Peano as the finitistic upper bound?

[Marshall]

I was reading this:

http://rationalargumentator.com/issue195/godel.html

Excerpt:

"... John von Neumann [...] held the view that 'Peano
Arithmetic already encompasses all that can be done
finitistically,'"

and

"According to Feferman, most mathematicians today
agree with von Neumann."

True? Has it been proven one way or the other?
Opinions, comments, etc. appreciated.

Thanks,

Marshall

[MoeBlee]

As far as I can tell (I'd welcome any needed correction or
refinement), commonly, PRA is identified with finitistic mathematics,
so, since first order PA encompasses PRA, a fortiori, PA encompasses
finitistic mathematics.

But since 'finitistic' is an informal notion, I don't see how one
would prove (in a formal sense) things about it; though, of course,
one may give various arguments and reasons for adopting different
views of what is or is not finitistic. No?

MoeBlee

[Rupert]

> MoeBlee- Hide quoted text -
>
> - Show quoted text -

I think that what MoeBlee says is correct.

This is an interesting discussion:

http://home.uchicago.edu/~wwtx/finitism.pdf

[Sergei Tropanets]

Hi!
Originaly, Hilbert and Bernays had not exactly define what the
finitary
meaningful reasoning and propositions are, maybe because they had
held over this question and worked on finding any consistency proof,
which is, of course, impossibly to do using any reasonable
intuitively
obvious methods. Since that time three famous logicians, Kreisel,
Parsons and Tait, did efforts to define finitary view strictly by
analysing Hilbert-Bernays intuitive explanations and obtained
different
results: PA, BRA (Bounded Recursive Arithmetic) and PRA
respectively. See chapter 2 from the Stanford Encyclopedia of
Philosophy article:

http://plato.stanford.edu/entries/hilbert-program/

and the works of mentioned logicians referenced there.

Best,
Sergei Tropanets