# Could there be more than a countably infinite number of propositions?

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...

# HERE IS QUITE A CONUNDRUM: Can we meaningfully speak of the "infinity-th" and "infinity+1-th" term of the sequence of natural numbers? If not, then what do we in fact mean by "all" (as distinct from "any" or "each") when applied to an "infinite" set? Given that a real number constructed via the diagonal construction on a F I N I T E set, of n reals, can always be added to the list at position n+1 to give a list of n+1 reals, why couldn't a real number constructed via the diagonal construction simply be included in the "infinite" list of reals at "position" "infinity+1" ??? (Which is to say that, in the "infinite" case, no real could be constructed outside the infinite list of reals at all!) Also, in the case of the natural numbers, if a number m, is defined as the sum from 1 to n of the first n natural numbers, then m is a natural number that is not in the list of the first n natural numbers. If you make this construction on the "entire" set of "all" natural numbers, then by construction, there is...

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...

# How do we account for the weird coincidence of math and science (e.g., physics)?

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.