Todo: Yablo's paradox

Infinities are one of the most complex and confounding topics in mathematics and they lead to an absurd number of paradoxes. However, many of the paradoxes turn out to be variations on the same theme once you dig into what is actually happening. I will provide informal hints on how surreal numbers could help us solve some of these paradoxes, although the focus on this post is primarily categorisation, so please don't mistake these for formal proofs. I'm also aware that simply noting that a formalisation provides a satisfactory solution doesn't philosophically justify its use, but this is also not the focus of this post. I plan to update this post to include new infinite paradoxes that I learn about, including ones that are posted in the comments.

Resolution Paradoxes

Infinitarian Paralysis: Suppose there are an infinite number of people and they are happy so there is infinite utility. A man punches 100 people and destroys 1000 utility. He then argues that he hasn't done anything wrong as there was an infinite amount utility before and that there is still an infinite utility after. What is wrong with this argument?

If we use cardinal numbers, then we can't make such a distinction. However, if we use surreal numbers, then n+1 is different from n every when n is infinite.

Paradox of the Gods: "A man walks a mile from a point α. But there is an infinity of gods each of whom, unknown to the others, intends to obstruct him. One of them will raise a barrier to stop his further advance if he reaches the half-mile point, a second if he reaches the quarter-mile point, a third if he goes one-eighth of a mile, and so on ad infinitum. So he cannot even get started, because however short a distance he travels he will already have been stopped by a barrier. But in that case no barrier will rise, so that there is nothing to stop him setting off. He has been forced to stay where he is by the mere unfulfilled intentions of the gods"

Let n be a surreal number representing the number of Gods. The man can move: 1/2^n distance before he is stopped.

Two Envelopes Paradox: Suppose that there are two sealed envelopes with one having twice as much money in it as the other such that you can't see how much is in either. Having picked one, should you switch?

If your current envelope has x, then there is a 50% chance of rec...

Investigations Into Infinity