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
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