Wouldnt It be Cool if These Were Equivalent?
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From: Frederick Williams on 1 Jul 2010 05:56 Charlie-Boo wrote: > > >> > > > 2. (allX) |- P(X) > > What is a value "external to the language"? > There are symbols (of the language) and there are things (not of the language) symbolized. Logic text books are wont to point out the difference between London (a place) and 'London' the name of the place. -- I can't go on, I'll go on.
From: Frederick Williams on 1 Jul 2010 09:49 Charlie-Boo wrote: > > In the meantime, if I just said that the universal set is N (which was > my intention, ... Look, (allX) |- P(X) is hard to understand _whatever_ the domain of quantification (is that what you mean by 'universal set'?). Does it mean something like whatever term is substituted for the free variable X in P, the result is provable or what? -- I can't go on, I'll go on.
From: Frederick Williams on 1 Jul 2010 10:19 Charlie-Boo wrote: > > Do you think that the quantifiers refer to values outside of the > universal set? How do you define truth when the expression has a > quantifier? > > (Hmmm . . . If ZFC can't have a universal set, how can it have even > quantifiers??) First order languages "have" quantifiers. Universes of set theory "have" sets (i.e. the sets are individuals in the universe). Are you muddling up these two kinds of "having"? The quantifiers refer to those individuals just as they do in any first order theory. Oh, just to muddle you up further (clearly you like being muddled): nothing says that ZFC doesn't have a set model. -- I can't go on, I'll go on.
From: Frederick Williams on 1 Jul 2010 10:33 Charlie-Boo wrote: > > On Jul 1, 9:49 am, Frederick Williams <frederick.willia...(a)tesco.net> > wrote: > > Charlie-Boo wrote: > > > > > In the meantime, if I just said that the universal set is N (which was > > > my intention, ... > > > > Look, > > > > (allX) |- P(X) > > > > is hard to understand _whatever_ the domain of quantification (is that > > what you mean by 'universal set'?). Does it mean something like > > > > whatever term is substituted for the free variable X in P, > > the result is provable > > > > or what? > > I thought I said it, but does this work: Read it left-to-right, > reading substrings that form familiar mathematical concepts. In this > case we have "(allX)" which is . . . etc. > > Yes, of course it means that. How would you express it as a formal > wff? Wff of _what_? If we are talking about PA then there is a provability predicate to express |-. Let's call it prov, then prov(n) iff |-phi where n is the numeral for the G"odel number of phi. -- I can't go on, I'll go on.
From: Frederick Williams on 1 Jul 2010 11:40
Charlie-Boo wrote: > > But more to the point. Does ZFC have a universal set? It may have. -- I can't go on, I'll go on. |