If the term 'proposition' is used to mean a sentence -- a string of symbols, be they spoken, written, gesticulated or whatever -- then I suppose there could be uncountably many propositions if we allow there to be propositions of infinite length. In that case, one ought to be able to diagonalise on them just as one does with the infinite decimal expansions of the real numbers. But I think it's reasonable to stipulate that we're only going to countenance finitely long sentences. After all, they wouldn't be much use in communication, if you could literally never get a sentence out. Alternatively, if the set of symbols is itself uncountable, then that will certainly lead to an uncountable infinity of strings of such symbols. But it seems reasonable to stipulate against that case too. Communication would once again be thwarted, because we don't seem to have the perceptual capacity to discriminate between uncountably many different symbols -- indeed, our discriminatory abilities probably only extend...

No, mathematicians haven't defined any meaning for expressions like "infinity-th" or "infinity+1-th". (The fact that they're so awkward to write should be something of a giveaway!). It's important to appreciate that infinity is not a number. Don't be misled by the fact that we can say things like "there are infinitely many natural numbers", which seems to have the same form as a sentence like "there are three coins in the fountain". The number sequence doesn't go: 0, 1, 2, 3,... 1,000,000, 1,000,001, 1,000,002, 1,000,003,... infinity, infinity+1, infinity+2, infinity+3.... Rather, infinity is a property of certain sets, such as that of the natural numbers. The infinity of that set consists in the fact that, for any member you might care to consider, there will be another member, which is larger than it but which is nevertheless still finite . And we can easily refer to that set as a whole, and we can even quantify universally over its members. We can say, for instance, that they all have...

Given that mathematics is a body of universal and necessary truths, how could science (or anything else for that matter) not coincide with it? If the physical world was to violate the principles of mathematics, now that would be weird.