Torkel Franzen on truth
in [Logic]
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From: george on 8 Nov 2007 09:47 On Nov 8, 1:01 am, Newberry <newberr...(a)gmail.com> wrote: > I am not sure that I understand what Franzen is saying. Don't panic; neither did he. > Is he saying that No. > a) We are absolutely certain about the truths of PA, There is no such thing as a truth of PA. PA is an axiom-set. You prove things from it. There are THEOREMS of PA, things that are PROVABLE from PA. Everything else IS FALSE in AT LEAST ONE model of PA, so there is no point in calling it a truth "of PA". > even those PA cannot prove If PA cannot prove it, then there is a model of PA in which it is false, so it is not a "truth of PA". THEORIES *don't have* "truths". "Truth" comes from MODELS. THEORIES have THEOREMS. > b) The consistency of PA can be proven in ZFC Well, this is true, regardless of whether he meant it. But even there, you have to use epsilon_0 induction. > c) Therefore we can write a computer program emulating ZFC that can > generate the truths of PA No, this is false. Just because PA is consistent does NOT mean there is a computer program that can "generate" all its "truths", especially since there aren't any. There is a program that can recursively enumerate all of PA's THEOREMS, yes, but you don't need as much as ZFC to do *that*. That program is not that complicated (unless you want it to be efficient). > d) We are absolutely certain about the truths of ZFC, even those ZFC > cannot prove Again, ZFC, like PA, IS A THEORY, so it does NOT HAVE "truths". > e) There is a theory X in which we can prove the consistency of ZFC Trivially, X=the-theory-whose-only-axiom-is-"ZFC-is-consistent". Torkel Franzen is not saying any of this (he knows better). Your paraphrases are confused.
From: Newberry on 8 Nov 2007 10:41 On Nov 8, 12:26 am, Peter_Smith <ps...(a)cam.ac.uk> wrote: > On 8 Nov, 06:01, Newberry <newberr...(a)gmail.com> wrote: > > > In "Gödel's theorem" Torkel Franzen disputes that the theorem > > indicates that the human mind surpasses any computer. > > > >> ... the mistaken idea that "Gödel's theorem states that in any consistent system which is strong enough to produce simple arithmetic there are formulas which cannot be proved in the system, but which we can see to be true." The theorem states no such thing. As has been emphasized, in general we simply have no idea whether or not the Gödel sentence of a system is true, even in those cases when it is in fact true. What we know is that the Gödel sentence is true if and only if the system is consistent, and that much is provable in the system itself. << p. 55 > > >> ... there is no doubt whatever about the consistency of any of the formal systems we use in mathematics. << p. 105 > > >> If the axioms of ZFC are manifestly true, they are obviously consistent. << p. 105 > > > I am not sure that I understand what Franzen is saying. > > The first quote you give, I take it is entirely clear (and correct!). > > Your second quote is misleading: What TF in fact wrote was "Nothing in > Gödel's theorem in any way contradicts the view that there is no doubt > whatever about the consistency of any of the formal systems we use in > mathematics." TF isn't there endorsing the view (as your truncated > quotation suggests), he is just pointing out that Godel's theorem > doesn't refute it -- a point evidently consistent with the first > quote. TF is not saying that Gödel's theorem does not contradict the view that the system is consistent. He says it does not contradict the view that there is no doubt. So who is the one that does not have any doubts? > > The third quote you give starts with an emphasized "If" in TF. It is a > triviality (any set of truths is consistent!). > > He is not, at least in those quotations, saying any of (a) to (g).
