How many numbers exist? Infinity proof moves math closer to an answer

Most importantly, the result strengthens the case against the continuum hypothesis, a hugely influential 1878 conjecture about the strata of infinities. Both of the axioms that have converged in the new proof indicate that the continuum hypothesis is false, and that an extra size of infinity sits between the two that, 143 years ago, were hypothesized to be the first and second infinitely large numbers.

“We now have a coherent alternative to the continuum hypothesis,” said Ilijas Farah, a mathematician at York University in Toronto.

The result is a victory for the camp of mathematicians who feel in their bones that the continuum hypothesis is wrong. “This result is tremendously clarifying the picture,” said Juliette Kennedy, a mathematical logician and philosopher at the University of Helsinki.

But another camp favors a different vision of infinite mathematics in which the continuum hypothesis holds, and the battle between these sides is far from won.