Not signed in

Want to take part in these discussions? Sign in if you have an account, or apply for one below

• Forgot your password?
• Apply for membership

1. • CommentRowNumber1.
   • CommentAuthorlaureltei
   • CommentTimeNov 12th 2020
   • PermaLink
   Author: laureltei
   Format: TextWhat is the status of undefinable numbers in category theory? Does category theory deny them like Skolem did in a sense? Or are we just adhering to Lawveres concept of cohesive types? How can we have numbers that we cannot define? This is paradoxical. I understand diagonal arguments were used to show the uncountability and that this has been generalized by Lawvere, but is it really a matter of infinite algorithm steps needed that make numbers undefinable or are they just plain not able to even be talked about? If they cannot be talked about and most of math is just a tiny sliver of the numbers then what are they? Is Chaitins constant which is frequently given as an example of an undefinable number really even a number? Kleene and Skolem seem to think the Skolem paradox and model theory in general show that there is no absolute notion of counting. How is this categorically understood? Thank you for all pointers ahead of time

   What is the status of undefinable numbers in category theory? Does category theory deny them like Skolem did in a sense? Or are we just adhering to Lawveres concept of cohesive types? How can we have numbers that we cannot define? This is paradoxical. I understand diagonal arguments were used to show the uncountability and that this has been generalized by Lawvere, but is it really a matter of infinite algorithm steps needed that make numbers undefinable or are they just plain not able to even be talked about? If they cannot be talked about and most of math is just a tiny sliver of the numbers then what are they?
   
   Is Chaitins constant which is frequently given as an example of an undefinable number really even a number?
   
   Kleene and Skolem seem to think the Skolem paradox and model theory in general show that there is no absolute notion of counting. How is this categorically understood?
   
   Thank you for all pointers ahead of time
2. • CommentRowNumber2.
   • CommentAuthorDavidRoberts
   • CommentTimeNov 12th 2020
   • PermaLink
   Author: DavidRoberts
   Format: MarkdownItex>Does category theory deny them like Skolem did one of these things is not like the other. Category theory is not a person who can take a philosophical position. A better comparison would be to compare catgory theory with first-order logic, or ZFC, or Peano Arithmetic. Category theory per se is a framework, a language, a foundation, depending on your point of view. Vanilla category theory says nothing about numbers a priori, it's about objects and morphisms, functors, natural transformations and so on. If one gets to the point of defining real numbers in some kind of category-theoretic foundation or framework, then one might as well be working in some other foundation, since all these things are interconvertible. ETCS+R says the same things about real numbers as ZFC. Real numbers in a more general topos behave the same as in a constructive foundation like [IZF](https://en.wikipedia.org/wiki/Constructive_set_theory#Intuitionistic_Zermelo%E2%80%93Fraenkel). >Is Chaitins constant which is frequently given as an example of an undefinable number really even a number? It's perfectly well-defined in the standard definition of real numbers, yes. I don't know about how it works in constructive logic.

   Does category theory deny them like Skolem did

   one of these things is not like the other. Category theory is not a person who can take a philosophical position. A better comparison would be to compare catgory theory with first-order logic, or ZFC, or Peano Arithmetic. Category theory per se is a framework, a language, a foundation, depending on your point of view. Vanilla category theory says nothing about numbers a priori, it’s about objects and morphisms, functors, natural transformations and so on. If one gets to the point of defining real numbers in some kind of category-theoretic foundation or framework, then one might as well be working in some other foundation, since all these things are interconvertible. ETCS+R says the same things about real numbers as ZFC. Real numbers in a more general topos behave the same as in a constructive foundation like IZF.

   Is Chaitins constant which is frequently given as an example of an undefinable number really even a number?

   It’s perfectly well-defined in the standard definition of real numbers, yes. I don’t know about how it works in constructive logic.

