But no mare, meaning there is not any mare such that it has five (5) legs.
Thus the cardinality of "no mare" > the cardinality of "mare".
> It has two hind legs, aside from its fourlegs.
Hind legs are the opposite of forelegs so the statement is true but misspelled, and [---tortured logic here---] two hind legs plus "four" forelegs totals to six legs.
> Which means it has six legs... Now six is an even number, so it must have an even number of legs... But six is an odd number of legs for a horse to have, so it has an odd number of legs.
Six is an even number but it's "odd" - as in strange - for a horse to have six legs so horses [---tortured logic here---] must also have an odd - as in mathematical - number of legs.
> The only number that is both odd an even is infinity. Since a horse has both an odd an even number of legs, it must have infinite legs.
Infinity is a complicated "number" with different interpretations and [---tortured pure math here---] if you stretch some of them to the breaking point, the only horse that can have both an "even" and an "odd" number of legs is one that "must have infinite legs"
> But there is one horse, that not only has an infinite number of legs, it has legs that are infinitely long.
This plays on the ambiguity of "infinite legs" which could mean an "infinite number of legs" or "infinitely long legs".
All in all, thumbs up. Cost me some cherry switches thanks to projectile tea.
I am not aware of this math; can you elaborate? In pretty much any system, there is no number that is both even and odd, and including infinite quantities in the system doesn't change that.
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.
Or something like that. It's probably more accurate to say Infinity is neither odd nor even. If you add 2+2+2+2... up to infinity, it might appear even. But that could also be equivalent to 1+3+1+3+1... to infinity, which would be alternating even and odd. Or 1+2+2+2+2... which would always be odd.
Infinite quantities bend or break a lot of typical rules that other rational numbers follow.
2+2+2+… is a different infinity than 2+1+2+1+…
You can construct infinite sets of numbers that do not overlap. Odd vs. Even infinity is the example our professor used in class to show the concept. I don’t remember enough to replicate here.
On the other hand, you reminded me that conditionally convergent series may converge to different sums depending on how you choose to group their terms, which is fun. (That is, [(s_1 + s_2) + s_3 + (s_4 + s_5) + s_6 + ...] may have a different sum than [(s_1 + s_2) + (s_3 + s_4) + (s_5 + s_6) + ...] does.)†
† I hope this is true; I can't find a cite for it. The more famous theorem is that you can get any sum you want by reordering the terms, but I think regrouping is good enough to get you different sums, if not arbitrary sums.
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.