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> Almost all real numbers that exist can never, even in principle, be written down or described in any meaningful way.

I don't think many people see that as a matter of philosophy.




Most people are uninformed.

This question strikes at the heart of the debate between Constructivism and Formalism. A debate about what it means for things to exist, statements to be true, and so on. This is very much a matter of philosophy.

To a Constructivist, most of classical mathematics is nonsense. And Constructivism is at least as logically consistent as classical mathematics.

More precisely any contradiction found in Constructivism necessarily will lead to a contradiction in classical mathematics. The converse is only partially true. Gödel did prove that a logical contradiction in the classical handling of infinity will lead to a contradiction in Constructivism. But a flaw in a specific set of classical axioms, such as ZFC, need not lead to a flaw in usual Constructivism.


I think you needed to be more precise. Most people, would read "almost all" in the context of the reals as dependent on the existence of an uncountable set, to make sense.


I was perfectly precise.

In real analysis you learn that "almost all" means that the exceptions are a set of measure 0. Since all countable sets have measure 0, the result is trivially true in classical mathematics.

In the constructible universe, you again have measure theory. Almost all still has a perfectly well-defined meaning. And all sets with enumerations again have measure zero, just like in classical mathematics. But "uncountable" now is a statement about self-referential complexity, not size. Next, "the set of all numbers with finite definitions" is not a well-defined set. And numbers without concrete definitions do not exist.




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