Thoughts of a cognitive scientist and artist.
I can prove this (by contradiction)If infinity is even, then there is some 2^n such thatinfinity--------2^nsuch that infinity is not even.For all n, the above will still be infinity, which we assumed was even. Therefore, it is already odd.Even taking the case of n=infinity, we have the above = 0, which is a useless answer.
actually, for n = infinity, we have the above = 1. However, this is based on the approximation thatinfinity approximately = 2^infinitySo, while this would be acceptable for a limit in the real number domain, when we are referring to integers it is not.There, infinity is still odd, in fact, it is prime.
One of the properties of even numbers is that they can be "evenly" divided into two equal subsets, and with an infinite set you can ALWAYS evenly divide it. Traditional models of infinity assume that half sets can be put into correspondence with each other: for example, if you knock out every second number in the series to get 1,3,5..."infinity minus all even numbers", you can still put each odd number in one-to-one correspondence with some number from the original series to see that both sets have the same magnitude. So here clearly you can divide an infinite set in half and get two equally sized sets, so infinity "feels even".Similarly, you can do interesting things with doubling: for example, if you multiply the infinite series of whole numbers 1,2,3...infinity by 2 to get 2,4,6..."infinity*2", you can put each element in correspondence with each other to get a series of the same size. Here we've doubled a set and shown that the doubled, therefore presumably "even" set is infinite in size.Moreover, an odd set might be defined as an even number plus "one man left out"; however, in an infinite set you can always shift one of the sides to cancel out the odd man out. For example, let's say that we divided our infinite numbers into the two sets 2,4,6..."even infinity" and 3,5,7..."odd infinity" with the odd man out being the number 1. However, we can always put the set 4,6,8..."even infinity" in one-to-one correspondence with 2,4,6..."even infinity", leaving 2 to pair with our odd man out 1 and having again an infinite even set.Unfortunately the last argument can be run full speed in reverse with no damage to the transmission. So it's best to say that mathematically infinity is neither odd, nor even: it's just plain weird.
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