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It seems for me that set theory has an absolute view of information. I will try to express this idea. This sentence are contradictory for me if we take into account the information that you have and you can have from an object from the context of observation of that object:

1) There exist infinite undefinable real numbers.
2) A set is made of distinct objects (the set of reals are all distinct).
3) I have choice function in the set of reals.

My point of view is the following:

It is possible to conceive a set of elements that are distinct from each other but that you are not able to distinguish from your context of observation. That would be the interpretation of an undefinable number in a set theory that is sensitive to the limits of information.

But if I have a magic choice function that (even if it is a black box that I'm not able to know how to build it myself) can select any element from the set of reals then I'm able to distinguish myself (from my context of observation) any number.

So there would not be undefinable objects for me since I could differentiate this object from all others object of the set that I know and in principle with this magic choice function I could select any object so I could know any element of the real set.

I know that probably this is not a typical question since I'm not asking for a solution. But I would be glad to know the point of view of an expert in set-theory about this thoughts.

Some definitions that are required to understand my point of view:

definable number: A number that can be expressed uniquely with respect to other numbers. Here I'm not even concerned about the issue with finite or infinite description. In whatever framework that you use the number needs a unique and distinct definition from all others.

undefinable number: A number that can not be defined in a distinct manner (uniquely) from other. Normally in the framework of set-theory it is common to use the "exhausted number of formulas" argument to say that there are undefinable reals since the real set is uncountable and the number of FOL formula is countable.

In this context, the "meta-property" that a number must have to be defined is to be distinguishable in some way from the other defined numbers with whatever properties you can define in your framework (here set-theory and FOL).

J.F.
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    There is a problem: what do you mean by "definable number" or "undefinable number"? I say, mathematically speaking, what's your definition? – Crostul Mar 09 '15 at 12:43
  • @Crostul thanks, I updated my question with some definitions – J.F. Mar 09 '15 at 13:01
  • You may wish to ask this in the Philosophy Stack Exchange. – Zubin Mukerjee Mar 09 '15 at 13:04
  • I'm glad you've added some definitions, but you've missed the key one: What do you mean by "a choice function on the set of reals"? What are the domain and codomain of this function, and what properties is it supposed to have? (This is not just a request for greater precision; I honestly have not got the foggiest idea what you've got in mind.) – WillO Mar 09 '15 at 14:06
  • @WillO I suppose (correct me if I'm wrong) that the axiom of choice allow me to (even in principle in a non constructive manner) select in a unique way any object of the real set. – J.F. Mar 09 '15 at 15:34
  • @B.E. Rather than "suppose", you might want to look up the actual content of the axiom of choice. What you've supposed looks pretty much meaningless to me, but even if there's a meaning I'm not grasping, it appears to have essentially nothing in common with the axiom of choice. – WillO Mar 09 '15 at 17:46
  • @WillO From wikipedia: For any set A, the power set of A (with the empty set removed) has a choice function Authors who use this formulation often speak of the choice function on A, but be advised that this is a slightly different notion of choice function. Its domain is the powerset of A.. With this alternate notion of choice function, the axiom of choice can be compactly stated as Every set has a choice function reference: "Suppes, Patrick (1972) [1960]. Axiomatic set theory" – J.F. Mar 09 '15 at 18:17
  • B.E. Yes, the Wikipedia definition is accurate, and therefore axiom of choice implies in particular that there is a choice function for the real numbers. This, of course, has absolutely nothing to do with "selecting any element from the set of reals", whatever that might mean. – WillO Mar 09 '15 at 18:23
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    @B.E. In other words, I still don't know what you mean by "choice function". I know what it means in ordinary mathematical discourse, e.g. as it's used in the Wikipedia definition, but you appear to be using it to mean something entirely different, and I can't figure out what. – WillO Mar 09 '15 at 19:16
  • @WillO I'm thinking of an uncountable choice. – J.F. Mar 09 '15 at 19:33
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    @B.E. : That's every bit as mysterious as everything else you've written. – WillO Mar 09 '15 at 20:03

1 Answers1

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A number of observations:

First, your assumption that there are undefinable real numbers is not necessarily true. There are models of set theory in which every object is uniquely definable (in the sense that there is a finite formula that is true for that set and false for everything else), and for all we know we could be living in such a universe. See Is it possible that every set can be specified? and references therein.

(Note that the countability argument you sketch doesn't prevent this, because the relation between formulas and the objects they define doesn't need to exist as a set inside the model).

Secondly, you can't have two different real numbers that you're not able to distinguish. For every two distinct real numbers there is a rational (and all rationals ought to be definable in the "intended model", though there are non-standard models where they aren't) which is greater than one and smaller than the other. If there are undefinable reals, then what makes them undefinable is that no matter how well you try to narrow them down there will be infinitely many of them that satisfy the property you have so far. (In particular any finite set of undefinable reals in the "intended model" must itself be undefinable).

Third, a magic choice function will not necessarily allow you to select any element -- you have to put a set of reals into the choice function in order to get anything out of it. It may well be that it's only some of the reals that are in the image of a definable set under the choice function.

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hmakholm left over Monica
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  • "Note that the countability argument you sketch doesn't prevent this": My point of view is that the information that we have of the object is the only thing that can define that object. The countability argument seems to be true for me in this case because if you the relation between the formula and the object is undefinable or undecidable I don't see how you can make this information equivalent to the object. – J.F. Mar 09 '15 at 13:27
  • "If there are undefinable reals...there will be infinitely many of them that satisfy the property you have so far": From the perspective of information that you can have to define that object this is exactly saying that they are indistinguishable from my point of view. – J.F. Mar 09 '15 at 13:28
  • I don't understand the third point since you seem to say that the axiom of choice doesn't allow me to (even in principle in a non constructive manner) select any object of the reals. – J.F. Mar 09 '15 at 13:29
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    @B.E. It appears that your concept of "definable" is not the standard one. With your concept, what makes you conclude that undefinable numbers exist at all? Every real number is uniquely determined by knowing which of the rationals are less than it, for example. – hmakholm left over Monica Mar 09 '15 at 13:57
  • "With your concept, what makes you conclude that undefinable numbers exist at all" That is the heart of the contradiction (that I think exist) with reals (uncountable set of numbers) if we take into account in set-theory the information that we can have from an object and make this information equivalent to this object – J.F. Mar 09 '15 at 14:03