An Ultrafinite Set Theory
in [Theory]
|
Prev: Now Win a laptop computer from (EZ laptop) free of charge
Next: Prizes worth 104,000 INR at stake for RAD. Certificates to all participants.
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!
From: Barb Knox on 5 Mar 2010 01:27 In article <IqWdnfyuwsZcexDWnZ2dnUVZ_s-dnZ2d(a)earthlink.com>, Patricia Shanahan <pats(a)acm.org> wrote: > Transfer Principle wrote: [snip] > > Also the standard theorists Patricia Shanahan > > William Eliot have also criticized RE for trying to steal > > terminology from the standard theories (such as ZFC and PA) > > and use them in his own theory. > > I'm not a theorist at all, standard or otherwise. I'm a practical > programmer and computer architect. I do think it would reduce confusion > if the term "natural numbers" were used for a structure that does > conform to the Peano Postulates, and other terms were used for > structures that don't. Just a pedantic point, but "natural numbers" should be reserved for the structure that is the *standard* model for the Peano Postulates. There are other (non-standard) structures which also conform. > There are examples of errors in algorithms that may have been due to > thinking "integer", and applying a formula that works for integers, when > the reality is a bounded range number type. > > Patricia -- --------------------------- | BBB b \ Barbara at LivingHistory stop co stop uk | B B aa rrr b | | BBB a a r bbb | Quidquid latine dictum sit, | B B a a r b b | altum videtur. | BBB aa a r bbb | -----------------------------
From: Frederick Williams on 5 Mar 2010 12:09 Frederick Williams wrote: > > 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)? Wrong! Hilbert and Bernays (1934). See below. > > 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, Found him. The sixth paragraph of section 49 reads: Especially difficult is the matter of a correct statement of the rule of substitution for functional variables. An inadequate statement of this rule for the pure functional calculus of first order appears in the first edition of Hilbert and Ackermann (1928). The are better statements of this rule in Carnap's _Logische Syntax der Sprache_ and in Quine's _A System of Logistic_ (1934), but neither of these is fully correct. In the first volume of Hilbert and Bernays's _Grundlagen der Mathematik_ (1934) the error of Hilbert and Ackermann is noted,^{459} and a correct statement of a rule of substitution for functional variables is given for the first time. Church's footnote 459: A revised statement of the rule is given also in the second edition of Hilbert and Ackermann's book (1938, see pp 56-57), but it is still open to to some objection. In the third edition the rule is correctly stated (1949, see pp. 60-61). > 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... That makes no sense, it should be: ... the authors failed to include an essential condition relative to bound variables 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... Sorry. > 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!
From: MoeBlee on 5 Mar 2010 12:23 On Mar 4, 6:42 pm, Frederick Williams <frederick.willia...(a)tesco.net> wrote: > H&A's Grundzuge der theoretischen > Logik Is that the same as 'Principles Of Mathematical Logic'? MoeBlee
From: MoeBlee on 5 Mar 2010 12:30
On Mar 5, 11:09 am, Frederick Williams <frederick.willia...(a)tesco.net> wrote: > ... the authors failed to include an essential condition relative > to bound variables 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... Hmm, I have the '56 version of Church, so I should look there to see the whole discussion. Thanks, MoeBlee |