It was usual to argue as follows: From the set A to be well-ordered take by arbitrary choice an element and denote it as a_0, then from the set A \ {a_0} an element a_1, then an element from the set A \ {a_0, a_1} and so on. If the set {a_0, a_1, a_2, …} is not yet the complete set A, we can choose from A \ {a_0, a1, a_2, …} an element a_omega, then an element a_ (omega+1), and so on. This procedure must come to an end, because beyond the set W of ordinal numbers which are mapped on elements of A, there are greater numbers; these obviously cannot be mapped on elements of A.

This naiveté is reported as late as in 1914 by Felix Hausdorff, obviously without reservations because he remarks: “We cannot share most of the doubts which have been raised against this method.” Hausdorff only deplores the undesired impression of a temporal process but confirms that the element a_omega is fully determined in the sense of transfinite induction and claims that every single action of choosing an element as well as their order has to be understood as timeless. “In order to support this timeless approach E. Zermelo has got the lucky idea to choose from the scratch from every non-empty subset A’ of A one of its elements a’ = f(A’), such that we do no longer have to wait until it is the turn of A’ but for every set, whether or not it will come up, an element is available prae limine. The system of successive choices has been replaced by a system of simultaneous choices which in practical thinking of course is as unfeasible.” [F. Hausdorff: “Grundzüge der Mengenlehre”, Veit, Leipzig (1914); reprinted: Chelsea Publishing Company, New York (1965) p. 133f]

Of course the original method as well as the “timeless” one introduced by Zermelo are wrong. Timeless nonsense is nonsense nevertheless. The mistake committed by Zermelo’s in his “proof” is his assumption that there is always “a first” element following a given one. That is only possible in a sequence, i.e., in a countable set. The details can be found here: “An invalid step in Zermelo’s proof of well-ordering” in https://www.hs-augsburg.de/~mueckenh/Transfinity/Transfinity/pdf

Regards, WM

]]>When you have to resort to hyperbole, an intelligent reader knows that it doesn’t actually make any useful contribution to the debate, and can see that the notion that I don’t understand ** any** of what I’m talking about is ridiculous.

You seem to think that formal systems cannot include any assumptions. But of course they can, in fact they must, otherwise there could be no formal axioms. And of course that means that there isn’t any way of proving that the axioms of your formal systems are indubitably “correct”.

You keep banging on about your chosen formal systems as if your chosen formal systems are the only possible “correct” formal systems, saying: “there is no room for opinion”. But you can’t prove that the axioms of your chosen formal systems are the “right” ones – and so there is no reason why anyone should accept your formal axioms as the “right” ones – and hence there is no reason to accept your chosen formal systems as the “right” ones. You can say that I don’t understand anything of what I’m talking about – but the evidence of who understands what they are taking about lie within the words on this page.

You started off by insisting on ZF set theory. Fine. But now you are talking about all sorts of other stuff and simply asserting that it can all be formalised. But doing that isn’t really any different to talking informally, because now no-one can be sure what assumptive axioms and rules of inference you would be using if it was all put in strict formal terms.

Of course you can use your chosen formal systems and obtain the results you desire. And we can accept that you will include whatever assumptions you make as axioms of your formal systems. But that does not mean that we have to accept those axiomatic assumptions as some sort of gospel that may not be challenged. And it does not mean that we have accept those axiomatic assumptions if they lead to a circular argument. Accepting Platonist assumptions that are assumptions of the “existence” of mathematical entities independently of any finite definition is akin to a faith based belief in a deity – as is assuming that all propositions have a definite “true” or “false” value independent of any means of determining such values.

But the above presumes that your claim is correct – that your formal systems can “prove” the “existence” of a number that has no finite representation. But the fact is that all that a formal system can prove is that the ** symbol sequences** that is Chaitin’s definition – i.e., that the

You are asking me to accept that you can have a formal system that can say, in effect, where Omega is Chaitin’s definition in that system:

“Omega is a real number, and there exists no y such that y = Omega, where y is a finite symbol sequence for a real number (in this system).”

But that would be absurd, since Omega ** is** a symbol sequence of that system. The idea that a formal system can say something about the “existence” of a number that “exists” outside of that formal system is absurd. Of course, people can attach interpretations onto the statements of formal systems, but here we are dismissing such interpretations, since this page is about why formality is important in maths.

Finally, you brazenly claim that you have proved that there exists a real number with no finite representation. But you haven’t, and you completely ignored the key point that I have already pointed out ** twice** – which is that the definition of Chaitin’s number does not prove that there does not exist an alternative definition of that number that makes no reference to Turing machines/programs, and hence cannot be used to solve the halting problem, since there is no way of identifying that number as the Chaitin number.

You didn’t even refer to either instance of the two times I made that argument, so the presumption must be that you do not find any flaw in that argument.

]]>You specifically challenged me to, using ZFC set theory, prove that there exists a real number with no finite representation.

I did that.

You’re now quibbling with an objection that amounts to “I don’t know what it means to define something in ZFC set theory”.

I’m going to repeat this one last time.

The whole argument of the post that started this reply thread is: in mathematics, we use formal notation, formal language, and formal reasoning for a reason: because that kind of precision is crucial to doing math.

