Fringe Math

[Categories+Sheaves]

We were taught about countably infinite and uncountably infinite sets in my first year at university

This is a weaker point than what I'm aiming for. Cantor's diagonalization proof doesn't need the axiom of choice; we suppose the bijection f: N -> P(N) exists and is well-defined, and we define the set S in relation to the "diagonal." To answer the question "Is the element a in N an element of S?" we need only look at the f we defined, and see whether a is in f(a). This process can be performed without regard to whether any b in N (such that b != a ) is in S; our 'counterexample set' S can be constructed without AC. And we need to already understand the difference between countable and uncountable sets in order for us to interpret what that 'countably additive measure' stuff is supposed to even mean... (see earlier links)

Using standard notation to try and represent the first number that comes after 0 gets us into even more trouble...

If this is in relation to the statement about a well-ordering on the reals: no. The standard order relation on the reals is not well ordered. But by the axiom of choice, there is some well-ordering. Such a well-ordering must disregard the standard order relation on the reals. And again, it 'exists' but cannot be constructed using finite strings in a finite language.

since 0.000....1 is an invalid number

Regardless of what we're going to take 'valid number' to mean, many nonstandard formulations of analysis posit the existence of nonzero infinitesimals (the most readable one I know of is here). You're 100% right in the standard construction of the reals but... this is thread is supposed to be about the fringe, not the canon.

...and ultimately leads to all the lovely "impossible" examples of mathematics...

As I was saying before, there's a gap in the required axioms. Some things from the list, esp. B-T, can be done with a weaker formulation of AC, but... well, you know what I mean.

Since you seem to know your math; what about large cardinals, i.e. the cardinality of a proper class, i.e. the cardinality of the set of all sets? There are some formulations of class-set theory (adding the continuum hypothesis as an axiom and a fistful of other reasonable-sounding things too) that are able to disprove the existence of 'large cardinals'. But the Hardin & Taylor paper (about the infinite hat problem) states some results that require the existence of large cardinals. Does this 'split' matter to us? Don't we still want there to be some sort of 'truth' on the matter?

As far as 'truth' goes, large cardinals either exist or don't exist. We have that business about the excluded middle to thank for that. But the truthfulness of these sorts of statements may be completely unrelated to the "natural" or "intuitive" machinery of math (e.g. the continuum hypothesis is independent of ZFC). Are these curiosities some sort of 'nightmare' in our mathematical imagination that we need to dispense with and free ourselves from? Or are these things that should be embraced, and studied as objects in-themselves?

These questions should cut into the teleology of math; what deserves study/acceptance, and why? Do we consider ourselves citizens of the world we construct, or the world we intuit?