Do you know Mr. Show’s sketch on the highest number? A group of gangsters discusses whether 24 is actually the highest number or not. Very funny! It is such a crazy idea that 24 could be the highest number… or not?!
When you start counting, you seemingly can go forever. But eventually you will hit the biggest number; and then, when you add 1 to it, you go back to zero.
Interviewer’s reply seems quite surprised:
Interviewer — You… go back to zero?!
Zeilberger — Yeah.
Interviewer — How is that possible?
Zeilberger — How is it not possible? Have you ever been there?
Ultrafinitism — as the name suggests — is an extreme form of finitism. Finitists believe that a mathematical object exists only if it can be construed from natural numbers in a finite number of steps. Ultrafinitists, in addition, deny the existence of the infinite set of natural numbers and pose further physical restrictions in constructing finite mathematical objects. For example, they usually reject that theoretic functions like exponentiation over natural numbers could be total.
An interesting question is: Who bears the burden of the proof? If I claim, for example, that large cardinals exist and you deny it, who has the duty to convince the other? In zoology, if I say that unicorns exist, I have to give convincing proofs of that. This seems to suggest that supporters of large cardinals should give convincing evidences of their claims. On the other side, if I am a physicist and I say that by bombarding a target of bismuth-209 with accelerated nuclei of nickel-64 we can synthesize an atom of rontgenium-272, I must prove that it is really feasible. Zeilberger’s claim seems to be similar to the previous physicist’s claim, since in both cases the question is the possible result of a specific operation. Thus, according to this line of reasoning, he seemingly cannot unload the burden of the proof on his opponents.
The question can be formulated in different terms: which is the less paradoxical position? The apparently more paradoxical position should give convincing proofs that, in spite of the appearances, that is the right position. But even thus the question is not easy to answer. What do you think it is more paradoxical: that there is a natural number whose successor is 0, or that you can break a sphere into a finite number of parts and, by employing those same parts, recompose two identical copies of the original sphere (Banach-Tarski theorem)?