From: kleptomaniac666_ on 8 Nov 2007 15:30 On Nov 8, 3:41 pm, Newberry <newberr...(a)gmail.com> wrote: > On Nov 8, 12:26 am, Peter_Smith <ps...(a)cam.ac.uk> wrote: > > > > > > > On 8 Nov, 06:01, Newberry <newberr...(a)gmail.com> wrote: > > > > In "Gödel's theorem" Torkel Franzen disputes that the theorem > > > indicates that the human mind surpasses any computer. > > > > >> ... the mistaken idea that "Gödel's theorem states that in any consistent system which is strong enough to produce simple arithmetic there are formulas which cannot be proved in the system, but which we can see to be true." The theorem states no such thing. As has been emphasized, in general we simply have no idea whether or not the Gödel sentence of a system is true, even in those cases when it is in fact true. What we know is that the Gödel sentence is true if and only if the system is consistent, and that much is provable in the system itself. << p. 55 > > > >> ... there is no doubt whatever about the consistency of any of the formal systems we use in mathematics. << p. 105 > > > >> If the axioms of ZFC are manifestly true, they are obviously consistent. << p. 105 > > > > I am not sure that I understand what Franzen is saying. > > > The first quote you give, I take it is entirely clear (and correct!). > > > Your second quote is misleading: What TF in fact wrote was "Nothing in > > Gödel's theorem in any way contradicts the view that there is no doubt > > whatever about the consistency of any of the formal systems we use in > > mathematics." TF isn't there endorsing the view (as your truncated > > quotation suggests), he is just pointing out that Godel's theorem > > doesn't refute it -- a point evidently consistent with the first > > quote. > > TF is not saying that Gödel's theorem does not contradict the view > that the system is consistent. He says it does not contradict the view > that there is no doubt. So who is the one that does not have any > doubts? > > > > > > > The third quote you give starts with an emphasized "If" in TF. It is a > > triviality (any set of truths is consistent!). > > > He is not, at least in those quotations, saying any of (a) to (g).- Hide quoted text - > > - Show quoted text -- Hide quoted text - > > - Show quoted text - Consistency of the system at hand is just another arithmetical statement, like "every prime of the form 4k+1 is a sum of two squares" or "the sum of the divisors of the nth positive integer is less than or equal to Hn + exp(Hn)log(Hn) where Hn is the nth harmonic number". As far as I am aware, the epistemological issues for determining the truth of the consistency of the theory are no different from the issues for those statements I just mentioned. As for doubts, doubt and certainty are human emotions. One could be certain of con(PA) if one could prove that theorem from a list of arithmetical axioms which one felt certain were true. The same as for any other theorem.
From: george on 8 Nov 2007 15:41 On Nov 8, 3:30 pm, kleptomaniac6...(a)hotmail.com wrote: > Consistency of the system at hand is just another arithmetical > statement, like "every prime of the form 4k+1 is a sum of two squares" > or "the sum of the divisors of the nth positive integer is less than > or equal to Hn + exp(Hn)log(Hn) where Hn is the nth harmonic number". No, it is NOT just like THOSE. THOSE are THEOREMS. THOSE are PROVABLE from the axioms of PA and therefore true in all models of PA. The consistency statement for PA is not provable from/in PA.
From: MoeBlee on 8 Nov 2007 15:42
On Nov 7, 10:01 pm, Newberry <newberr...(a)gmail.com> wrote: > In "Gödel's theorem" Torkel Franzen disputes that the theorem > indicates that the human mind surpasses any computer. > > >> ... the mistaken idea that "Gödel's theorem states that in any consistent system which is strong enough to produce simple arithmetic there are formulas which cannot be proved in the system, but which we can see to be true." The theorem states no such thing. As has been emphasized, in general we simply have no idea whether or not the Gödel sentence of a system is true, even in those cases when it is in fact true. What we know is that the Gödel sentence is true if and only if the system is consistent, and that much is provable in the system itself. << p. 55 > >> ... there is no doubt whatever about the consistency of any of the formal systems we use in mathematics. << p. 105 > >> If the axioms of ZFC are manifestly true, they are obviously consistent. << p. 105 > > I am not sure that I understand what Franzen is saying. Is he saying > that > > a) We are absolutely certain about the truths of PA, even those PA > cannot prove What do you mean by "the truths of PA"? There are theorems of PA, and they are true in any model in which the axioms of PA are true. And, for each model for the langauge of PA, there is the set of truths in that model; specifically, there is the set of truths in the standard model for the language of first order PA. > b) The consistency of PA can be proven in ZFC ZFC proves first order PA is consistent. I would think Franzen agrees. > c) Therefore we can write a computer program emulating ZFC that can > generate the truths of PA What is a "program emulating ZFC"? And, again, you say, "truths of PA". If you mean the theorems of first order PA, and if by "generate" you mean recursively enumerate, then, yes, there is a recursive enumeration of the theorems of first order PA. If you mean the sentences true in the standard model of the language of first order PA, then it follows from the incompleteness theorem that there is no recursive enumeration of the set of sentences true in the standard model of the language of first order PA. > d) We are absolutely certain about the truths of ZFC, even those ZFC > cannot prove What do you mean by a "truth of ZFC"? If you mean the theorems of ZFC, then note that "a theorem of ZFC that cannot be proven in ZFC" is an oxymoron. (And there is no sentence S of any language such that there is no theory that proves S.) So only you can say what you mean by "the truths of ZFC". > e) There is a theory X in which we can prove the consistency of ZFC For any theory T (even an inconsistent T) there exists theories that prove the consistency of T. That's trivial. "Theory X proves the consistency of theory T" is not necessarily a very "substantive" claim. > f) Therefore we can write a computer program emulating X that can > generate the truths of ZFC Again, what are "the truths of ZFC"? However, since ZFC is a recursively axiomatized theory, there is a recursive enumeration of the theorems of ZFC. > g) We are not certain about the truths of X > ?? Again, for a theory X, what do you mean by "the truths of X"? MoeBlee |