Let N be the largest positive integer. Then either N = 1 or N > 1.
If N > 1 then N2 > N, which breaks our definition of N as the largest integer. Therefore N = 1.
“The implications of this paradox are devastating,” writes Laurence Chisholm Young. “In seeking the solution to a problem, we can no longer assume that this solution exists. Yet this assumption has been made from time immemorial, right back in the beginnings of elementary algebra, where problems are solved by starting off with the phrase: ‘Let x be the desired quantity.'”