This thread is a spawn of the "Riemann Hypothesis" thread, which led to a discussion on model theory. Before we start getting onto it, I propose that we get our terminology clear:

"True" and "False" should be used semantically, ie. referring to whether a formula is semantically true or false in a specific model of a theory, whereas "provable" and "refutable" should be used syntactically, ie. referring to whether a specific formula can be deduced in some theory.

Also "theory" should be defined: I propse the definition that a (deductive) theory is a set of sentences in a formal language which is closed under logical consequence.

Furthermore I propose to distinguish between a model and an interpretation of a theory - an Interpretation shall be called model iff. any sentence of the theory is semantically true in that interpretation.

For starters, because this is spawned from the RH thread, and in order to ensure a passionate discussion, I will throw this question out:

Are hypercomplex numbers a model of complex analysis? What models of complex analysis are there and how many of them?

Cheers for starting this thread. I like the definitions you have provided. One subtlety to mention is that we are restricting ourselves to first-order logic, which is okay, because first-order set theory is assumed sufficient for all of maths. But if we go higher-order, we'll lose the completeness theorem.

Another issue is deciding what theories we are talking about. I don't know of any theory of real analysis or complex analysis. There is, to be sure, a (decidable) theory for real numbers, but it cannot be used to do any analysis since it cannot say anything about whether the field is Archimedean. You need natural numbers for that, and so we are talking about a theory at least as expressive as Peano Arithmetic. Considering all the other classical mathematics you'd need to have a robust theory of complex analysis, ashley is probably right to focus on something as expressive as set-theory. But then we're talking about complex numbers which are defined not by axiom systems, but by a construction within a more powerful theory.

So I'd like to formulate the question as: can you recover the hypercomplex numbers by considering a model of set theory in which ω is not isomorphic to the set of natural numbers?

Which brings up another subtlety which I'd like to have cleared up before we really start getting into this: Do we, by ω mean the set that is postulated to exist by the axiom of infinity (as in ZF or NBG) or do we mean the minimal set that is closed under the successor operation, ie. the intersection of all sets that are closed under the successor operation?

cateye wrote:Which brings up another subtlety which I'd like to have cleared up before we really start getting into this: Do we, by ω mean the set that is postulated to exist by the axiom of infinity (as in ZF or NBG) or do we mean the minimal set that is closed under the successor operation, ie. the intersection of all sets that are closed under the successor operation?

Definitely the latter. ω specifically refers to the first infinite ordinal, and ω + 1 to the second, ω + 2 to the third and so on. The first infinite ordinal is the natural numbers.

In that case I will add this question aswell: can you recover the complex numbers by considering a model of set theory in which ω is not isomorphic to the set of natural numbers?

Here I am.

cateye wrote:This thread is a spawn of the "Riemann Hypothesis" thread, which led to a discussion on model theory. Before we start getting onto it, I propose that we get our terminology clear:

"True" and "False" should be used semantically, ie. referring to whether a formula is semantically true or false in a specific model of a theory, whereas "provable" and "refutable" should be used syntactically, ie. referring to whether a specific formula can be deduced in some theory.

Also "theory" should be defined: I propse the definition that a (deductive) theory is a set of sentences in a formal language which is closed under logical consequence.

Furthermore I propose to distinguish between a model and an interpretation of a theory - an Interpretation shall be called model iff. any sentence of the theory is semantically true in that interpretation.

Right with you. I am used to using the word "structure" for a set with an interpretation.

A sentence is "undecidable in theory T" when neither it nor its negation are in T.

A theory is consistent if it doesn't contain both a sentence and its negation.

For starters, because this is spawned from the RH thread, and in order to ensure a passionate discussion, I will throw this question out:

Are hypercomplex numbers a model of complex analysis? What models of complex analysis are there and how many of them?

Problem is, I'm not sure what a first-order theory of complex analysis would look like.

It's very important that all the theories we look at be first-order, which means that variables only range over objects in the model. So you are not allowed to say "for all functions C->C...", or "for all sets of complex numbers...". Most of model theory does not apply to languages powerful enough to say things like that. In particular, the completeness theorem that was relied on in the other thread, which says that they is a model for any consistent theory, definitely will not be true.

