Talk:Definable real number

message: The formula φ(x) is not restricted to first-order statements.
There are restrictions on φ, though, thaks to Berry's paradox.  Should this be mentioned here?
----
Good point; yes, I think we should mention Berry's paradox and explain which φ's are allowed. Which ones are allowed? --AxelBoldt
:I think that's the problem with the whole page.  If you nail down what "definable" means, you'll find that the set of definable reals itself is not definable in that sense.  "Definable" is a moving target; the whole field of descriptive set theory is in some sense an exercise in watching the target move. --Trovatore 04:00, 19 July 2005 (UTC)
replies: text: So it's not very clear to me what should be done with the page, even if it should be kept at all.  I'm torn because it exposits a useful notion, but it doesn't say anything very precise about just what the notion is, and in some sense it never can.  Moreover since there isn't any fixed notion standardly called a "definable number", it's almost a neologism page.  Maybe it could be saved by showing various precise notions of definability without parameters and observing that they all give rise to countable sets of "definable reals", albeit different countable sets. --Trovatore 04:00, 19 July 2005 (UTC)

text: And this passage above doesn't even make sense.  What does it mean to prove in ZFC that φ(a) is true?  We don't even have a definition for a, in advance; that's what φ is supposed to be giving us.  It does make sense to talk about proving that there exists a unique real number that makes φ true -- but now we don't have a around anymore.  We can add it back in by requiring, in addition, that φ(a) really be true, semantically, i.e. in the Von Neumann universe V.  But now what do we need the uniqueness proof in ZFC for?  Isn't it enough that φ have a unique real solution?  No, the whole thing is just horribly muddled; needs major surgery. --Trovatore 04:24, 19 July 2005 (UTC)
replies: text: I don't see what's wrong with it. I don't understand anything you're saying about proving versus "be true"/"have a unique solution". The wording you say does make sense is just a rewording as far as I can see.--MarSch 09:37, 19 July 2005 (UTC)
replies: text: For an arbitrary real number a, there is no notion of "proving" φ(a) from ZFC, because to prove φ(a), you first have to be able to write it down--and we don't know how to write down a. --Trovatore 16:38, 19 July 2005 (UTC)

text: The defect here seems obvious; not that a is the unique real number  but that there is a unique real number a. Emending text accordingly. Are there other factuality disputes? Septentrionalis 19:51, 19 July 2005 (UTC)
replies: text: Bunches of 'em.  I'll start a new section below, with a list.--Trovatore 20:18, 19 July 2005 (UTC)
thread_title: Berry's paradox, definability of definability

message: I've taken a first cut at a rewrite, to address the issues above.  It may be viewed at User:Trovatore/Sandbox/Definable number.
Some things to note:
* I've deleted the claim that the definable numbers contain all reals that can ever be unambiguously described, and added a section that explains why this isn't true.
* I've also deleted the claim about calculus.  I'm not opposed to its being there--if someone can explain just what it means, and give references.  It looks like it may have been motivated by some sort of neo-logicist project; if so, the project should be named.  (I don't really see how you're going to get, say, the Lebesgue integral, just with definable reals, but maybe someone has managed--certainly it wouldn't be the standard treatment, though, so "all the theorems of calculus" seems to be overstating the case.)
* I've left the bit about "standard" numbers, but I don't understand it, and would like someone to explain it.
** So when I got around to following the link, it made a certain amount of sense.  Not sure it belongs just here, though. --Trovatore 07:46, 20 July 2005 (UTC)
replies: text: I'm not sure I'm convinced by your diagonalization argument. You are talking about an explicit diagonal process that will, given a list of denumerably many Gödel numbers of well-formed definitions of real numbers, produce another Gödel number of a well-formed definition of a real number. This is a complicated enough idea that you will probably need to give an example of such a procedure--I have never seen an example even of a diagonalization producing a natural number from a list of natural numbers. --Ian Maxwell 03:02, 19 October 2005 (UTC)
replies: text: No, actually, I'm not talking about doing what you say in your second sentence above--at least, I don't claim to recover a Gödel number of a definition in the same language. The language is not rich enough to define definability in the language, or at least not to definably get from a definition to the thing defined. That's kind of the whole point. It's analogous to Tarski's result on the undefinability of truth. --Trovatore 03:09, 19 October 2005 (UTC)

