Recent content by Nagase

Since Hilbert wanted to convince Brouwer that classical mathematics was consistent, he actually proposed to employ as a background logic only means that Brouwer accepted. So no, formalists are not tied to classical logic, though, depending on your purposes, that may not be much of a difference...

Actually, formalists generally do not use any set theory. Hilbert, for instance, employed a very weak form of arithmetic, enough to code a sufficient amount of syntax. Hilbert did not specify exactly what theory he was employing, but William Tait has argued rather forcefully that it's probably a...

Just a quick comment: most strict finitists or ultrafinitists actually work in much weaker systems than (say) Z+the negation of infinity. In fact, they generally work in a version of Robinson arithmetic, since even Peano Arithmetic is too strong for their tastes (it allows induction over...

About those opposed to Cantorian set theory, I gave you a (partial?) answer in post #35; do you have any further questions about that? As for when Cantor's ideas became mainstream, I'd say probably in the 30's, when it became clear the limits of logic set theory in general (due to Gödel's...

You're welcome. Fortunately, the book is available as a very cheap Dover paperback, so it won't cost you much!

I suggest reading Tarski, Mostowski, and Robinson's Undecidable Theories, which shows that (a) Robinson's arithmetic is the weakest finitely axiomatizable theory which is essentially undecidable (cf. Theorem 11, where they prove that the omission of any of its seven axioms makes the theory...

A couple of things: 1) First, let's remember what "after" and "before" mean in this context (I'll just assume we're working with von Neumann ordinals). ##x## is after ##y##, that is, ##x > y## iff ##y \in x##; we also say in this case that ##y## is before ##x##. 2) Now, given an ordinal ##x##...

I'm not sure what "right before it" means. Supposedly,you mean that ##x < \omega## and such that there is no ##y \not = x## such that ##x < y < \omega##? If so, nothing satisfies these conditions, so 3. is the answer. Also, what do you mean by "well ordered continuedness does not work backwards"?

As far as I know, Wittgenstein never changed his mind regarding his finitism (if I'm not mistaken, he only accepted as legitimate a very weak form of arithmetic known as Primitive Recursive Arithmetic); similarly, I don't think Poincaré changed his mind about his "intuitionism", which...