[FOM] Alternate arithmetic

[Daniel Méhkeri]

Andrei Rodin made a rather different proposal: that a falsification 
of platonism could occur by exhibiting an alternative form of 
arithmetic from practical work with very large (but finite) numbers.
He gave an analogy with non-Euclidean geometry. He says it would 
show there is no such thing as *the* natural numbers.

I agree. This would be extremely damaging to even finitism.

The ultrafinitist program has exhibited a system of "feasible 
arithmetic" in which the exponential function is not totally defined. 
The system is inconsistent, but the inconsistency is infeasible 
in a certain sense. 

What has not been done is to exhibit systems in which the exponential
can take on different values.

For instance I have seen it claimed (but I can't remember where) that

   5^{5^{5^5}} + 7^{7^{7^7}} + 1 is prime

is rather like the continuum hypothesis. But this analogy is faulty.
We can force the continuum to be aleph_1 or something else. But so
far the ultrafinitist can at best claim the question not meaningful.
We don't have a pair of systems where this number is provably prime 
in one and provably composite in the other. Absent these, we can 
still claim that the question of whether this number is prime or not 
has only one possible answer. Therefore it is an objective question, 
unlike CH. 

If it were shown otherwise, I think I would doubt that the good 
lord really did make the integers. 

I believe such an alternate exponential is impossible. Even if 
exponentiation isn't feasible, modular exponetiation is. Any 
alternate exponential would be very strongly constrained by this 
alone. And there are surely other strong constraints of this type.

It also seems to me that it is sufficient to consider alternate
exponentials, since given an exponential we have the Kleene 
universal predicate. So if there are no alternate exponentials, 
there shouldn't be any alternate arithmetic at all.


Daniel Mehkeri


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