3. • CommentRowNumber3.
   • CommentAuthorlaureltei
   • CommentTimeNov 13th 2020
   • PermaLink
   Author: laureltei
   Format: TextI see your point, BUT and im sorry for my ignorance. I have to ask. It seems to me by reading Lawvere's book on ETCS and topos theory that numbers dont really even exist. But, on the flip side, you have someone like Ramanujan who said that every number is unique and has differing properties. Is category theory or like-wise perspective taking the position that what defines each number is just a structure itself? Like Ramanujan's famous number 1729. Is it just a product of a structure which is really not even just about "2-ness" in a platonic sense? Secondly, could you please enlighten me if you would on what undefinable number means in standard set theory? I have been asking this on other forums and consulted other people, and they seem to have understandings that go something like: "an undefinable number is something thatis a consequence of having only a countable number of descriptions, N, to describe an uncountable set, R. therefore, an undefinable number can be such that given a description X, there are infinitely many 'things' in R that satisfy that same condition." Is this accurate? Or are there other ways to understand this? I have a description of finite strings of elements of N, no matter how long, it is still finite, and there are an infinite number of 'things' in R that satisfy it. While '2' in N is only satisfied by one other unique hting in N? Is this a tenable understanding? thank you

   I see your point, BUT and im sorry for my ignorance. I have to ask. It seems to me by reading Lawvere's book on ETCS and topos theory that numbers dont really even exist. But, on the flip side, you have someone like Ramanujan who said that every number is unique and has differing properties. Is category theory or like-wise perspective taking the position that what defines each number is just a structure itself? Like Ramanujan's famous number 1729. Is it just a product of a structure which is really not even just about "2-ness" in a platonic sense?
   
   Secondly, could you please enlighten me if you would on what undefinable number means in standard set theory? I have been asking this on other forums and consulted other people, and they seem to have understandings that go something like:
   
   "an undefinable number is something thatis a consequence of having only a countable number of descriptions, N, to describe an uncountable set, R. therefore, an undefinable number can be such that given a description X, there are infinitely many 'things' in R that satisfy that same condition."
   
   Is this accurate? Or are there other ways to understand this?
   
   I have a description of finite strings of elements of N, no matter how long, it is still finite, and there are an infinite number of 'things' in R that satisfy it. While '2' in N is only satisfied by one other unique hting in N?
   Is this a tenable understanding?
   
   thank you
4. • CommentRowNumber4.
   • CommentAuthorDavidRoberts
   • CommentTimeNov 13th 2020
   • PermaLink
   Author: DavidRoberts
   Format: MarkdownItex>numbers dont really even exist This is beyond the scope of mathematics. There are plenty of mathematicians that are not Platonists, or even anti-Platonists, and there are plenty of mathematicians who are strongly Platonist, even happy to say that arbitrarily large infinite sets "exist". One can be agnostic on this and just ask what theorems follow from what axioms, and this no one disagrees on, once the proof is pinned down sufficiently. >But, on the flip side, you have someone like Ramanujan Ramanujan was talking about integers, IIRC, not real numbers. When you say "numbers" you need to be more specific. >Secondly, could you please enlighten me if you would on what undefinable number means in standard set theory? Set theory is a red herring here. It's about having a formal language and that we only write down finitely many symbols drawn from a countable pool. Even allowing for things like non-halting Turing machines that calculate successive, improving, approximations to a real number, or convergent infinite sums and so on, these are still finite descriptions. "Definability" in set theory also applies to something else that is not about real numbers at all, and involves transfinite recursion.

   numbers dont really even exist

   This is beyond the scope of mathematics. There are plenty of mathematicians that are not Platonists, or even anti-Platonists, and there are plenty of mathematicians who are strongly Platonist, even happy to say that arbitrarily large infinite sets “exist”. One can be agnostic on this and just ask what theorems follow from what axioms, and this no one disagrees on, once the proof is pinned down sufficiently.

   But, on the flip side, you have someone like Ramanujan

   Ramanujan was talking about integers, IIRC, not real numbers. When you say “numbers” you need to be more specific.

   Secondly, could you please enlighten me if you would on what undefinable number means in standard set theory?

   Set theory is a red herring here. It’s about having a formal language and that we only write down finitely many symbols drawn from a countable pool. Even allowing for things like non-halting Turing machines that calculate successive, improving, approximations to a real number, or convergent infinite sums and so on, these are still finite descriptions.

   “Definability” in set theory also applies to something else that is not about real numbers at all, and involves transfinite recursion.