A Finite Model of Peano Arithmetic
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From: Aatu Koskensilta on 4 Mar 2010 09:54 Frederick Williams <frederick.williams2(a)tesco.net> writes: > I know, but Russell is taking about FOL and I was suggesting the most > modest, that I am aware of, addition to FOL. Your awareness is sadly limited -- perhaps you've led a sheltered life. Here's a more modest addition: 0 + 1 =/= 0. > Nothing "comes equipped with" any G�del numbering, does it? I can > only guess at what Russell's logical needs are. The Hello Kitty alarm clock I got last Christmas came with batteries included. As for Russell's logical needs, it's a bit pointless to have a random guess at them and then go on about it in news, don't you think? Your actual point is well taken, of course. Which formula defining primality has the smallest G�del number depends on the numbering in question. On any usual numbering, such as we meet in the literature, the obvious formula will probably be among those with the smallest G�del number. -- 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 4 Mar 2010 09:56 Aatu Koskensilta wrote: > Boolos's _The Unprovability of Consistency, An Essay in Modal Logic_ is > a standard reference. I've told you before, my little darling, that's out of date. You(*) want Boolos's The Logic of Provability, CUP. (* That's you in the sense of "people in general but not necessarily Aatu Koskensilta".)
From: Aatu Koskensilta on 4 Mar 2010 10:00 Frederick Williams <frederick.williams2(a)tesco.net> writes: > Yes, in first order _arithmetic_, but not in first order _logic_ (which > is what I read "FOL" as being). Yes, yes, and all that. I was just wondering what you had in mind with your claim that "this statement can't be proven" can't expressed in first-order logic "because provability, being applicable to formulae, isn't first order". The reason we can't express "this statement can't be (logically) proven" in the language of first-order logic -- containing, say, an infinite supply of predicate variables of all arities -- is that nothing whatever can be expressed, in the sense relevant here, in first-order logic, except logical trivialities; or, in other words, the language of pure first-order logic has no intended interpretation. -- 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 4 Mar 2010 10:05 Frederick Williams <frederick.williams2(a)tesco.net> writes: > Aatu Koskensilta wrote: > >> Boolos's _The Unprovability of Consistency, An Essay in Modal Logic_ is >> a standard reference. > > I've told you before, my little darling, that's out of date. You've told me alright! I went out of my way and checked at amazon.com to make sure I had the more up-to-date version, but, alas, my inexcusable ineptitude proved my ultimate undoing in this instance. > You(*) want Boolos's The Logic of Provability, CUP. With utter disregard for your starry footnotes, yes, I want that book. -- 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 4 Mar 2010 10:47
Aatu Koskensilta wrote: > > Frederick Williams <frederick.williams2(a)tesco.net> writes: > > > Yes, in first order _arithmetic_, but not in first order _logic_ (which > > is what I read "FOL" as being). > > Yes, yes, and all that. I was just wondering what you had in mind with > your claim that "this statement can't be proven" can't expressed in > first-order logic "because provability, being applicable to formulae, > isn't first order". The reason we can't express "this statement can't be > (logically) proven" in the language of first-order logic -- containing, > say, an infinite supply of predicate variables of all arities -- is that > nothing whatever can be expressed, in the sense relevant here, in > first-order logic, except logical trivialities; or, in other words, the > language of pure first-order logic has no intended interpretation. Agreed. Just suppose we had what might be called an "applied" FOL with names a0, a1, a2, ... for which the intended interpretations are the formulae phi0, phi1, phi2, ... taken in some order. There may be other names besides. FAFOL(*) also has a one-place predicate P(x) for which the intended interpretation is "the formulae named x is provable". There may be other predicates besides. Axioms are added: P(x) for each provable x and not-P(x) for each non-provable x. Then one could consider the statements P(an) where an is the name of P(an). (Fixed points, so to speak.) But then what, I don't know. (* "Fred's Applied FOL". I rather like the abbreviation, it is pronounced "faffle".) |