In a Consistent System, Can a True Sentence Imply a False One?
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From: George Greene on 7 Aug 2010 17:20 > >How exactly do you think 0 GOT INTO the language if not via an > >axiom??? On Aug 6, 2:35 pm, stevendaryl3...(a)yahoo.com (Daryl McCullough) wrote: > Somebody said something along the lines: > > Consider the language with one constant symbol, 0, one unary > function symbol, S, two binary function symbols, plus and times, > and one binary relation symbol, =. > > Let T be the theory in this language with the single axiom: > "Ax x=x". You are JUST LYING. I'm sorry, NOBODY EVER said this EXCEPT in some VACUOUS context LIKE THIS one. YOUR WHOLE CONTRIBUTION to this debate (it was constructive) was to highlight that SOME SITUATIONS REALLY ARE VACUOUS, REALLY ARE CONTENT-FREE, really ought to be IGNORED A LOT HARDER than they have historically been ignored. E.g.,"Is the empty relation reflexive?" (that's one of those range-vs.-codomain partial-vs.-total questions). IN ANY *REAL* investigation, 0 WOULD HAVE SOME PROPERTIES, it would be something that somebody cared about, it would be either mentioned in the axioms or constructed from something mentioned in the axioms, and if it weren't, ANYTHING YOU COULD PROVE ABOUT 0 WOULD AUTOMATICALLY be universally generalizable. > Ithaca, NY
From: Nam Nguyen on 10 Aug 2010 12:10 Nam Nguyen wrote: > Aatu Koskensilta wrote: >> Nam Nguyen <namducnguyen(a)shaw.ca> writes: > >> The notion of the truth of a formula being impossible to know in some >> absolute sense is not unproblematic, and requires elucidation. > > But what kind of problem would you think there be? If you could give > an example of such problem, I'll try to elucidate. > >> A necessary first step to this line of investigation is to clarify for >> ourselves just what sort of absoluteness and relativity is in >> question. We don't need -- and indeed can't in general expect -- any >> formal explication, but we need a clear informal explanation, detailed >> and precise enough that we can be sure we know what we're talking >> about. > > Similarly, what I've been saying in one way or the other is that the > concept of the natural numbers overall _seems_ to comprise of 2 parts: > the statements [formulas in L(PA)] that we know for certain the truth > values, e.g. 0=0, Ax[~(x=0) -> x > 0], etc..., while the truth value > of cGC appears to be assumable without contradicting anything as far > as FOL is concerned. > > At least that's what I'm proposing as a thesis to be accepted as valid > in reasoning in FOL. > Let me repeat your comment above: >> A necessary first step to this line of investigation is to clarify for >> ourselves just what sort of absoluteness and relativity is in >> question. and add a few more comments of my own on "what sort of absoluteness and relativity is in question". Two facts about modern FOL are well known: a) formal systems with infinitely many non-logical symbols are permissible and b) Godel's work would require, among other things, the underlying formal systems to have only finitely many non-logical symbols. Two difficult questions about modern FOL seem not often reflected: 1) why was there such limit in the Incompleteness and 2) what would Incompleteness be like if we allow the systems to have *non-finitely* many non-logical symbols? (Note I'm careful to use only the word "non-finite", as oppose to "infinite"). Fwiw, I don't have all the details of my own reflections on the questions but the long and short of it is that I see while the "absoluteness" of the concept of the naturals is tolerable in the class of systems with finitely many non-logical symbols, I also see that such concept of the naturals must necessarily be an relative notion in the other class. And to the extend we can't get rid of "non-finitely" many non-logical-symbol systems, overall the concept of arithmetic truths about the naturals would have to be a relative notion. -- ----------------------------------------------------------- Normally, we do not so much look at things as overlook them. Zen Quotes by Alan Watt -----------------------------------------------------------
From: Aatu Koskensilta on 10 Aug 2010 12:15 Nam Nguyen <namducnguyen(a)shaw.ca> writes: > Two facts about modern FOL are well known: a) formal systems with infinitely > many non-logical symbols are permissible and b) Godel's work would require, > among other things, the underlying formal systems to have only finitely many > non-logical symbols. The second well-known fact is simply false. -- Aatu Koskensilta (aatu.koskensilta(a)uta.fi) "Wovon man nicht sprechen kann, dar�ber muss man schweigen" - Ludwig Wittgenstein, Tractatus Logico-Philosophicus
From: Nam Nguyen on 10 Aug 2010 12:25 Aatu Koskensilta wrote: > Nam Nguyen <namducnguyen(a)shaw.ca> writes: > >> Two facts about modern FOL are well known: a) formal systems with infinitely >> many non-logical symbols are permissible and b) Godel's work would require, >> among other things, the underlying formal systems to have only finitely many >> non-logical symbols. > > The second well-known fact is simply false. Well, it doesn't seem to be false to me according to Godel's own remark on his paper (though I have only a translated version). -- ----------------------------------------------------------- Normally, we do not so much look at things as overlook them. Zen Quotes by Alan Watt -----------------------------------------------------------
From: Aatu Koskensilta on 10 Aug 2010 12:31
Nam Nguyen <namducnguyen(a)shaw.ca> writes: > Well, it doesn't seem to be false to me according to Godel's own > remark on his paper (though I have only a translated version). What remark is that? The incompleteness theorems certainly apply to e.g. primitive recursive arithmetic, which has infinitely many non-logical symbols. -- Aatu Koskensilta (aatu.koskensilta(a)uta.fi) "Wovon man nicht sprechen kann, dar�ber muss man schweigen" - Ludwig Wittgenstein, Tractatus Logico-Philosophicus |