The author's argument relies on "indescribable-ness" somehow blocking the ability to "choose" an element. But that trades on a vague, intuitive understanding of "choosing", which doesn't capture what the AC is about. The AC is about the existence of certain functions. Since the "indescribable-ness" of the domain of a function is not relevant to its existence, the author's argument doesn't successfully attack the AC.
But that is not at issue. What is at issue is whether or not a function exists whose domain is an (infinite) set of indescribable numbers and whose range is a member of the set. The existence of such a function is far less clear.
Of course you can postulate the existence of such a function. But you can also postulate the existence of an oracle for the halting problem. My gut tells me that these two ideas have a lot in common.
> Remember that I am not here trying to argue for the truth of the AC; I'm arguing against the author's argument.
Yep, I get that.
> The author's argument, as I read it, relies on an intuition that "choosing" an element that is not-finitely-describable is problematic.
Not quite. Choosing an element from a set where all of the members are not-finitely-describable is problematic.
> However, the mathematical meaning of "choosing", in the context of the AC, is a statement about the mathematical existence of a certain function.
Yep, I get that.
> To respond to the author's intuition requires only showing that not-finitely-describable domains are not a "problem" for the existence of functions.
No, you would have to show that defining a function on this particular set is not a problem. Note that this is not the same thing as defining a function on elements of this set. That is obviously not a problem. But defining a (non-trivial) function on the set itself is.
It doesn't have anything to do with AoC per se, it has to do (I claim) with people's intuitions about AoC. Specifically, an infinite set of arbitrary subsets of the set of indescribable reals is something for which a reasonable person might be a tad suspicious about postulating the existence of a choice function.
https://mathoverflow.net/questions/44102/is-the-analysis-as-...
The point is there is a lot of nuance and no matter what one believes regarding choice there will be unintuitive results. For instance, when the axiom of choice fails you can have an infinite set with not countably infinite subset. The reals can be a countable union of countable sets. There is a set that can be partitioned into strictly more equivalence classes than the original set has elements, and a function whose domain is strictly smaller than its range. In fact, this is the case in all known models.
A similar result is that there are models of the reals in ZFC that are countable, even though the reals are not countable. How can this be? Well it's similar to the above argument, those models do not represent the actual real numbers.
All this means is that it's not possible to categorically define real numbers in first order ZFC. Any axiomatic definition of the reals in first order ZFC will have some models that satisfy those axioms but are not actually real numbers.
The reason I brought this up is because when you said that there are models of ZFC where every real is definable, those reals are only definable from outside of the model. From within that model there are still undefinable reals and hence I don't think it matters too much to the point lisper is making.
Maybe to put it another way, there is no model of the reals where within that model all reals are definable, and hence lisper's point stands.
I'm not a math expert, but "populating" a set strikes me as an imperative programming mindset. In math, I understand a set to more like a function returning a boolean that indicates membership. If the contents of the set change, then it's just a new set.
Set A is the set of all widgets that have a frozz. The existence of Set A doesn't imply that there exists a widget that has a frozz. Perhaps the existence of such a widget is an open question. But the set is perfectly well defined.
By the way, it is worth keeping in mind how Gödel proved the consistency of the axiom of choice. Roughly speaking, the steps are: start with a model of ZF, build from it an inner model where all sets are definable (in a sense) in terms of ordinals, that model (called the "constructible universe") satisfies the axiom of choice. In other words, the axiom of choice holds as soon as you assume that all sets are constructible.
OK, but how are you going to construct a bijection from an arbitrary infinite set of undefinable numbers onto a set with a choice function without the AoC?
Ultimately, you're saying that if we have an indescribable x, then x+1 is conditionally describable (to abuse terminology). But I think that just sidesteps the question. Even then, if we're taking a computability lens, then indescribable(x) implies indescribable(x+1), which would seem to imply that applying your function to indescribable x's is a moot point.
The fact that your function is finitely describable even though we can't describe each of its applications is admittedly a mindfuck.
edit: Hm... thinking about it. The finitely-many-symbols argument also only holds if you consider finitely many interpretations of those symbols and if there actually are only finitely many symbols.
But it does not mean the set is well defined. The description is not necessarily clear an unambigious. It might not be consistent under the generalizations we come up with.
> "the set of all undescribable primes"
Interesting example, because the set is trivially empty under the assumption that we can describe any finite natural number by writing down its decimal expansion. (Based on that you can either argue that the primes are a subset of the natural numbers or that the primes are countable.)
I see no reason to believe that "undescribable" is a well defined property so far.
Well, obviously such a set must be defined relative to whatever system you're using to describe the numbers. But once you have it, you can simply declare that a number is in your set if and only if there does not exist a finite description in your system of choice. (In other words, you enumerate the countable many finite descriptions in your system, put them in the set, then define the new set as the complement.) How is that ambiguous?
> Interesting example, because the set is trivially empty under the assumption that we can describe any finite natural number by writing down its decimal expansion. (Based on that you can either argue that the primes are a subset of the natural numbers or that the primes are countable.)
Whoops, you're right, all integers are describable. Perhaps I should amend that to "the set of all undescribable real numbers with a prime integer part", or something along those lines.
The problem I see is that we do not have such a rigurous system and instead can introduce new notation, and how we decide on the notation may change the contents of the set of undescribable numbers. If there is no ambiguity, because we are using a very limited description language that can not be extended, we may be too restrictive in the language so the set becomes meaningless, e.g., because we could describe a number within it outside of the system which represents a trivial choice function.
* ZFC does not have a unique model that satisfies the axioms.
