Same topic as https://thephilosophyforum.com/discussion/15304/mathematical-truth-is-not-orderly-but-highly-chaotic/

Is it ok if we just copy-and-paste the replies? Or should we link to them?

The overwhelmingly vast majority of truth cannot be expressed by language — Tarskian

This is not right. Perhaps "the vast, vast majority of subsets of natural numbers cannot be expressed by language," but judgments of true or false only apply to propositions. Propositions are linguistic entities - they can all be expressed in language. If it can't be expressed in language, it isn't a proposition and if it isn't a proposition, it can't be true or false.

judgments of true or false only apply to propositions — T Clark

The following is a legitimate proposition:

The set {6,8,11} is a subset of the natural numbers.

It is true or false.

If it can't be expressed in language, it isn't a proposition — T Clark

The following proposition is tautologically true:

Every subset of the natural numbers is a subset of the natural numbers.

The problem is that most individual subsets of the natural numbers cannot be expressed by language. Some can but most cannot.

The ineffable propositions are still true propositions because all of them are true given the tautology mentioned above.

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. For example, the general case of "Subset X of the natural numbers is a subset of the natural numbers" is true, irrespective of whether X can be expressed by language or not.

Propositions that can be expressed by language are indeed linguistic entities. The ones that cannot be expressed by language, however, are not. — Tarskian

There are no propositions that can't be expressed in language.

Subset X of the natural numbers is a subset of the natural numbers — Tarskian

This is just a restatement of the tautological proposition "All subsets of the natural numbers are subsets of the natural numbers."

I can see you and I are not going to agree on this. I'll give you the final word.

I can see you and I are not going to agree on this. — T Clark

The distinction between countable and uncountable infinity, originally introduced by Georg Cantor, has always been controversial.

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.

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?

Any thoughts on why? Is it a blow to people's egos to face the limitations of human thought? — wonderer1

In my opinion , it decisively divorces mathematical reality from physical reality, which is otherwise its origin.

Humans, but also animals, have quite a bit of basic arithmetic and logic built into their biological firmware, if only, for reasons of survival. To the extent that mathematics stays sufficiently close to these innate notions, people readily accept its results.

There is no notion of infinity in physical reality. In that sense, Cantor's work is rather unintuitive. You have to learn to think like that. It does not come naturally.

Human language is countably infinite because:

its alphabet is finite
every string in human language is of finite length — Tarskian

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.

What is one example of a subset of the natural numbers that cannot be expressed by language?

Also note that mathematical notation is a kind of extension to the natural languages.

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?