Torkel Franzen on truth
in [Logic]
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From: george on 5 Jan 2008 16:14 On Jan 4, 1:46 am, herbzet <herb...(a)gmail.com> wrote: > That's what's "a new thought to me": that an arbitrary structure > is a model of some non-null theory. I don't happen to know whether > that's true. Of course it's true. Every structure decides every sentence. Pick 1 sentence. Assert the theory with that 1 sentence decided the way the structure decides it as an axiom. Obviously, the structure is a model of that theory. But this depends rather sillily on how you go about describing a structure BEFORE you know what names the axioms are going to use, before you know what signature the language is going to have. Perhaps you have to attach that, first, too. > > > Shall we agree as to what sets of sentences constitute "a theory" first? > > > For simplicity, let's confine to classical first order. > > > Authors such as Enderton take a theory to be any set of sentences > > closed under entailment (which, thanks to the completeness theorem, is > > a set of sentences closed under provability). That is the canonical definition. "Canon" meaning what it means, that is the ONLY definition you are going to get to use, without first explicitly attacking that definition. > > I adopt Enderton's definition. > > OK. It's NOT ok. It does violence to the word (theory). If you can't tell what's an axiom and you therefore can't tell even APPROXIMATELY *whether* some sentence is "in" (declared "provable") by the theory OR NOT, then you don't ACTUALLY HAVE any coherent *theory* (NON-local-technical sense) of what makes sentences "in" the theory true! > > > 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.
From: george on 5 Jan 2008 16:48 > > On Jan 5, 11:48 am, Alan Smaill <sma...(a)SPAMinf.ed.ac.uk> wrote: > >> there are simnply no grounds there for saying that *TF* believed that > >> the axioms of ZFC are clearly true, as you suggest. > > > OF course there are. > > Then please tell me where you find these grounds, > because they are not to be found in the passage cited. OF course they are. Besides, the issue isn't even "truth" to begin with. The issue is TF's insistence on treating "truth" like it is coherent or reasonably understood. > > Even worse, there are grounds for thinking that TF endorses > > the position that "manifestly true" is even meaningful, which > > is, in the context of formal languages, stupid. > > He is not talking about formal languages here; I can't help it if you can read the book carefully and closely and still lie about it. The axioms ARE WRITTEN IN a formal language. He therefore CANNOT avoid being ABOUT formal languages. That IS his subject matter. That IS SO TO what he was talking about. If you are actually reading the book and still disputing this then there is little hope of an actual discussion. > he is assuming, for the > purposes of discussion, that the someone adopts the position that the > axioms are true statements This DOES NOT CHANGE the fact that those statements ARE WRITTEN IN A FORMAL LANGUAGE. > about the (assumed) world of sets. But assuming the existence of any such world is, I repeat, STUPID. If anyone tries to assume THAT then the simple refutation is "what is a set? I don't get these "sets" you're trying to talk about." ALL anyone CAN do by way of replying to that is GIVE A FORMAL LANGUAGE, which IS going to have MULTIPLE interpretations. That is just the end of it.
From: herbzet on 6 Jan 2008 01:19 herbzet 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. > > The further question would be whether every structure is a model > of some non-null recursively axiomatized theory. Assuming again that not every sentence of L is a validity or its negation, take any contingent sentence phi as the sole axiom of theory T1, and take ~phi as the sole axiom of theory T2. Both theories are non-null and finitely axiomized, hence recursively axiomatized, and one of the two theories will be modeled by any structure for L. -- hz
From: herbzet on 6 Jan 2008 01:19 george wrote: > > On Jan 4, 1:46 am, herbzet <herb...(a)gmail.com> wrote: > > That's what's "a new thought to me": that an arbitrary structure > > is a model of some non-null theory. I don't happen to know whether > > that's true. > > Of course it's true. > Every structure decides every sentence. > Pick 1 sentence. Drat. You beat me to "publication". > Assert the theory with that 1 sentence decided the > way the structure decides it as an axiom. The structure could decide it's a false sentence. Picky, picky. > Obviously, the structure is a model of that theory. Or of the theory of the negation of that sentence. > But this depends rather sillily on how you go about > describing a structure BEFORE you know what names > the axioms are going to use, before you know what > signature the language is going to have. > Perhaps you have to attach that, first, too. Yuh, better pick a language/signature first, then define a structure for that. > > > > Shall we agree as to what sets of sentences constitute "a theory" first? > > > > > For simplicity, let's confine to classical first order. > > > > > Authors such as Enderton take a theory to be any set of sentences > > > closed under entailment (which, thanks to the completeness theorem, is > > > a set of sentences closed under provability). > > That is the canonical definition. > "Canon" meaning what it means, that is the ONLY definition > you are going to get to use, without first explicitly attacking > that definition. > > > > I adopt Enderton's definition. > > > > OK. > > It's NOT ok. > It does violence to the word (theory). > If you can't tell what's an axiom and you therefore can't > tell even APPROXIMATELY *whether* some sentence is "in" > (declared "provable") by the theory OR NOT, then you don't > ACTUALLY HAVE any coherent *theory* (NON-local-technical sense) > of what makes sentences "in" the theory true! > > > > > 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. -- hz
From: Aatu Koskensilta on 6 Jan 2008 07:35
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. -- Aatu Koskensilta (aatu.koskensilta(a)xortec.fi) "Wovon man nicht sprechen kann, daruber muss man schweigen" - Ludwig Wittgenstein, Tractatus Logico-Philosophicus |