An Ultrafinite Set Theory
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From: MoeBlee on 4 Mar 2010 13:10 On Mar 3, 11:41 pm, Aatu Koskensilta <aatu.koskensi...(a)uta.fi> wrote: > MoeBlee <jazzm...(a)hotmail.com> writes: > > Thanks. You've said 'in a strictly logical sense' a few times. Would > > you amplify what you mean by that in this context? > > Do we need the Lorentz group in our physical blather about relativity? > Not in any strictly logical sense, in that in physical applications we > can explain away any general reference to the group, by concentrating on > the concrete physical situation at hand. What we need, in a strictly > logical sense, in our physical thinking, are those basic mathematical > principles -- of a theory conservative over PA, say, in which we can do > stuff with sets of naturals, functions on naturals, reals, what have you > -- without which it is impossible to derive the (particular applications > of) the mathematics we make use of in our analysis of concrete physical > situations. (Here "concrete" is to be understood widely, in a rather > attenuated sense.) This observation, in the philosophy of mathematics, > is essentially a counter to the Quinean idea that classical mathematics > is justified because it is a part of and presupposed in our best > scientific stories, and hence we should accept e.g. infinitary set > theory. Unless one endorses such Quinean follies the observation is of > course perfectly consistent with the view that e.g. large large Thanks. I think I get the gist of it (without prejudice as to whether Quine's notions do or don't hold up). Another question, unrelated to this. Somewhere else you mentioned that there was an overlooked technical problem with some rule of logic (substitution? replacement of some sort?). What specifically were you referring to? And would you relate some more about the historical details? MoeBlee
From: MoeBlee on 4 Mar 2010 13:16 On Mar 4, 6:55 am, "Jesse F. Hughes" <je...(a)phiwumbda.org> wrote: > Transfer Principle <lwal...(a)lausd.net> writes: > > (Of course, the easiest way to stop posting "lies" about MoeBlee > > on Usenet is just to stop posting on Usenet, period. But calling > > me a "liar" isn't going to make me disappear that easily, any more > > than calling someone a "crank" makes "cranks" stop posting.) > > An alternative is to stop generalizing. When Moe says something, he > speaks for Moe. And, selfishly speaking, I stress that when "standard theorists" or anyone else says something, then they speak for themselves and not necessarily for Moe. MoeBlee
From: Frederick Williams on 4 Mar 2010 13:51 MoeBlee wrote: > Another question, unrelated to this. Somewhere else you mentioned that > there was an overlooked technical problem with some rule of logic > (substitution? replacement of some sort?). What specifically were you > referring to? And would you relate some more about the historical > details? Is this about the error in Hilbert & Ackermann (1928) that wasn't fixed until Church, Intro. to mathematical logic (1944)?
From: MoeBlee on 4 Mar 2010 19:05 On Mar 4, 12:51 pm, Frederick Williams <frederick.willia...(a)tesco.net> wrote: > MoeBlee wrote: > > Another question, unrelated to this. Somewhere else you mentioned that > > there was an overlooked technical problem with some rule of logic > > (substitution? replacement of some sort?). What specifically were you > > referring to? And would you relate some more about the historical > > details? > > Is this about the error in Hilbert & Ackermann (1928) that wasn't fixed > until Church, Intro. to mathematical logic (1944)? I don't know. Tell me about that please. (Or is there a link to it?) Thanks. MoeBlee
From: Frederick Williams on 4 Mar 2010 19:42
MoeBlee wrote: > > On Mar 4, 12:51 pm, Frederick Williams <frederick.willia...(a)tesco.net> > wrote: > > MoeBlee wrote: > > > Another question, unrelated to this. Somewhere else you mentioned that > > > there was an overlooked technical problem with some rule of logic > > > (substitution? replacement of some sort?). What specifically were you > > > referring to? And would you relate some more about the historical > > > details? > > > > Is this about the error in Hilbert & Ackermann (1928) that wasn't fixed > > until Church, Intro. to mathematical logic (1944)? > > I don't know. Tell me about that please. (Or is there a link to it?) > Thanks. > > MoeBlee Unfortunately I can't find my Church at the moment, but the Editor's Preface to the English translation of H&A's Grundzuge der theoretischen Logik (the second u needs an umlaut) has this: ... the authors failed to include in their statement of the rule of substitution for predicate variables [Rule alpha 3), pp. 69f. of the translation]. This error was pointed out by Professor Alonzo Church in his monograph of 1944 (Introduction to Mathematical Logic) and has been corrected... The bit in square brackets *isn't* my interpolation. The rule alpha 3) is two thirds of a page long and I am dissuaded from typing it since I feel like going to bed! |