ZFC is a *foundation* on which we define other theories of math. It’s a key part of all of modern mathematics. It’s not all of math: it’s the *foundation* of all of math.

You challenged me to produce a number with no finite representation using ZFC set theory.

How does a mathematician define something like a number with no specific representation?

They provide a set-theoretic definition that precisely specifies the number they’re talking about.

How do they know that the thing they defined actually exists? Because they define it in ways that are valid within the foundational axioms of the system of mathematics.

How did I define Chaitin’s Ω?

I started by talking about a recursive (or effective) computing system. That’s a construct well-defined using the foundation of set theory.

Then I defined a set of values: . Halting is a concept defined in the theory of computation. It’s a *predicate*. Using simple ZFC, I can define a set using a predicate, according to the axiom of specification. That set – is absolutely a valid, well-defined set under ZFC set theory. And an expression that sums over the elements of a set of integers? Absolutely valid under ZFC set theory.

But you’re playing a game by trying to add new rules – new rules which you neither stated nor defined. Under ZFC set theory, Chaitin’s number exists. Mathematically, Chaitin’s number exists.

Your argument against it is:

But there can be an assumptive notion that the Chaitin number “exists” if one takes together both that definition and the assumption that for any Turing machine/program, there “exists” a Platonist “true” or “false” value for whether it halts, and which is completely independent of whether there exists any finite method that can determine if that Turing machine/program halts. That is very different to asserting that the Chaitin definition of itself is sufficient to determine a specific number – the definition itself without that assumption does not give a determination of any specific number.

But the whole point of formal mathematics is that it defines exactly what you can and can’t do: whot, under a given set of definitions, exists; what, under a given set of definitions is provable.

We’re talking in terms of the mathematics based on ZFC set theory. What defines whether or not something exists is the axioms of ZFC, and the constructs that we build using those axioms. The whole point of that formalism is to address exactly this argument: there’s no room for opinion here. The mathematical foundations and the mathematical reasoning process tell us exactly what we need to know, without a scrap of ambiguity. If we’re talking about mathematics? Chaitin’s number exists.

And one last quibble:

It’s nice to see that at last you admit that, actually, it’s better to use all sorts of other stuff besides ZF set theory to do all sorts of math.

I don’t admit anything remotely like “It’s better to use all sorts of other stuff besides ZF set theory”. What I said in that previous comment, which you’re trying to twist here, is that ZFC is the toolbox that we use to built all of the theories of mathematics that we use every day. ZFC isn’t all of mathematics: it’s the foundation that pretty much all of modern mathematics is built on. When I’m doing number theory, I haven’t stopped doing ZFC: I’m working in a theory who’s model and underpinnings are defined with ZFC. ZFC is still a fundamental part of it.

When I’m doing the theory of computation, whether I’m doing it via the mechanistic version, or via recursion function theory, I haven’t stopped doing ZFC set theory. I’m still working in a theory that’s built using ZFC as a foundation, and I’m still using ZFC every step of the way.

I’m not doing something *besides* ZFC. I’m doing something *built with* ZFC.

I never used the term “infinite representation”. That is why I did not define it, and why I ignored your definitions of it. The term I actually used was “finite representation”, whose meaning is perfectly clear to you, as used in your blogs

http://www.goodmath.org/blog/2009/05/15/you-cant-write-that-number-in-fact-you-cant-write-most-numbers/ and

http://www.goodmath.org/blog/2014/05/26/you-cant-even-describe-most-numbers/

You say:

“Chaitin’s Ω can absolutely be determined from that definition.”

No, it can’t. But there can be an assumptive notion that the Chaitin number “exists” if one takes *together* both that definition and the assumption that for any Turing machine/program, there “exists” a Platonist “true” or “false” value for whether it halts, and which is completely independent of whether there exists any finite method that can determine if that Turing machine/program halts. That is very different to asserting that the Chaitin definition of itself is sufficient to determine a specific number – the definition itself without that assumption does not give a determination of any specific number.

Let’s pretend for a moment that your argument that Chaitin’s number does actually provide an example of a real number that has no finite representation. Then your “proof” that there “exists” a real number for which there is no finite definition that can determine its value relies on the assumption that there “exists” “true” and “false” values for which there are no finite definitions that can determine such values. Don’t complain if anyone rejects that argument on the grounds that it is absurdly circular.

But, anyway, I don’t need to point out the circularity of the argument. I only need to reiterate the point of my previous comment that you simply ignored. Just because you have an assumption together with Chaitin’s definition that gives you an assumed number R that you cannot determine by any finite method, that does not mean that there is no alternative finite definition B that gives that assumed number R, where the definition B makes no mention of Turing machines or computer programs. Knowing the definition B does not enable you to solve the halting problem, since there is no method of determining if any given definition (including B) is a definition that gives the assumed number R. Hence there is no proof that R does not have a finite representation.

It’s nice to see that at last you admit that, actually, it’s better to use all sorts of other stuff besides ZF set theory to do all sorts of math.

