Mathematicians who are finitists.

[polymath257]

After thinking a bit, this does not quite prove the convergence since you need to show that
|x_n^2 -2|<eps for ALL n>=p where p is that smallest exponent.

Is there a finitistic definition that gets around this issue?

Yes, you're right, but that use of a quantifier is still legal.

n >= p implies that |x_n^2 - 2| <= 1/2^n <= 1/2^p < ε.

OK, so this particular unbounded quantifier is allowed? Why is that? Are quantifiers of the form
for all n>=p, P(n)
allowed?
If so, take p=1 and we have quantification over all of N.

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Why not accept Rosen's definition (page 127), "A set is said to be infinite if it is not finite."  Seems reasonable to me, especially in light of Cantor's diagonalization proof (page 183) that the cardinality of some infinite sets is greater than others.

A good definition, but in a purely finitistic world, there are no infinite sets. We cannot even talk about the set of natural numbers, let alone the set of real numbers, so the diagonalization argument can't even be started.

Once you accept that there *is* a set of real numbers, then you can do the Cantor trick. But it is an *axiom* that there even exists an infinite set. Without that axiom (which the finitists reject), there may only be finite sets.