Peano as the finitistic upper bound?
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From: Marshall on 3 Jan 2010 15:30 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
From: MoeBlee on 3 Jan 2010 17:49 On Jan 3, 12:30 pm, Marshall <marshall.spi...(a)gmail.com> wrote: > 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. 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
From: Rupert on 5 Jan 2010 00:16 On Jan 4, 9:49 am, MoeBlee <jazzm...(a)hotmail.com> wrote: > On Jan 3, 12:30 pm, Marshall <marshall.spi...(a)gmail.com> wrote: > > > > > > > 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. > > 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- 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
From: Sergei Tropanets on 7 Jan 2010 13:12
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 Marshall wrote: > 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 |