Spoiler for

It's true that if the cutter makes a slice at X degrees then he will never land at X degrees again because the cut size is irrational. But the slice positions move when a piece of cake containing a previous slice gets cut out and flipped. So he doesn't need to land on X degrees, he needs to land on wherever the slice from X degrees will be moved to when it gets cut out and flipped.

After a cut is made at any arbitrary point X, the cutter will eventually make it around the cake and cut out a piece containing X and flip that point into a new position Y such that the next time around the cake the cutter will land on Y. Y is irrational, but it's equal to a rational number (an integer in fact) times the irrational slice size, so it will be reached in a finite number of cuts. It's possible to have two irrational numbers that are integer multiples of each other, such as sqrt(2) and sqrt(8), so that 2 cuts of size sqrt(2) would reach the point sqrt(8).

After a cut is made at any arbitrary point X, the cutter will eventually make it around the cake and cut out a piece containing X and flip that point into a new position Y such that the next time around the cake the cutter will land on Y. Y is irrational, but it's equal to a rational number (an integer in fact) times the irrational slice size, so it will be reached in a finite number of cuts. It's possible to have two irrational numbers that are integer multiples of each other, such as sqrt(2) and sqrt(8), so that 2 cuts of size sqrt(2) would reach the point sqrt(8).

- 1