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Qiaochu_Yuan comments on Second-Order Logic: The Controversy - Less Wrong
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This isn't my objection personally, but a sufficiently ultra finitist rejects the principle of induction.
I think it's more that an ultrafinitist claims not to know that successor is a total function - you could still induct for as long as succession lasts. Though this is me guessing, not something I've read.
Either that, or they claim that certain numbers, such as 3^^^3, cannot be reached by any number of iterations of the successor function starting from 0. It's disprovable, but without induction or cut it's too long a proof for any computer in the universe to do in a finite human lifespan, or even the earth's lifespan.