In set theory one tries to get round this by making sets themselves the objects. That is you have a first-order theory involving a binary relation that mimics set membership. But they cannot really get around the limitation because they cannot guarantee that every set of things in the model actually has an object whose "members" are precisely the members of that set. In fact with Russell's paradox you prove that there is actually a definable set of objects in the model that can't have an object corresponding to it. So you have to be careful. If you have some sentence of ZF "for all sets, blah", the fact that that sentence is true in some model doesn't mean "blah" will be true of all sets of things in the model. It only has to be true of all sets that have objects corresponding to them. I hope I'm being understandable.

So back to complex analysis. It is not obvious that you can have a first-order theory just of complex analysis. For example the definition of integrals over a curve is usually something like "the number such that for all epsilon, then for any sufficiently fine finite partition of the curve, such-and-such a sum is always within epsilon of that number". I doubt it is possible to make a first-order language that is not a set theory and can talk about "any partition".

EDIT: Damn it, now VazScep has swooped in and already said half of this.

cateye wrote:Are hypercomplex numbers a model of complex analysis?

What do you mean by "complex analysis"? Complex analysis is first of all a branch of mathematics (normally performed within the framework of some theory like ZFC), not a formal theory.

Since complex analysis includes complex numbers, functions and expressions like "every derivative (of some function)", if you want to come up with some theory tailored specifically to complex analysis, you'd presumably have to use many-sorted logic (i.e. some form of type theory).

What models of complex analysis are there and how many of them?

There are infinitely many models of "complex analysis" (assuming by "complex analysis" you mean some first-order theory with an infinite model), at least one for each infinite cardinality.

VazScep wrote:So I'd like to formulate the question as: can you recover the hypercomplex numbers by considering a model of set theory in which ω is not isomorphic to the set of natural numbers?

Meaning? In set theory, ω is defined as the set of natural numbers (finite ordinals).

Preno wrote:

cateye wrote:Are hypercomplex numbers a model of complex analysis?

What do you mean by "complex analysis"? Complex analysis is first of all a branch of mathematics (normally performed within the framework of some theory like ZFC), not a formal theory.

Since complex analysis includes complex numbers, functions and expressions like "every derivative (of some function)", if you want to come up with some theory tailored specifically to complex analysis, you'd presumably have to use many-sorted logic (i.e. some form of type theory).

What models of complex analysis are there and how many of them?

There are infinitely many models of "complex analysis" (assuming by "complex analysis" you mean some first-order theory with an infinite model), at least one for each infinite cardinality.

VazScep wrote:So I'd like to formulate the question as: can you recover the hypercomplex numbers by considering a model of set theory in which ω is not isomorphic to the set of natural numbers?

Meaning? In set theory, ω is defined as the set of natural numbers (finite ordinals).

Yeah, I think it's confusing to just jump into this thread out of context. You should have a quick glance at the Riemann Hypothesis thread to get an idea of what we're up to. Just saying..

Welcome to the forum, btw!

Preno wrote:Since complex analysis includes complex numbers, functions and expressions like "every derivative (of some function)", if you want to come up with some theory tailored specifically to complex analysis, you'd presumably have to use many-sorted logic (i.e. some form of type theory).

We're leaving out type theory for now (though you might guess from my avatar that I am very much fond of it!). We want the completeness theorem. But that's okay. We have first-order set theory in which functions and quantification over functions and properties is just quantification over sets.

VazScep wrote:So I'd like to formulate the question as: can you recover the hypercomplex numbers by considering a model of set theory in which ω is not isomorphic to the set of natural numbers?

Meaning? In set theory, ω is defined as the set of natural numbers (finite ordinals).

Right. But if you embed PA in ZFC using the finite ordinals and then construct the Godel sentence of ZFC, you're forced to conclude that there are non-standard models. I'd need to think about this a little more carefully to be absolutely sure, but in these non-standard models, the interpretation of "ω", by which I mean the appropriate constant symbol definable in ZFC, would not be isomorphic to ω, by which I mean the set of natural numbers.

cateye wrote:Yeah, I think it's confusing to just jump into this thread out of context. You should have a quick glance at the Riemann Hypothesis thread to get an idea of what we're up to. Just saying..

