# Talk:WikiJournal Preprints/Can each number be specified by a finite text?

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## Article information

Tsirelson, B.

PDF available: File:Definable.pdf. It is hyperlinked, which is not visible here; download it and open in an appropriate pdf viewer.   Boris Tsirelson (discusscontribs) 19:26, 21 December 2018 (UTC)

The PDF is now updated, it corresponds to this version of the preprint. Boris Tsirelson (discusscontribs) 20:18, 7 April 2019 (UTC)

Self-published on arXiv. Boris Tsirelson (discusscontribs) 03:26, 26 September 2019 (UTC)
Discussions broke out the same day (and died out the next day) on Twitter and Reddit.

• (Uncanny as this is similar to something I was planning to tweet about - but this article will be a lot better than anything I could have written.)   (@sigfpe).
• My mind is being blown by section 8, the value of Chaitin's constant depends on which model of ZFC you're in!   (@luqui)
• I thought it was trivially true. Thanks for sharing. Another pdf in my todo read list. On the other side the concept of "definable" is quite fuzzy in my mind. Hopefully the doc would clarify my misconceptions a little.   (@DuduRyer)

...and from Reddit
• So in fact we can't say that not every real is definable, because we can't even ask the question. [...] The problem is that it's impossible to define what it means for a formula to hold true of a given real number.   (Oscar_Cunningham)
• Great article. Also interesting to see so many commenters miss the point. Let me try to summarize...
Alice: There are uncountably many reals, but only countably many definitions in any finite language, so undefinable reals must exist. I win!
Bob: By Lowenheim-Skolem, there's a countable model of ZFC. Moreover, there's a model of ZFC whose every member is definable. You were saying?
Alice: Wait, doesn't ZFC prove that reals are uncountable?
Bob: Sure it does. You're just mixing up levels. The reals of that model do have a one-to-one mapping with the integers, but that mapping isn't part of the model.
Alice: That's a pathological model then!
Bob: But you can't tell it from a normal one using ZFC. That's what being a model means.
Alice: Ok, screw this ZFC bullshit. My argument works in truth.
Bob: If your system of truths about reals is non-contradictory, then by Lowenheim-Skolem it has a countable model too. And I don't think it could prove the existence of reals undefinable in that system.
Alice: What's then the status of my argument?
Bob: I'd have to say it's neither true nor false. You know how the consistency of {union of all your truths} feels like a truth, but can't be part of the union? The existence of undefinable numbers is in the same kind of limbo. Philosophy of math is weird.   (want_to_want)
• [...] "defining" something is a delicate process. There's a potential difference between a definition in English words and one in mathematical symbols.   (avocadro)
• If an issue like this is resolved in the paper, don't tell me. I'm gonna go read it now before I say anything else potentially wrong/misinformed.   (point_six_typography)
• Read it to find out what the subtleties are!   (completely-ineffable)
• Premise 2 is too vague to a be a theorem, as it depends heavily on what counts as a “definition”. This is what the article is challenging.   (CheekySpice)
• "exercises are waiting for you"   (jmdugan)

Boris Tsirelson (discusscontribs) 19:56, 24 October 2019 (UTC)

## First review

Review by Laureano Luna ,
This review was submitted on , and refers to this previous version of the article

Minor remarks

-In the introduction, the author says that P is a statement. In fact, it is a predicate or a Russellian propositional function but not by itself a proposition or a statement.

-In the section titled “From Predicates to Relations”, the author defines A={(x_1,…,x_(n+1) )∈R^n (…)} Note that n+1 doesn’t match R^n.

-In the section ‘Beyond the Algebraic’, this expression can be found: ∀ε>0 ∃n∈N ∀m∈n(m≥n→-ε^(n^n )<(n+1)^n-e^n<εn^n )

Note that ‘m’ plays no role in the consequent. As a simpler alternative, I would suggest:

∀ε>0 ∃n∈N ∀m∈n(m≥n→|(1+1⁄m)^m-e|<ε)

More substantial remarks

-In ‘Beyond the Algebraic’, the author writes:

“On the other hand, if we choose a number between 0 and 1 at random, according to the uniform distribution, we almost surely get an indefinable number because the definable numbers are a countable set”.

Here the author assumes that all real numbers are given and available once forever. This is an important part of the problem of the definability of the reals. Many definitionists (or predicativists) don’t believe the reals to be all simultaneously given; for many, the reals are indefinitely extensible. I think the author could profit from taking a look at this (for all I know unpublished)dissertation: A Defence of Predicativism as a Philosophy of Mathematics, by Storer: https://core.ac.uk/download/pdf/1322419.pdf

-In ‘Definable but Uncountable’, the author assumes the set R of all real numbers to be well-defined, when he claims the following number to be well-defined: (z=0 ∧ ¬CH) ∨ (z=1 ∧ CH)

Again this is not clear in philosophy of mathematics: many predicativists (famously, Solomon Feferman in this paper: https://philpapers.org/rec/FEFITC) tend to believe CH is not a well-posed problem.