text: I've been reading this article and the discussion, and I'm was trouble understanding how we get an "unambiguously described" number from the diagonalization argument.  I'm going to attempt to clarify it.  —Preceding unsigned comment added by Maxsklar (talk • contribs) 00:19, 1 March 2009 (UTC)
replies: text:  I tried to do this, but I'm still not happy with it.  I have a feeling that there's some larger idea or paradox here. It would be nice if someone could explain it more thoroughly and add a citation. Maxsklar (talk) 01:07, 1 March 2009 (UTC)
thread_title: Proposed rewrite

message: Having heard no objection, I have copied the rewrite to the main page.  Comments solicited. --Trovatore 05:10, 21 July 2005 (UTC)
thread_title: Rewrite in place

message: Here's the proof that it's consistent with ZFC that there are only countably many OD reals.  Someone may want to check it (or finding a ref would be even better).
Start with L, in which every real (in fact every set) is OD.  Now force to collapse \aleph_1 (the forcing conditions are finite partial functions from \aleph_1  to \omega).
:(Actually, make that finite partial functions from \omega to \aleph_1--the other direction might work but it's harder to see.)--Trovatore 07:11, 22 July 2005 (UTC)
thread_title: Countably many OD reals

message: This is the problem with this page as I see it:
* Intuitively, "Definable number" means "there exists a formula uniquely identifying"
* However, in what language? Due to Godel's theorem, whenever you tie down the definition to a particular formal language, there are going to be numbers which are not definable in that formal language, but which are definable in the intuitive sense, definable in 
* So really, the intuitive notion of "definable" is impossible to formalise
* So, the attempt of this article to provide a formal definition of the intuitive concept of "definable number" is to some extent pointless (unless that particular formal definition has some other value)
* Really, this intuitive concept of "definable number" belongs not to mathematics, but the philosophy of mathematics -- it is unformalizable
This is the most correct definition of definable number I can conceive:
:A number is definable if there exists a proposition capable of being conceived by a mind of at least human-level intelligence which is true for that number and false for every other number.
Obviously, this definition is unformalisable due to Godel's theorem. But it is I think the true one.
Another definition might be:
:A number x is definable if, for some formal language F and associated model M, there exists a statement S(x) in F such that S(x) is true in M and for all numbers y S(y) is true in M iff x=y.
That perfectly captures in my view the concept of a definable number, but by it is inexpressible in a formal language, since Godel's theorem does not permit a formal language to quantify over every possible formal languages and every possible model. Also, I think its informalizable because we won't not just any old model M but the "correct" one, a notion which I don't believe is formalizable either.
--SamKat
:Much of your analysis is right on. A quibble: Gödel's theorem per se is not really relevant; that's about syntax, whereas when we talk about a real that's not definable in a language, we've moved beyond syntax and are talking about the real number, semantically, as an object. So it's more like Tarski's theorem on the undefinability of truth. But they are very similar in style.
:But if you want to see a page with much worse problems along the lines of the ones you discuss, take a look at the page before I got to it. That should make my criticisms on this talk page more understandable.
:Your criticisms still have merit, but I don't really see a solution short of deleting the page. --Trovatore 16:48, 20 October 2005 (UTC)
replies: text: Oh, to avoid confusion I should also say that your attempt above with the language F and model M isn't going to work; it would make every real number definable. Just put a function symbol f in the language, and now for any real x that you want to make definable, choose your model M so that it interprets f to assign to each natural number n the nth decimal digit of x. Details to be filled in by you. You're thinking along the right lines, though. --Trovatore 17:17, 20 October 2005 (UTC)
thread_title:  Problem with this page 

