Torkel Franzen on truth
in [Logic]
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From: george on 8 Nov 2007 15:42 On Nov 8, 3:30 pm, kleptomaniac6...(a)hotmail.com wrote: > 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. Not really. The whole point about theorems is that truth doesn't even MATTER for them. Even if the axioms are false, the theorems ARE STILL provable from them.
From: MoeBlee on 8 Nov 2007 16:03 On Nov 8, 7:41 am, Newberry <newberr...(a)gmail.com> wrote: > On Nov 8, 12:26 am, Peter_Smith <ps...(a)cam.ac.uk> wrote: > > > >> ... there is no doubt whatever about the consistency of any of the formal systems we use in mathematics. << p. 105 [Newberry's quote of Franzen] > > 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? To say "there are no doubts" may be understood as a casual way of saying "there is no reasonable basis for doubt" as opposed to an unnecessarily extremely literalistic interpretation that there does not exist in ceratin people the psychological experience of doubt about the constinency of certain formal theories. Franzen's book is written at a very informal level and it is grossly missing the point to split hairs about a non-technical use of such expressions as "there is no doubt", just as when in, say, a debate, someone says, "So there is no doubt whatever that the proposed amendment is too costly", it is not meant literally that there are not people who experience doubt whether the the amdendment is too costly. And, then, with that more reasonable sense ('a reasonable basis for doubt' as opposed to a sweeping claim as to what psychological experiences people have), still Franzen in that particular passage did not say that there are not reasonable bases for doubt but rather that the incompleteness theorem itself does not provide any such reasonable bases. MoeBlee
From: kleptomaniac666_ on 8 Nov 2007 16:05 On Nov 8, 8:41 pm, george <gree...(a)cs.unc.edu> wrote: > 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. Woah. OK con(PA) is different in that it can be proven in PA. But be careful, the second of those two statements is (equivalent to) the RIEMANN HYPOTHESIS, and it is an important unsolved problem in mathematics. We don't know if it is provable in PA or not (wow, usenet bickering is so much fun!). Actually I wasn't specifically talking about con(PA), even if it may have seemed like it. What I was trying to say was that consistency statements in general, though they are intimately related to Godel's theorem, have no particular epistemological relevance compared to other arithmetical statements. As for PA, the fact that con(PA) is not derivable from the axioms of PA is interesting, but it has no different epistemological status to all the other statements which are not provable in PA.
From: kleptomaniac666_ on 8 Nov 2007 16:11 On Nov 8, 8:42 pm, george <gree...(a)cs.unc.edu> wrote: > On Nov 8, 3:30 pm, kleptomaniac6...(a)hotmail.com wrote: > > > 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. > > Not really. The whole point about theorems is that truth > doesn't even MATTER for them. Even if the axioms are false, > the theorems ARE STILL provable from them. How could you possibly disagree with that statement? Maybe I could substitute "the same as for being certain of the truth of any other theorem" for "The same as for any other theorem." Happy now?
From: Newberry on 8 Nov 2007 23:59
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. > > 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). Here is another quote from TF: >> We do of course know the Gödel sentence of, for example PA, to be true since we know PA to be consistent. << p. 117 TF does make the point that the incompleteness theorem does not contradict the view that we know PA/ZFC to be be consistent with absolute certainty. But he also does endorse the view that we know PA to be consistent with absolute certainty (since the axioms are manifestly true.) He clearly believes that the we know PA/ZFC to be consistent with the same certainty as any mathematical theorem i.e. we can prove G. So the question arises how we can construct a machine that can do the same. obviously not by emulating PA/ZFC. |