* Each of those models have different "undescribable numbers" with different properties.
But it was a long time I touched model theory, and it was just an optional extra course at uni.
However, there is a way around it, by taking as axiom that ZF is consistent, choosing some model M of ZF, and then talking about the set S of numbers inside M that cannot be described.
[0] https://en.wikipedia.org/wiki/Tarski%27s_undefinability_theo...
> Informally, the theorem says that the concept of truth of first-order arithmetic statements cannot be defined by a formula in first-order arithmetic. This implies a major limitation on the scope of "self-representation". It is possible to define a formula True(n){\displaystyle True(n)} whose extension is T∗,{\displaystyle T^{*},} but only by drawing on a metalanguage whose expressive power goes beyond that of L.L. For example, a truth predicate for first-order arithmetic can be defined in second-order arithmetic. However, this formula would only be able to define a truth predicate for formulas in the original language L.L. To define a truth predicate for the metalanguage would require a still higher metametalanguage, and so on.
It’s easy to pretend math is some universal language but it’s really several mutually exclusive languages.
Infinity is not a number. You can pretend it is and there are several ways to build a mathematical system around such a choice, but it isn’t inherently correct. So in your example f(x) may no longer include negative infinity and there is no paradox, it may include multiple infinite numbers and the lowest of them are now excluded, or you can say x and f(x) are the same etc.
It depends on what you think a number is. Most people who think about it agree a number is the name that we give to equinumerate sets, so two eggs and two ducks are equinumerate, we can give a bijection between these sets; they have the same property, "twoness" and we abstract twoness to the number 2. If so, then infinity is a number, it this the number-ness of the set {1, 2, 3, ... }
Let’s not mix up teminology and semantics. What a number here is is quite clear, we’re all talking about relatively standard math, the standard axiom of choice, and not some opinion on numbers. Sure, you can also say a planet is a mammal »depending on what you think a mammal is«, but that doesn’t lead to any sort of fruitful discussion.
My point is that infinity is certainly not an individual number. It represents quite a few concepts, all of which have some number like qualities.
Sure, maybe. I think saying "infinity is a number" is wrong regardless. I'm happy with aleph-0 being a number, and I'm happy with it having a label of infinity. I'm just not happy with treating infinity as a single thing. We're totally on the same page in every way that matters though. I'm also certainly with you on questioning Retric's comment for a number of reasons, haha.
edit: Actually...I'm not so sure they mean aleph-0. I think frequently they mean the infinity of the extended real line. Maybe it depends on the context of the conversation.
Note that this view is common but not universal: https://cs.nyu.edu/pipermail/fom/2023-January/023710.html
See this exchange for details
https://math.stackexchange.com/questions/2812355/does-infini...
EDIT: There is no least real number under the standard ordering. There is a least real number for a given well ordering. If you are asking me to define a well ordering on the reals then your request question should have been stated differently.
I’ve seen some truly bizarre definitions of the reals which are internally consistent and include things like the lowest possible number.
The widely accepted first order axioms of the natural numbers have nonstandard models. Of course each such model has an isomorphic initial segment. There are two things going on. There is the set of axioms and there are models for those axioms. I think in your usage: “define real number” corresponds to “give the axioms for the real numbers”. I’m not sure though.
> Anyway, axiom/models is the common terminology though I recall a rather long rant that postulate is better once you start talking about multiple incompatible systems.
Model has a very specific definition in this area of mathematics. This sentence is nonsense.
It's not a real number, true. But that has no bearing at all on the validity of f(x) = x + 1, since that function is well-defined with both its domain and its range as all real numbers and no others. You don't need infinity to be a number to define that function.
If it's possible to construct the set at all, then there is by definition no function that can be described in a finite number of symbols that can select an element from the set (since if a finitely describable function can define a member of the set, it shouldn't be in the set - you'd define the element as being the result of the function).
I don't know what the scholarly view is on relying on functions that cannot be defined in any human language, including mathematics. I would intuitively lean towards not allowing them in proofs, but perhaps they're useful mathematically?
Another interesting aspect - these are numbers, so presumably there is a smallest one, but identifying that is definitionally ruled out too, since 'smallest element in the set of numbers indescribable in a finite sequence of symbols', is itself a finite sequence of symbols. It makes me think that using 'select the smallest from set X' may be an illegitimate operation without further explanation about the set X, even if we know it contains numbers.
It should be noted that there are models of ZFC in which each real number is definable. You may be interested in reading the top answer on this mathoverflow.net question:
https://mathoverflow.net/questions/44102/is-the-analysis-as-...
This is all crazy stuff, but it would be consistent:
No axiom of choice -> no well-ordering -> no smallest number.
However, remember that the axiom of choice is technically a statement about the existence of certain functions, specifically functions that map from any set of non-empty sets to elements of those sets. Whether or not the existence of those functions says something about the possibility of "choosing" is more of a terminological question about how to connect informal natural language to mathematics than it is a substantive mathematical question.
Turning to the mathematical question itself, it seems to me that the author's intuitions fare poorly. The author argues that we apparently cannot "choose" an element from a set that contains only numbers that are not finitely describable. But the mathematical question is about the existence of a mapping function, and it seems that whether or not the numbers in each set are finitely describable is irrelevant to this question:
Consider the function f(x) = x + 1. This function is defined for all real numbers x. It maps each real number to another real number. Note that the domain of the function includes even those real numbers that are not finitely describable. The meaning of the function itself is nevertheless clear and the function is easy to write down. The fact that the domain of the function includes indescribable numbers is not relevant to the question of whether such a function exists mathematically.