the basics for quite a while longer.

Edited by uncool

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On 2018/11/30 at 10:55 PM, studiot said:

If you want to study this stuff, read this preface to

Computability and logic by

Boolos and Jeffrey

It will give you a roadmap or guide of what to look for.

Thanks

23 hours ago, uncool said:

1) There is a school/philosophy of mathematics that TheSim is referencing, called constructivism. It is a relatively rare view.

Thanks

23 hours ago, uncool said:

2) To be honest, you don't have the experience, knowledge base, or understanding to deal with the differences between constructivism and the standard view. I would highly recommend that you study the basics for quite a while longer.

I just want to know the name of this philosophy.

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