message: All _real numbers_ are definable and countable --- just pick any irrational number (say, pi) which is definable and any real number can be defined by how it differs digit-for-digit with pi with their place-value-positional-numeral-system (say, decimal or binary) expansion point aligned.
First, consider:  1234567890123… --- this is not an integer (hence, not a rational nor real number) because integers have a finite count of digits!
Next, consider: 1234567890.123… --- this is now a “real number” that lies  between 1234567890.1230 and 123456789.1239.  Or is it?  What made it a “real number”? --- the decimal point!?  Well, does the decimal point sufficiently make it a “real number”?  If it is a “real number” then is it equal to 1234567890.1230, 1234567890.1231, 1234567890.1232, …, or 123456789.1239?  Well, let us not forget 1234567890.12301, 1234567890.12302, …, 1234567890.12309, 1234567890.123011, 1234567890.123012, …, etc., etc., etc.      
Isn’t 1234567890.123… actually an “interval” and not a “real number point”?  Again, what makes 0.5, square root of 2, e and pi real numbers? --- all their digits are well-defined (so they are definable --- that is, they are constants)!  Thus, real numbers are constants!  But what is an “interval”?   Well, 1234567890.123… could really be defined as 1234567890.1230 <= x <= 1234567890.1239.  Aha!  The notorious “x” symbol! --- that is, 1234567890.123… is in fact an _interval_ --- that is, 1234567890.123… is a _variable_ --- that is, 1234567890.123… is not a constant --- that is, 1234567890.123… is not a true real number!!!
Every one in the mathematical world agrees that, say, the 5th decimal digit of pi is “9” but what is the 5th decimal digit of 1234567890.123…?  If you can answer this last question then you can truly define 1234567890.123… (of course, that is if you could also answer what its nth digit is for any natural number n) --- thus, what makes a sequence of decimal digits with a decimal point a real number is that _all_ the digits are definable!  Thus, “Undefinable real number” is a _self-contradiction_! --- so is “Cantor’s diagonal argument” untenably used to “prove” the uncountability of real numbers and many other flawed claims of “modern mathematics” (please read my Wikipedia discussion notes on this article as well as on “Cantor’s theorem”, “Cantor’s first uncountability proof”, “Ackermann’s function”, and “Entscheidungsproblem”)
Cantor’s anti-diagonal “number”, Borel’s “number”, Chaitin’s “number”, etc. are _variables_ (or, at best, they are arbitrary constants) whose claim for being “real numbers” merely emanates from their prefixed decimal point.  They have the “form” of a fractional real number (that is, they lie between 0 and 1 only because of the prefixed decimal point) but they do not have “substantive” information of a true “real number” (a constant!) --- only that of an interval --- to convey.
BenCawaling@Yahoo.com [09 December 2005]BenCawaling 06:20, 28 March 2006 (UTC)
thread_title:  “Undefinable real number” is a self-contradiction 

message: The section Ian Maxwell edited probably could use some touching up (in particular, getting rid of "in the sense of this article") but the edit I reverted introduced a syntax/semantics confusion. Objects are definable "over structures", not "in theories"; it doesn't make sense to ask if a particular object is definable in a theory. Given an undefinable object, how do you ask if a particular theory can define it? The theory can't even talk about it. --Trovatore 21:01, 15 December 2005 (UTC)
replies: text: Oh, and a minor point: The change from Gödel numbers to Gödel numbers was quite unnecessary; the first form works fine and the source is easier to read. --Trovatore 21:03, 15 December 2005 (UTC)
thread_title: Explanation of revert

message: While reading the article a got an idea like this:
The set of definable reals(,complex numbers,and probably even R^n) is countable, therefore there exists a bijection to natural numbers. Now you can use induction to prove theorems about definable numbers. Of course they do not apply to all reals but will still apply to almost every number you will ever use.
Im guessing it would be a bit hard to actually construct a bijection that is useful, but is there something else wrong with the idea?
85.156.185.105 01:39, 31 August 2006 (UTC)
thread_title:  Induction 