-The author depicts a hierarchy of mathematical languages with increasing definitional power. Each mathematical language defines only countably many real numbers but there are uncountably many levels in the hierarchy of languages. This is correct.

The author adds the remark that all those mathematical languages must be introduced by means of a natural language, which is also correct. In addressing this problem, we surely have to reach to the natural language: as the author, remarks, there is no mathematical definition of ‘mathematically definable’. However, the question immediately arises how could a natural language give birth to uncountably many mathematical languages. The problem of the limited number of definitional resources poses itself anew for the natural language.

Surely, as the author points out, natural language goes beyond the concern of mathematics, strictly understood, but it is a bit surprising that the author stops at this point after so many and so detailed mathematical descriptions of definitional resources. More so because the author refers to a paper in which a solution is proposed (Luna, Laureano; Taylor, William (2010). ""Cantor’s proof in the full definable universe"". The Australasian Journal of Logic 9: 10-26). The solution proposed by these authors is, shortly put, as follows: whatever our language, our universe of discourse is always extensible along an uncountable hierarchy of language levels, so that on each level our quantifiers mean differently and the same syntactical expressions are endowed by the logical context with new referents, so that more numbers can be defined than syntactical expressions exist. I think the author should at least mention this proposal. To this end, the author could profit from reading: Luna, Laureano (2017). Rescuing Poincaré from Richard’s Paradox. History and Philosophy of Logic 1, 38: https://www.tandfonline.com/doi/full/10.1080/01445340.2016.1247322

-As the author is willing to deal with a real philosophical problem, he should perhaps take into account the main philosophical issues involved and particularly the question of indefinite extensibility versus unrestricted quantification. I suggest this book, already a classical: Rayo, Uzquiano editors (2006). Absolute Generality. OUP

Or for a more succinct overview:

Florio, Salvatore (2014). Unrestricted Quantification. Philosophy Compass 9/7 (2014): 441–454.

General Remark

The work merits publication although I would recommend a more balanced relation between the mathematical and the philosophical parts, which could be achieved as a result of addressing the mentioned philosophical issues.

Response

Thank you. On one hand, being a mathematician, not philosopher, I feel unable to balance the two parts (mathematical and philosophical). On the other hand, I do read the sources you provided, and I do hope to improve the article accordingly (which will take a time, of course). Anyway, my preprint is not intended to contribute to philosophy, nor to mathematics, nor to be an encyclopedic article. It is rather an explanatory essay intended for non-experts. Am I willing to deal with a real philosophical problem? No, my intention is more modest: to guide the reader from naive and incorrect approaches, through some relevant (and correct) mathematics, till the border between mathematics and philosophy, and say: here we stop; for good reason we fail to mathematically define "THE set of all definable numbers" (the "THE" means: ultimate, canonical, not relative to a model etc), and therefore WITHIN mathematics (as of now) we cannot ask, whether or not "this set" is countable, or contains all real numbers, etc.
(To be continued later.) Boris Tsirelson (discusscontribs) 08:08, 25 March 2019 (UTC)
The review contains three minor remarks (denote them M1, M2, M3), four "more substantial" remarks (S1, S2, S3, S4) and one general remark (G1).
M1
Fixed ("not a statement when ${\displaystyle x}$  is just a variable, but a statement whenever a real number is substituted for the variable").
M2
Fixed (${\displaystyle \mathbb {R} ^{n+1},}$  not ${\displaystyle \mathbb {R} ^{n}}$ ).
M3
Fixed (${\displaystyle m,}$  not ${\displaystyle n}$ ).
S1, S2
A clarification is added to Section 2 ("From predicates to relations"):
All mathematical objects (real numbers, limits, sets etc.) are treated in the framework of the mainstream mathematics, unless stated otherwise. Alternative approaches are sometimes mentioned in Sections 9, 10.
S3
The matter is now treated in Section 9.2 ("Transfinite hierarchy") near the end,
Countability or uncountability of ${\displaystyle {\tilde {R}}}$  matters for model dependence...
This matter is closely related to the position of Laureano Luna [37]...
Assuming existence of large cardinals...
and Section 10 ("Conclusion") near the end,
Another problem manifests itself as model dependence for a mathematical language, and context dependence for a natural language...
S4 and G1
New sources, added to the bibliography ([16], [18]–[21], [30]–[37]), are used in Section 9.1 ("Finite set theory") near the end,
Several possible such "alternative airfields" are examined by mathematicians and philosophers [30], [31], [32], [33], [34], [35], [36].
Note 12,
Alternative, more constructive approaches attract the attention of mathematicians and philosophers [18], [19], [20], [21]. Maybe the collection of all subsets of an infinite set should be treated as a class rather than a set
and Note 28. But still, the two parts (mathematical and philosophical) are not balanced. In particular, "absolute generality" is not included, since it is not (yet?) treated mathematically (to the best of my knowledge).
(The response is now completed.) Boris Tsirelson (discusscontribs) 08:49, 2 April 2019 (UTC)
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