The set {56, 134, 255, 533} is a subset of the natural numbers.
http://www.sci.brooklyn.cuny.edu/~noson/True%20but%20Unprovable.pdf
This brings to light an amazing limitation of the power of language.
The collection of all subsets of natural numbers is uncountably infinite while the set of expressions describing subsets of natural numbers is countably infinite. This means that the vast, vast majority of subsets of natural numbers cannot be expressed by language.
Some true mathematical facts are expressible while the vast, vast majority of mathematical facts are inexpressible.
The overwhelmingly vast majority of truth cannot be expressed by language — Tarskian
judgments of true or false only apply to propositions — T Clark
If it can't be expressed in language, it isn't a proposition — T Clark
Propositions are linguistic entities — T Clark
Propositions that can be expressed by language are indeed linguistic entities. The ones that cannot be expressed by language, however, are not. — Tarskian
Subset X of the natural numbers is a subset of the natural numbers — Tarskian
I can see you and I are not going to agree on this. — T Clark
https://en.wikipedia.org/wiki/Controversy_over_Cantor%27s_theory
Controversy over Cantor's theory
Initially, Cantor's theory was controversial among mathematicians and (later) philosophers.
As Leopold Kronecker claimed: "I don't know what predominates in Cantor's theory – philosophy or theology, but I am sure that there is no mathematics there."
Many mathematicians agreed with Kronecker that the completed infinite may be part of philosophy or theology, but that it has no proper place in mathematics.
Logician Wilfrid Hodges (1998) has commented on the energy devoted to refuting this "harmless little argument" (i.e. Cantor's diagonal argument) asking, "what had it done to anyone to make them angry with it?"
When first confronted with the matter, I do not think that anybody right in his mind agrees on this. It is just too controversial. The first reaction is usually, disgust. It takes quite a while before someone can actually accept this kind of thinking. — Tarskian
Any thoughts on why? Is it a blow to people's egos to face the limitations of human thought? — wonderer1
You seem to be forgetting that languages can evolve and it's use can be arbitrary. We can always add more letters to the alphabet and we only communicate what is relevant. Why would we need a word for every natural number if we never end up finding a use for those numbers? If the universe is finite then there is no problem here. If it isn't then the universe at least appears to be consistent in that the physical laws are the same no matter where you go in the universe. Novelty would be the only aspects of the universe needing new terms to describe them.Human language is countably infinite because:
its alphabet is finite
every string in human language is of finite length — Tarskian
What is one example of a subset of the natural numbers that cannot be expressed by language? — hypericin
Richard's paradox
The paradox begins with the observation that certain expressions of natural language define real numbers unambiguously, while other expressions of natural language do not. For example, "The real number the integer part of which is 17 and the nth decimal place of which is 0 if n is even and 1 if n is odd" defines the real number 17.1010101... = 1693/99, whereas the phrase "the capital of England" does not define a real number, nor the phrase "the smallest positive integer not definable in under sixty letters" (see Berry's paradox).
There is an infinite list of English phrases (such that each phrase is of finite length, but the list itself is of infinite length) that define real numbers unambiguously. We first arrange this list of phrases by increasing length, then order all phrases of equal length lexicographically, so that the ordering is canonical. This yields an infinite list of the corresponding real numbers: r1, r2, ... . Now define a new real number r as follows. The integer part of r is 0, the nth decimal place of r is 1 if the nth decimal place of rn is not 1, and the nth decimal place of r is 2 if the nth decimal place of r[n] is 1.
The preceding paragraph is an expression in English that unambiguously defines a real number r. Thus r must be one of the numbers r[n]. However, r was constructed so that it cannot equal any of the r[n] (thus, r is an undefinable number). This is the paradoxical contradiction.
Assuming this statement is true, what do you think is its philosophical significance? — 180 Proof
If you look at the epistemic JTB account for knowledge as a justified true belief, it means that the overwhelmingly vast majority of true beliefs are ineffable and cannot possibly be justified. — Tarskian
They are ineffable, so they have no opportunity to be beliefs at all, and therefore no occasion to be justified. — hypericin
But all of these ineffable truths seem quite irrelevant too. — hypericin
https://institucional.us.es/blogimus/en/2022/01/is-infinity-really-necessary
One of the axioms of mathematics is that there exists an infinite set. Without this axiom our mathematics would be much weaker. Many of our theorems would fall like a house of cards. Newton or Gauss would probably have hesitated to accept our axiom (although without being aware that they were using it). We have accepted it for our comfort. Faith, that some people say …
As Leopold Kronecker claimed: "I don't know what predominates in Cantor's theory – philosophy or theology, but I am sure that there is no mathematics there."
Perhaps your OP topic only indicates that "the epistemic JTB account" is inadequate in some way. — 180 Proof
Now, can you give an example of one those the truths? — Banno
Now, can you give an example of one those the truths? — Banno
Give an example of one of these unstatable true sentences... — Banno
https://en.m.wikipedia.org/wiki/Richard%27s_paradox
The preceding paragraph is an expression in English that unambiguously defines a real number r. Thus r must be one of the numbers r[n]. However, r was constructed so that it cannot equal any of the r[n] (thus, r is an undefinable number). This is the paradoxical contradiction.
The overwhelmingly vast majority of truth cannot be expressed by language — Tarskian
Assuming this statement is true, what do you think is its philosophical significance? — 180 Proof
The paragraph expresses a number, not an unstateable truth. — Banno
https://iai.tv/articles/most-truths-cannot-be-expressed-in-language-auid-2335
Most truths cannot be expressed in language
14th December 2022
Noson S. Yanofsky | Professor of computer science at Brooklyn College
There are more true but unprovable, or even able to be expressed, statements than we can possibly imagine, argues Noson S. Yanofsky.
Human language is countably infinite because:
its alphabet is finite
every string in human language is of finite length — Tarskian
There's noting novel in the natural numbers not being enumerable. What this shows is that the list from which r is derived cannot be constructed....a member of the natural numbers/not being a member of the natural numbers, as based upon the unresolvable paradox. — ucarr
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