As a side note, I personally consider a theory “safe” if it fits with my intuitions and can be proved consistent using primitive recursive arithmetic and transfinite induction up to an ordinal I can picture. So I think of Peano arithmetic as safe, since I can picture epsilon-zero.
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.
I like your proposal, but why not just standard finitism? What is your objection to primitive recursive arithmetic?
As a side note, I personally consider a theory “safe” if it fits with my intuitions and can be proved consistent using primitive recursive arithmetic and transfinite induction up to an ordinal I can picture. So I think of Peano arithmetic as safe, since I can picture epsilon-zero.
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.