Okay, I just the last couple of pages, but tbh I still don't understand what complex analysis (as a formal theory) is supposed to be, or what the distinction between ω and the set of natural numbers is supposed to be.

Welcome to the forum, btw!

Thanks.

Preno wrote:

cateye wrote:Yeah, I think it's confusing to just jump into this thread out of context. You should have a quick glance at the Riemann Hypothesis thread to get an idea of what we're up to. Just saying..

Okay, I just the last couple of pages, but tbh I still don't understand what complex analysis (as a formal theory) is supposed to be, or what the distinction between ω and the set of natural numbers is supposed to be.

Welcome to the forum, btw!

Thanks.

The (more general) question we're trying to figure out, is if we can assume to have the usual complex numbers (and everything that's proven about them) in any model of set theory or not, or if in some models we necessarily end up having hypercomplex numbers. Ashley brought up the idea that in nonstandard models constructing the rational numbers from ω in the usual way (equivalence classes of pairs, etc. ) will lead to hyperrational and therefore eventually to hypercomplex numbers, which would have implications on the decidability of the RH. At least thats my (limited) understanding.

Preno wrote:I still don't understand ... what the distinction between ω and the set of natural numbers is supposed to be.

The natural numbers are the real natural numbers that Platonists would say really exist. ω is a thing in set theory that tries to formalise them, but since set theories are first-order theories they can only guarantee that ω satisfies all the right first-order formulae of the language of arithmetic, which is not the same as being isomorphic to the naturals. We are contemplating models of set theory in which ω is not actually the natural numbers.

cateye wrote:The (more general) question we're trying to figure out, is if we can assume to have the usual complex numbers (and everything that's proven about them) in any model of set theory or not, or if in some models we necessarily end up having hypercomplex numbers.

Didn't you mention Skolem's paradox in the other thread? There the whole model is countable so clearly its C can't be the usual C. I can sketch a proof that there are models with complex numbers whose absolute value is bigger than any of the usual natural numbers if you like.


I've thought of a good example of a bad argument analogous to the one that was being made in the other thread and has a more obviously wrong conclusion.

Take a Turing machine and an input. Formalise the notion of Turing machine in ZF. Either we can prove in ZF that the machine halts or we can't. If we can't, then there must be a model of ZF where it does halt. But if it halts we can show it halts by running it. So it must be decidable. So we can always decide whether a Turing machine will halt.

And this fails for quite a similar reason. When we formalised Turing machines we will have done something along the lines of constructing for each machine T and input i a function fT,i : ω -> ω that we can prove is always primitive recursive and has 0 in its image iff the machine halts on the input. It would have to be something like that. Then we took a model in which 0 is in the image of fT,i. But this model must have had non-standard naturals in ω, so that even though 0 is in the image you would never find it. So it wasn't true to say that if it halts according to some model, we can decide whether the actual Turing machine will halt.

Come to think of it, we can probably use the Weierstrass factorization theorem to construct a meromorphic function that has zeros off the line Re(z)=1/2 iff the machine halts.

cateye wrote:The (more general) question we're trying to figure out, is if we can assume to have the usual complex numbers (and everything that's proven about them) in any model of set theory or not, or if in some models we necessarily end up having hypercomplex numbers. Ashley brought up the idea that in nonstandard models constructing the rational numbers from ω in the usual way (equivalence classes of pairs, etc. ) will lead to hyperrational and therefore eventually to hypercomplex numbers, which would have implications on the decidability of the RH. At least thats my (limited) understanding.

