I watched this program the other day

BBC iPlayer - Horizon: 2009-2010: To Infinity and Beyond and while I didn't like most of it, I was intrigued with what Doron Zeilberger had to say (21 minutes 30 seconds into the video). He said that there is no such thing as infinity and that it has no place in mathematics, which on it's own is a reasonable statement, but then actually asserts that there is a finite number that is the largest number that is logically possible to have in mathematics, and then makes the even weirder statement that once you have found that number, adding 1 to it would bring you back to 0. I don't want to know whether anyone thinks that this is right or not, I'd just like to know what could have led Zeilberger to the conclusion that there is a "largest" finite number which it is impossible to go higher than. A search on google didn't find me anything other than links to his research papers which I'm sure I'm not educated enough to understand, so if anyone knows of any form of possible justification for holding this belief then it would be greatly appreciated if you could tell me