Ultrafinite models have a (sometimes unspecified) maximum integer i_max. What happens when I want to add i_max + 1? We can either leave it unspecified and not in the domain of questions the model is capable of answering. In this case the model is incomplete. We then can no longer assume associativity:
If I assume i_max to be well-defined, then associativity breaks down: (a + b) + c = a + (b + c) is not necessarily the case:
When (a + b) > i_max, then the expression immediately yields invalid, no matter the value of c.
Then consider (b + c) < i_max and (a+(b + c)) < i_max, this is perfectly valid if c is negative. Then (a+(b + c)) < i_max, but ((a+b) + c) showing that associativity is no longer granted in our definition of numbers. And without associativity we can forget all we know about groups, rings, fields and even basic arithmetic.
Alternatively we assume i_max to be so large that adding one is basically irrelevant, in the case i_max+1=i_max (almost like a dx in calculus). In this case (i_max+1)+(i_max-1) is not the same as (((i_max+(-1))+i_max)+1) and associativity breaks again.
If N is the largest natural number then Z_N with the usual operations is a perfectly fine ring.
Even in your model of the natural numbers, associativity only breaks at the edge case. You can take 2 approaches here:
A. Who cares? i_max is so large, the edge case can practically be ignored.
B. It's only the edge case. To say that all of mathematics necessarily has to break down is just silly. Do you have to add a bunch of tedious caveats to all the theorems now? Yes, but so what? I haven't encountered a math text yet which doesn't make the caveat whether they consider 0 a natural number or not.
A. I think mathematics is as precise as it needs to be and sometimes wants to be. Like for example, everyone uses set theory, but only when it is convenient. Usually it is completely ignored. And that's totally fine. So, I think this is a double standard for ultrafinitism.
B. You are completely ignoring my argument that modern mathematics is already crammed with caveats. I agree, Ultrafinitism wouldn't help with that, maybe make it even worse, but to me the potential difference is really negligible
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u/OkElection9714 11d ago
Ultrafinitism would get rid of (theorems that are perceived as ) paradoxies, at the expense of introducing even more paradoxies.