2 added 145 characters in body

I think the assertion is basically correct. A finitist in Tait's (or Simpson's, or Hilbert's) sense would not object to a quantifier-free theorem in RCA_0 because there is a quantfier-free (in fact logic-free) proof of it in primitive recursive arithmetic. It would not extend to first-order proofs. A constructivist will not object. Kreisel's argument that finitism should extend to $<\epsilon_0$ still only includes the quantifier-free part, NOT all of first-order PA. In this sense finitism is definitely a subset of constructivism.

It is not contradictory for Hilbert to be in the same category as Brouwer because his point would then be that RCA_0 (or ACA_0 per Kreisel) is then justified as a conservative extension of quantifier-free arithmeticfinitist methods (not that they are themselves finitist).

There is of course a looser sense of "finitist" which is all of PA, because its domain of discourse is just the natural numbers, and not sets. But there are looser senses of "constructive" as well. Sometimes a result in classical set theory is called constructive if it just avoids the axiom of choice!

1

I think the assertion is basically correct. A finitist in Tait's (or Simpson's, or Hilbert's) sense would not object to a quantifier-free theorem in RCA_0 because there is a quantfier-free (in fact logic-free) proof of it in primitive recursive arithmetic. It would not extend to first-order proofs. A constructivist will not object. Kreisel's argument that finitism should extend to $<\epsilon_0$ still only includes the quantifier-free part, NOT all of first-order PA. In this sense finitism is definitely a subset of constructivism. RCA_0 (or ACA_0 per Kreisel) is then justified as a conservative extension of quantifier-free arithmetic.

There is of course a looser sense of "finitist" which is all of PA, because its domain of discourse is just the natural numbers, and not sets. But there are looser senses of "constructive" as well. Sometimes a result in classical set theory is called constructive if it just avoids the axiom of choice!