]]>Chaitin’s Ω can *absolutely* be determined from that definition. Once again, you’re not understanding a mathematical definition. Chaitin’s Ω *defines* a halting probability; it isn’t *defined by* a halting probability. It’s defined by a simple fact: given a computing system and a program , does halt?

Of course, you’ve got a convenient way to wiggle out of that. I give you a number which meets your requirements. But you sneer and say it doesn’t count, because I’m only allowed to use pure ZF, which consists of nothing but sets. But that kind of weaseling gives you a universal out. *Real numbers* aren’t part of pure ZF – they’re part of a construction on top of it. The concept of a decimal expansion isn’t part of pure ZF. Anything I can produce will, necessarily, depend on constructions built on top of ZF.

I can’t talk about real numbers in pure ZFC set theory, because real numbers aren’t a concept there. Real numbers are part of a mathematical theory constructed using ZFC set theory. I can talk about them using set theory, because the theory of real numbers is built using ZFC as a foundation. The real numbers are a set – and I can prove that using ZFC, but not *only* ZFC.

To meaningfully define the real numbers, I need to start by defining the natural numbers. I can do that by defining their mathematical properties using the axioms of Peano arithmetic written in first order predicate logic, and then I can show that there is a model for that theory in sets using ZFC. Once I’ve done that, I can use all of the axioms of ZFC to construct sets of natural numbers and do proofs over those sets using the axioms of infinity, comprehension, and induction. That gives me a valid ZFC-backed theory of the natural numbers. Once I’ve done that, I can construct a theory of integers using a set of axioms, and show that there’s a model for that theory using ZFC and sets of natural numbers. Once I’ve done that, I can built sets and proofs about those sets using the axioms of ZFC like comprehension, infinity, and induction. Then I’ve got a valid ZFC-based theory of the integers. I can then define the rational numbers using a series of axioms, and show that there’s a model for those using ZFC and sets of integers. That gives me a valid ZFC-based theory of the rational numbers. Then I can define a set of axioms defining the real numbers, and show that there’s a valid model for those using ZFC and sets of rational numbers. That gives me the theory of the real numbers.

I need all of that construction before I can really talk about the set of the real numbers. That doesn’t get to the necessary background that’s needed to talk about decimal representations and digit expansions.

The thing is, this is what math is about. It’s about building stacks of intricate and carefully defined structures. You start from a very simple base, and build the world on top of it.

ZFC isn’t a great theory because it includes everything. It’s a great theory because it provides a platform and a set of building blocks that you can use to construct almost anything.

As I said in the previous post: I can define recursive function theory, but it’s not something that can be defined in a blog post. It’s something that requires a textbook. Likewise for just about any interesting thing. I can’t define a transcendental number in pure set theory. I can’t define π in pure set theory, because *numbers* don’t exist in pure set theory. But I can build numbers using set theory.

I can define Chaitin’s numbers, because I have set theory as a foundation. I can use set theory to define natural number theory, and then natural number theory to define recursion function theory. First order predicate logic defines the rules, and set theory provides the model. That gives me the objects I need to talk about computation. Using set theory, I can define objects and sets using ZFC axioms. Look at the definition of Ω: it’s a valid set theoretic construction, built on the axioms of comprehension, infinity, and induction, operating over the objects of recursive function theory.

]]>Need I elaborate on this basic framework, and to what end?

]]>Two of Mark’s definitions:

http://www.goodmath.org/blog/2009/05/15/you-cant-write-that-number-in-fact-you-cant-write-most-numbers/

http://www.goodmath.org/blog/2014/05/26/you-cant-even-describe-most-numbers/

Mine: A real number having a finite representation;

There is a definition in a given formal system that can be written down with a finite number of symbols, and which precisely defines the entire expansion of that number (to a given base).

A real number not having a finite representation;

There is no definition in any formal system that can be written down with a finite number of symbols, and which precisely defines the entire expansion of that number (to a given base).

Why is Chaitin’s number irrelevant? Several reasons, here’s two.

1) because for the umpteenth +1 time, I asked for a proof there exists real numbers with no finite representation within ZF set theory. You simply assert that Chaitin’s number can be defined within ZF and expect everyone to simply take your word for it. I don’t. The only entities in ZF are sets; Chaitin’s number is defined in terms of Turing machines that stop or don’t stop. But you provide no reason why I should believe that ZF can refer totally WITHIN itself to Turing machines stopping or not stopping.

2) According to you, Chaitin’s number “exists” without any finite representation, but the only evidence you provide for that is Chaitin’s definition, which is such that the number cannot be completely derived from that definition, since it is defined in terms of halting probability. From that you conclude there cannot be any finite representation, since that would solve the halting problem. But that does not mean that there cannot be an alternative finite definition of the number, and which makes no reference to halting probability. Since you haven’t yet proven that there is any real number that has no finite representation, then you cannot assume that there is no such finite definition. So one such finite definition can be the definition of the same decimal expansion as Chaitin’s number, but you have no way of proving that it is Chaitin’s number, since you can’t deduce the digits of

Chaitin’s number indefinitely, but you could for the alternative definition – and so you still can’t solve the halting problem. So you still haven’t proved that there exists a real number that has no finite representation.