message: For your information, the article fr:Nombre définissable has been deleted today, due to the same problems you mentioned here. We'll possibly replace it with an article focused on Borel's "inaccessible numbers". --Michel421 (talk) 20:41, 28 February 2010 (UTC)
replies: text: I am not familiar with these — can you give me a pointer?  I did take a look at the deletion discussion (here: :fr:Discussion:Nombre définissable/Suppression) and I saw that various people had the same concern I had, that inaccessible number would most probably be taken to mean inaccessible cardinal.  One contributor linked three papers, in all of which (from a brief scan) it appeared that inaccessible number or translation thereof was being taken to mean either "inaccessible cardinal" or "weakly inaccessible cardinal".  But certainly this is not an insurmountable obstacle, as the name can always be made clear. --Trovatore (talk) 22:26, 28 February 2010 (UTC)
thread_title:  WP:fr article has been deleted 

message: What can be said about the relation of definable numbers to elements of the ring of periods of Zagier? Tkuvho (talk) 16:08, 29 July 2010 (UTC)
replies: text: According to that article, all periods are computable; the computable numbers are a proper subset of the definable numbers. — Carl (CBM · talk) 19:37, 29 July 2010 (UTC)
replies: text: Thanks, I should have realized that.  This takes some steam out of the remark that the periods provide an countable intermediate system between Q-bar and C. Tkuvho (talk) 02:30, 30 July 2010 (UTC)
thread_title:  Relation to ring of periods? 

message: Substituted at 01:59, 5 May 2016 (UTC)
thread_title: Assessment comment

message: A recent edit called my attention to this text:
:Because the notion of definability in the language of set theory cannot itself be expressed in the language of set theory, the collection of real numbers definable in the language of set theory may not form a set.  However, assuming they do form a set, the real numbers definable in the language of set theory over a particular model of ZFC form a field.
Where does this come from?  Is this really what Hamkins thinks?  Informally, it's obvious nonsense.  Every subcollection of a set is a set.
The only way I can see that this claim is at all defensible is if you're thinking formalistically; that is, you're making reference to the fact that you can't apply the separation axiom to show the existence of such a set, because there's no first-order property that defines it.  But then I don't see how you formulate the claim at all; there's no way to talk about the collection in the first place.
We really need to fix this.  Right now it looks really bad and obviously nonsensical.  Paging User:CBM; we've had discussions in this neighborhood in the past. --Trovatore (talk) 21:11, 10 April 2017 (UTC)
replies: text: That sentence would most naturally be read as "the real numbers definable in a model of set theory may not be a set in that model" - which certainly seems correct to me. We have to talk about the definable reals of a given model from outside that model, as usual. The first clause does seem to be referring to the separation axioms, but that clause could be omitted. The important point IMO is that we need to assume the definable numbers form a set in order to prove they form a field. Including that point should be unproblematic for a realist, because the assumption is trivial from a realist perspective; but including it helps to highlight that the situation with the definable numbers being a field is not their closure under the arithmetic operations, it's whether they are actually a set. The proof that the definable reals of a model form a field is also written from a perspective outside the model. — Carl (CBM · talk) 01:54, 12 April 2017 (UTC)
replies: text: I don't have a problem with writing "the real numbers definable in a model of set theory may not be a set in that model" (assuming you/we can find an example, probably easy but I'm not instantly thinking of one — obviously it's not the case for those countable models in which all reals are definable, because in that case the set of all reals definable in the model is just the set of reals of the model).  I don't agree at all that that's the natural reading of the text as it stands.  --Trovatore (talk) 02:09, 12 April 2017 (UTC)
thread_title:  "may not form a set" 

message: Is a real number undefinable with respect to first-order \mathcal L_\in formulae iff it's indiscernible? C7XWiki (talk) 03:58, 2 March 2021 (UTC)
replies: text: Per WP:TALK#TOPIC, we're not supposed to discuss that here unless it's part of a discussion of what should appear in the article.  For questions about the topic, we have the refdesks.  Why don't you ask at WP:RD/Math?  We could use some interesting questions there. --Trovatore (talk) 04:48, 2 March 2021 (UTC)
thread_title: Indiscernibility