> What is Infinity plus one? It must still be Infinity, since it's already infinitely high and can't go any higher.
This is a mistake. That is how the extended reals behave. There are other systems. The diagonalization argument will already tell you that if you're at infinity, you can still go higher. This is also true for ordinal numbers and nonstandard analysis. It is the general case; the behavior in the extended reals is a special case.
> But if you have an odd number and add one, you get an even number (and vice versa). For Infinity = Infinity + 1 to hold true, Infinity must be both odd and even.
This is much worse; your conclusion is unrelated to your premise. You would need an additional premise, that infinity is either even or odd, which you didn't bother to supply. The same argument will tell you that either (a) 1.3 is even and 2.3 is odd, or (b) 1.3 is odd and 2.3 is even. (a) and (b) are both false.
This is a mistake. That is how the extended reals behave. There are other systems. The diagonalization argument will already tell you that if you're at infinity, you can still go higher. This is also true for ordinal numbers and nonstandard analysis. It is the general case; the behavior in the extended reals is a special case.
> But if you have an odd number and add one, you get an even number (and vice versa). For Infinity = Infinity + 1 to hold true, Infinity must be both odd and even.
This is much worse; your conclusion is unrelated to your premise. You would need an additional premise, that infinity is either even or odd, which you didn't bother to supply. The same argument will tell you that either (a) 1.3 is even and 2.3 is odd, or (b) 1.3 is odd and 2.3 is even. (a) and (b) are both false.