Well, that depends on what you mean by "the usual complex numbers". Yes, in any model of ZF(C), one can construct the complex numbers as ordered pairs of reals, reals as the Dedekind completion of the rationals and rationals as ordered pairs of naturals, but the question is how you're equating the complex numbers of different models. Yes, everything that can be proven about the complex numbers in ZF(C) is true in any model of ZF(C). No, the complex numbers in one model don't have to be isomorphic to the complex numbers in another model, even if the models are elementarily equivalent. Yes, if ω (defined as the intersection of all inductive sets, i.e. sets which along with any set x also contain its successor x+{x}) contains non-standard members in a given model of ZF(C) - i.e. members not designated by any term - then naturally, the complex numbers will also contain non-standard members (i.e. members which aren't ordered pairs of the Dedekind completion of ordered pairs of standard naturals). I doubt this has any relevance to the RH, though, as any sum or product which involves a non-standard natural is necessarily non-standard (although I haven't been following that argument in detail).

ashley wrote:The natural numbers are the real natural numbers that Platonists would say really exist. ω is a thing in set theory that tries to formalise them, but since set theories are first-order theories they can only guarantee that ω satisfies all the right first-order formulae of the language of arithmetic, which is not the same as being isomorphic to the naturals. We are contemplating models of set theory in which ω is not actually the natural numbers.

Okay, but then you're not talking about a set, i.e. not using the word "isomorphism" in the set-theoretic sense.

ashley wrote:Didn't you mention Skolem's paradox in the other thread? There the whole model is countable so clearly its C can't be the usual C.

Okay, but in and of itself, that doesn't say anything interesting. If we consider the model where we replace each set x by the singleton {x} and define epsilon accordingly, we get complex numbers which are different from "the usual" (i.e.: absolute) C, but not in any interesting way. A countable model which is elementarily equivalent to the standard one also doesn't differ from "the usual C" in any relevant respect.

ashley wrote:
Didn't you mention Skolem's paradox in the other thread? There the whole model is countable so clearly its C can't be the usual C. I can sketch a proof that there are models with complex numbers whose absolute value is bigger than any of the usual natural numbers if you like.

Yes, I mentioned it. However I did not use it for any argument, it was just to illustrate the subtleties of "seen from inside the theory" and "seen from outside the theory". I'd be very interested in seeing that proof sketched, but I don't want to bother you too much either.

ashley wrote:
I've thought of a good example of a bad argument analogous to the one that was being made in the other thread and has a more obviously wrong conclusion.

Take a Turing machine and an input. Formalise the notion of Turing machine in ZF. Either we can prove in ZF that the machine halts or we can't. If we can't, then there must be a model of ZF where it does halt. But if it halts we can show it halts by running it. So it must be decidable. So we can always decide whether a Turing machine will halt.

And this fails for quite a similar reason. When we formalised Turing machines we will have done something along the lines of constructing for each machine T and input i a function fT,i : ω -> ω that we can prove is always primitive recursive and has 0 in its image iff the machine halts on the input. It would have to be something like that. Then we took a model in which 0 is in the image of fT,i. But this model must have had non-standard naturals in ω, so that even though 0 is in the image you would never find it. So it wasn't true to say that if it halts according to some model, we can decide whether the actual Turing machine will halt.

Come to think of it, we can probably use the Weierstrass factorization theorem to construct a meromorphic function that has zeros off the line Re(z)=1/2 iff the machine halts.

I will have to meditate on this a little bit - be patient with me, please

Preno wrote:

cateye wrote:The (more general) question we're trying to figure out, is if we can assume to have the usual complex numbers (and everything that's proven about them) in any model of set theory or not, or if in some models we necessarily end up having hypercomplex numbers. Ashley brought up the idea that in nonstandard models constructing the rational numbers from ω in the usual way (equivalence classes of pairs, etc. ) will lead to hyperrational and therefore eventually to hypercomplex numbers, which would have implications on the decidability of the RH. At least thats my (limited) understanding.

Well, that depends on what you mean by "the usual complex numbers". Yes, in any model of ZF(C), one can construct the complex numbers as ordered pairs of reals, reals as the Dedekind completion of the rationals and rationals as ordered pairs of naturals, but the question is how you're equating the complex numbers of different models. Yes, everything that can be proven about the complex numbers in ZF(C) is true in any model of ZF(C). No, the complex numbers in one model don't have to be isomorphic to the complex numbers in another model, even if the models are elementarily equivalent. Yes, if ω (defined as the intersection of all inductive sets, i.e. sets which along with any set x also contain its successor x+{x}) contains non-standard members in a given model of ZF(C) - i.e. members not designated by any term - then naturally, the complex numbers will also contain non-standard members (i.e. members which aren't ordered pairs of the Dedekind completion of ordered pairs of standard naturals). I doubt this has any relevance to the RH, though, as any sum or product which involves a non-standard natural is necessarily non-standard (although I haven't been following that argument in detail).

