Mathematicians who are finitists.

[Jehanne]

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.

[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.

---

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.

[A Toy Windmill]

OK, so this particular unbounded quantifier is allowed? Why is that? Are quantifiers of the form
for all n>=p, P(n)
allowed?

Quantifiers are allowed if they're all outermost. Such quantifiers are then dropped, and you're left with a statement that just has free variables.

When every statement in every proof has this form, you can treat them all as schema or templates. That means that, for my proof above, if someone has concrete numerals for the numerator and denominator of ε, I can plug them in and run that proof down to applied axiom schemes for primitive recursive arithmetic.

I can't do this if quantifiers ever appear left of an implication, as in, say, the principle of mathematical induction:

for any property P, if P holds of 0, and holding of n implies holding of (n+1), then P holds for all natural numbers.

Primitive recursive arithmetic cannot have this as an axiom, because the quantifier position is illegal. Instead, it takes induction as an inference rule.

[Jehanne]

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.

Without axioms, there is no mathematics.

[polymath257]

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.

Without axioms, there is no mathematics.

Yes, precisely. So the question is *which* axioms. How do we choose the axioms? And it seems different people have different opinions on how to choose.

[Jehanne]

Without axioms, there is no mathematics.

Yes, precisely. So the question is *which* axioms. How do we choose the axioms? And it seems different people have different opinions on how to choose.

Ultimately, consensus among experts. I doubt that finitism even has a chapter in any number theory, abstract alebgra or complex analysis textbook, but, I admit that I could be wrong on this one.

[polymath257]

Yes, precisely. So the question is *which* axioms. How do we choose the axioms? And it seems different people have different opinions on how to choose.

Ultimately, consensus among experts.  I doubt that finitism even has a chapter in any number theory, abstract alebgra or complex analysis textbook, but, I admit that I could be wrong on this one.

Well, I wouldn't expect to see much about finitism in such textbooks. I have, however, seen good discussions in introductory books on axiomatic set theory and the issues come up extensively in proof theory. Kunen's book on the Foundations of Mathemtics talks quite a bit about finitism, formalism, and Platonism in the context of introductory set theory. The point is that the meta-theory used to discuss proofs tends to be finitistic by nature.

[Jehanne]

Ultimately, consensus among experts.  I doubt that finitism even has a chapter in any number theory, abstract alebgra or complex analysis textbook, but, I admit that I could be wrong on this one.

Well, I wouldn't expect to see much about finitism in such textbooks. I have, however, seen good discussions in introductory books on axiomatic set theory and the issues come up extensively in proof theory. Kunen's book on the Foundations of Mathemtics talks quite a bit about finitism, formalism, and Platonism in the context of introductory set theory. The point is that the meta-theory used to discuss proofs tends to be finitistic by nature.

Fact is that most professional mathematicians accept ZFC and the proofs of Cantor, which was the whole point of my OP. I don't think that it is an either/or proposition -- one can accept ZFC while having intelligent conversations about finitism; without having to reject the former while working within the confines of the latter.

[polymath257]

Well, I wouldn't expect to see much about finitism in such textbooks. I have, however, seen good discussions in introductory books on axiomatic set theory and the issues come up extensively in proof theory. Kunen's book on the Foundations of Mathemtics talks quite a bit about finitism, formalism, and Platonism in the context of introductory set theory. The point is that the meta-theory used to discuss proofs tends to be finitistic by nature.

Fact is that most professional mathematicians accept ZFC and the proofs of Cantor, which was the whole point of my OP.  I don't think that it is an either/or proposition -- one can accept ZFC while having intelligent conversations about finitism; without having to reject the former while working within the confines of the latter.

Philosophically, I am a formalist. You tell me which axioms to work with, and I will work with them.

I *prefer* a set of axioms that has classes and a single choice function for all sets, but that's just me. I can live with ZFC.

That said, working in the hereditarily finite sets has advantages in some contexts.

[A Toy Windmill]

Fact is that most professional mathematicians accept ZFC and the proofs of Cantor, which was the whole point of my OP. I don't think that it is an either/or proposition -- one can accept ZFC while having intelligent conversations about finitism; without having to reject the former while working within the confines of the latter.

Yes, we can. Please try to do so next time, without making silly comparisons to creationism.