Torkel Franzen on truth
in [Logic]
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From: Aatu Koskensilta on 6 Jan 2008 07:48 On 2008-01-05, in sci.logic, Alan Smaill wrote: > The point of my post was simply to say that I find no evidence > that TF himself thought that the axioms of ZF are manifestly true. > > If you have evidence that suggests otherwise, I'd welcome > a view of that evidence. You will find in the archives several posts in which Torkel explicitly says he finds the axioms of set theory manifestly true. That is of course entirely irrelevant to the point he makes in the quoted passage. -- Aatu Koskensilta (aatu.koskensilta(a)xortec.fi) "Wovon man nicht sprechen kann, daruber muss man schweigen" - Ludwig Wittgenstein, Tractatus Logico-Philosophicus
From: herbzet on 6 Jan 2008 23:41 Aatu Koskensilta wrote: > > On 2008-01-05, in sci.logic, herbzet wrote: > > The further question would be whether every structure is a model > > of some non-null recursively axiomatized theory. > > The answer is yes, quite trivially: given a structure just take any finite, > and hence recursive, set of sentences true in it. Yes, thank you. At this point in the discussion, we have to add that one of the chosen sentences not be a validity. -- hz
From: herbzet on 6 Jan 2008 23:41 herbzet wrote: > george wrote: > > > > > I usually think of "a theory" as having a recursive (or at least r.e.) > > > > > set of axioms. > > > > THAT *IS* the *CORRECT* definition. > > As MoeBlee is explaining, it is NOT the standard one. > > The point is that the standard is just broken. > > Yes, well, it's of a piece with tolerating arbitrary subsets of N or > arbitrary functions from N to N or other "ideal" objects. Which I > don't have a doctrinaire stand on. Although there is, perhaps a slight difference in "level": The ideal objects of various theories presuppose an underlying logic. Their existence follow from given axioms in a given logic. In this sense they are non-problematical. The assumption of the existence of non-recursively specified theories seems not to be the consequence of formally given axioms, but in consequence of meta-theoretical reasoning about the formalisms in which our logical theories are couched. -- hz
From: george on 7 Jan 2008 12:02 On Jan 6, 11:41 pm, herbzet <herb...(a)gmail.com> wrote: > herbzet wrote: > > > > > > I usually think of "a theory" > > > > > > as having a recursive (or at least r.e.) > > > > > > set of axioms. > > Yes, well, it's of a piece with tolerating arbitrary subsets of N or > > arbitrary functions from N to N or other "ideal" objects. With all due respect, you're missing the point. The point is TERMinological, NOT ONTOlogical. > > Which I > > don't have a doctrinaire stand on. Well, I do. You can prove, once you have enough machinery to prove that recursive and rec.enumerable subsets exist, that non-r.e. subsets must exist. Any definable class SMALLER than arbitrary, you can diagonalize out of. So that FORCES you (no pun intended) to "tolerate arbitrary". That is NOT the issue. > The assumption of the existence of non-recursively > specified theories IT is NOT an ASSUMPTION! It is a DEFINITION! > seems not to be the consequence of formally given axioms, Well, it certainly is, in the sense that you can prove, from relevant axioms, the existence of non-r.e. sets. > but in consequence of meta-theoretical reasoning about the > formalisms in > which our logical theories are couched. Just because it's meta-theoretical does NOT mean it is NOT ALSO *theoretical*, SIMULTANEOUSLY. The point being that the SAME ZFC that we study as an object theory is powerful enough TO SERVE as a meta-theory. So the meta- distinction here is moot. THE POINT is that "A theory" is WHATEVER WE SAY a theory is. The point is that we need to start TALKING differently. Currently, you get to have a theory by closing ANY old set of sentences under consequence, EVEN if you CANNOT SPECIFY EITHER of the "before" or "after" sets. "Theory" is arbitrary, so you need another term for theories actually worthy of the name. The generally encountered term for r.e. theories is "formal theory". We just need to redefine "theory" to denote what is currently aimed at by "formal theory" and come up with a more complicated term for "arbitrary" collections of sentences. > > -- > hz
From: MoeBlee on 7 Jan 2008 12:35
On Jan 4, 11:28 pm, herbzet <herb...(a)gmail.com> wrote: > MoeBlee wrote: > For any structure S in a language L the structure assigns a truth-value > to every sentence of L. For every sentence phi of L, one of phi and > ~phi will be true and one will be false in S. On the assumption that > not all sentences of L are validities and their negations, then S will > be a model of some sentences that are not validities, and hence will > be a model of some non-null theory of language L. > > Perhaps you would like to clean that up a bit. It occurred to me Friday night that I spaced out the obvious and trivial: Let TH(M) be the set of sentences true in the model M. TH(M) is a complete, consistent theory and a proper superset of the set of validities. (I think what you wrote above might be along the same lines.) > The further question would be whether every structure is a model > of some non-null recursively axiomatized theory. Maybe just adopt one very simple axiom, such as an atomic sentence, or something like that? > > They are famous and important. Most famous in that way is perhaps the > > theory that is the set of sentences true in the standard model for the > > language of PA. It is a famous theorem that that theory is not > > recursively axiomatized (or 'not recursively axiomatizable' if you > > prefer). That there are theories that are not recursively axiomatized > > (or 'not recursively axiomatizable' if you prefer) is a famous and > > important subject in mathematical logic. > > Yes, I've heard of that theory. It still seems a little imaginary to > me, not being recursively axiomatizable. It may seem imaginary to you, but it is an extremely important subject in mathematical logic. > > > Yuh, I guess the argument turns on whether you want to affirm > > > > a) 2 = 1 + 1 is false in no model (of PA) > > > > or > > > > b) 2 = 1 + 1 is false in some models (of the language of PA). > > > I affirm both. > > But not, I hope, without the parenthetical qualifications. > That would be very confusing. What is confusing? If one knows the definitions, both a) and b) are clearly true. model of a theory vs. model for a language. Those are clearly distinct. > > > Actually, I find it even more annoying when someone is being a jerk > > > while being right! > > > I guess that's a joke or something. Of course, it's subjective what > > annoys a person. But it does seem disconnected to me to be more > > annoyed by someone who is at least correct. > > I can assure you from long experience, I'd rather play chess with > a poor loser than a swaggering, ungracious winner. Ah, yes, chess does have its own special poignant psychology. MoeBlee |