uncountable

I was coming dangerously close to conflating the fact of a sequence being uncountable with the proof that the sequence is uncountable being constructive.

Clearly if some sequence is uncountable it means that one of these criteria fails to hold.

Even though there isn't a constructive method of generating uncountable sequences, that doesn't mean that this also assumed from within the logic.

I just realised that in my mind uncountable (not in a one-to-one correspondence with the natural numbers) means non-constructive.

I think you didn't quite mean to say this, the real numbers being uncountable yet forming the basis for classical physics.

Basically, whenever you have uncountable sets, they come along with some topological structure which must be handled continuously.

In other words, having no constructive method for generating the reals is not at all the same thing, of course, as having a proof that the reals are uncountable.

By all means I invite you to come up with a better model, without using uncountable sets, for which there isn't a straightforward analogue, just as good, that uses uncountable sets.

So the fact that something is computable, per se, isn't exactly what makes the real numbers countable rather than uncountable.

In conventional mathematics you can also only write down countably many sequences, but the space of all sequences is still uncountable.

The reals are uncountable, ergo there is no constructive method for generating the reals.

There are uncountable ways to parse it and still come out with valid datetimes.

The proof of "the powerset of the natural is uncountable" (or equivalently, expanding out the definition of uncountability, "there does not exist a bijection between N and P(N)") is constructive, and will hold just as well in intuinistic logic.

[1] They almost feel like "uncountable" nouns, but I'm not sure.

Uncountable numbers of startups fail all the time in relative obscurity.

Uncountable sets are fundamentally unphysical.