We have established a pretty airtight case in the other thread that if the RH is false in the "standard" model, then it can be proven false in that model (ie. always assuming we're talking about the usual complex numbers). The thing that remains to be cleared up for me, is what follows from this, there were two possibilities discussed in the other thread (which seem mutually exclusive):

a) Since if it is false it can be refuted, if it can be shown to be undecidable it would infact be true. (that's du Sautoys argument)

b) Since if it is false it can be refuted, assuming it is undecidable will lead to a contradiction, since (by a similar argument to a) ) were it undicidable it is true, and we have infact decided it. Hence it must be decidable.

cateye wrote:We have established a pretty airtight case in the other thread that if the RH is false in the "standard" model, then it can be proven false in that model (ie. always assuming we're talking about the usual complex numbers).

If you have, that's no mean feat, since the phrase "proven false in a model" is meaningless. I understand what it means to be true in a model and to be provable in some theory, but I have no idea what it means to be provable in a model.

Preno wrote:

cateye wrote:We have established a pretty airtight case in the other thread that if the RH is false in the "standard" model, then it can be proven false in that model (ie. always assuming we're talking about the usual complex numbers).

If you have, that's pretty curious, since the phrase "proven false in a model" is meaningless.

I was being sloppy in terminology here: if it is false there exists a counterexample, ie. a non-trivial zero outside the Im(z)=1/2 line that we can find in finite steps. I am supposing to have all tools from complex calculus to do so, though. The question is if those tools are available in all models.

cateye wrote:

Preno wrote:

cateye wrote:We have established a pretty airtight case in the other thread that if the RH is false in the "standard" model, then it can be proven false in that model (ie. always assuming we're talking about the usual complex numbers).

If you have, that's pretty curious, since the phrase "proven false in a model" is meaningless.

I was being sloppy in terminology here: if it is false there exists a counterexample, ie. a non-trivial zero outside the Im(z)=1/2 line that we can find in finite steps.

What do you mean by being "found in finite steps"?

I skimmed the thread again, and I can't find the argument to that effect. The book of de Sautoy's which I just - let's use a neutral term - procured also contains merely the bare assertion that "if it is false then there is a zero off the critical line which we can use to prove it is false". Yeah, no. If the Goldbach hypothesis is false, then it's decidable, since we can find a term n referring to the counter-example (and since Peano Arithmetic - even Robinson Arithmetic, in fact - is "delta_0-complete", i.e. can prove every true statement which is simple enough syntactically). But no such thing is true of the real or complex numbers (not to mention that our notion of what it means to be true of the real numbers is much less clear than our notion of what it means to be true of the natural numbers).

Preno wrote:

cateye wrote:

Preno wrote:

cateye wrote:We have established a pretty airtight case in the other thread that if the RH is false in the "standard" model, then it can be proven false in that model (ie. always assuming we're talking about the usual complex numbers).

If you have, that's pretty curious, since the phrase "proven false in a model" is meaningless.

I was being sloppy in terminology here: if it is false there exists a counterexample, ie. a non-trivial zero outside the Im(z)=1/2 line that we can find in finite steps.

What do you mean by being "found in finite steps"?

I skimmed the thread again, and I can't find the argument to that effect. The book of de Sautoy's which I just - let's use a neutral term - procured also contains merely the bare assertion that "if it is false then there is a zero off the critical line which we can use to prove it is false". Yeah, no. If the Goldbach hypothesis is false, then it's decidable, since we can find a term n referring to the counter-example. But no such thing is true of the real or complex numbers (not to mention that our notion of what it means to be true of the real numbers is much less clear than our notion of what it means to be true of the natural numbers).

Would it be asked too much if you really read the thread then (I know, it's a painful one)? I don't want to derail this one too much, because I would merely copy-paste from the other thread. In any case, I'll have to go to bed now, maybe we can continue this discussion tomorrow.