An Ultrafinite Set Theory
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From: Aatu Koskensilta on 6 Mar 2010 05:06 MoeBlee <jazzmobe(a)hotmail.com> writes: > 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?). I've mentioned that the rule of substitution was incorrectly stated for some time, in news once or twice, and in a recent e-mail, I think. > What specifically were you referring to? And would you relate some > more about the historical details? Frederick has already posted a few pointers. -- Aatu Koskensilta (aatu.koskensilta(a)uta.fi) "Wovon man nicht sprechan kann, dar�ber muss man schweigen" - Ludwig Wittgenstein, Tractatus Logico-Philosophicus
From: Aatu Koskensilta on 6 Mar 2010 05:30 Barb Knox <see(a)sig.below> writes: > 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. This depends on what we mean by "Peano Postulates". "Peano arithmetic" almost invariably refers to first-order arithmetic, a formal theory that does have non-standard models. "Peano Postulates" often, and usually always outside logic, refer to the (essentially second-order) axioms presented by Peano -- and these are categorical. -- Aatu Koskensilta (aatu.koskensilta(a)uta.fi) "Wovon man nicht sprechan kann, dar�ber muss man schweigen" - Ludwig Wittgenstein, Tractatus Logico-Philosophicus
From: Frederick Williams on 6 Mar 2010 10:40 Frederick Williams wrote: > > MoeBlee wrote: > > > > 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'? > > Yes. More fully: what I tried to quote from was the 1950 Chelsea publication of the English translation of the second German edition from 1938. It's nice in that it's brief (c. 180 pages) but Church is superior.
From: Frederick Williams on 6 Mar 2010 10:52 RussellE wrote: > 2) Axiom of finiteness - the set U = {u0,u1,u2,...,uk) exists. > u is an element of U iff u is an urelement. What is k? Is 2) an axiom schema, i.e. one axiom for each k = 0, 1, 2, ....? Or is k fixed? If so, as what? What are you assuming about these numbers k (or this number k)?
From: Barb Knox on 6 Mar 2010 14:22
In article <87aaul213y.fsf(a)dialatheia.truth.invalid>, Aatu Koskensilta <aatu.koskensilta(a)uta.fi> wrote: > Barb Knox <see(a)sig.below> writes: > > > 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. > > This depends on what we mean by "Peano Postulates". "Peano arithmetic" > almost invariably refers to first-order arithmetic, a formal theory that > does have non-standard models. "Peano Postulates" often, and usually > always outside logic, refer to the (essentially second-order) axioms > presented by Peano -- and these are categorical. Only if one specifies the use of "the full second-order semantics", which IMO is a bigger hammer than just stipulating "the standard model of N" in the first place. -- --------------------------- | 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 | ----------------------------- |