details re Z-R |- PA consistent
in [Math]
|
From: Nam Nguyen on 6 Jul 2010 22:46 MoeBlee wrote: > William, I have some tough deadlines this week and next, so I can only > >> I am taking this introduction of >> "model theory" to be similar to the introduction of "complex number >> theory" to wrap up a proof in basic number theory > > Okay, let me sort that out: It seems to me that the heart of your > question is to distinguish between two things: > > (1) A Z-R proof of the consistency of PA, in which Z-R proof we refer > only to syntactical aspects of PA (its language syntax and proof > syntax) > > (2) A Z-R proof of the consistency of PA, in which Z-R proof we refer > not only to syntactical aspects of PA but also to semantical aspects > of PA (models). So the "proof of the consistency of PA" would require that PA has models? (May as well just require PA be consistent in the first place!). > > The answer is that in my proof here, I use also semantical aspects of > PA. Apparently you meant (2). Where did you get such an idea for a proof of consistency?
From: Chris Menzel on 7 Jul 2010 09:03 On Tue, 6 Jul 2010 13:15:27 -0700 (PDT), Transfer Principle <lwalke3(a)lausd.net> said: > On Jul 5, 8:34 pm, William Hale <h...(a)tulane.edu> wrote: >> I understand that you [Aatu] don't like the term "work in ZFC" since >> mathematicians don't work in ZFC as such. But, I am using the term >> "work" as a shorthand for what is meant when we say that ZFC serves as a >> foundation for proving things in standard mathematics. >> By standard mathematics, I was limiting myself to the areas of real >> analysis, complex analysis, algebra, topology, and (differential) >> geometry. That is, areas of mathematics discussed before 1900. I didn't >> include logic since its main results were done after 1900. >> Let me give some more on what I mean by "work in ZFC". I think some >> non-mathematicians think that a mathematician comes up with an informal >> proof and that it may or may not be formalized. > > I myself use the phrase "work in ZFC" all the time. But what > do I mean by "work in ZFC"? > > Well, I consider those who _don't_ "work in ZFC" to be those > who contradict it. If you simply prefer to work in a weaker theory, e.g., Z, you are not working in ZFC but you are also not contradicting it either.
From: Aatu Koskensilta on 7 Jul 2010 13:53 Transfer Principle <lwalke3(a)lausd.net> writes: > MoeBlee, Aatu, and the others can say however many times they want to > about how PA is obviously consistent, and yet that hasn't stopped > Nelson from searching for a proof that PA is instead inconsistent. Why would my saying this or that stop anyone from doing anything? > In conclusion, Nelson's work has the potential to vindicate those whom > MoeBlee calls "hopeless cranks" across the board. No it doesn't. Suppose I say the Riemann hypothesis is true because, you see, if we look very carefully at the zeta function, and squint our eyes a bit, we find it's in reality made out of paper clips, hotdog buns and naval lint. Am I vindicated if someone proves the Riemann hypothesis? -- Aatu Koskensilta (aatu.koskensilta(a)uta.fi) "Wovon man nicht sprechan kann, dar�ber muss man schweigen" - Ludwig Wittgenstein, Tractatus Logico-Philosophicus
From: Aatu Koskensilta on 7 Jul 2010 13:57 "Jesse F. Hughes" <jesse(a)phiwumbda.org> writes: > Transfer Principle <lwalke3(a)lausd.net> writes: > >> In this "secret" thread which he tried in vain to keep hidden[...] > > Which he tried to *what*? MoeBlee tried to keep this "secret" thread hidden by publicly posting it on a Usenet newsgroup regularly read by those I presume lwalke thinks he wanted to hide it from. This nefarious plan was foiled by lwalke's vigilance! -- Aatu Koskensilta (aatu.koskensilta(a)uta.fi) "Wovon man nicht sprechan kann, dar�ber muss man schweigen" - Ludwig Wittgenstein, Tractatus Logico-Philosophicus
From: MoeBlee on 7 Jul 2010 14:09
On Jul 7, 12:53 pm, Aatu Koskensilta <aatu.koskensi...(a)uta.fi> wrote: > Transfer Principle <lwal...(a)lausd.net> writes: > > In conclusion, Nelson's work has the potential to vindicate those whom > > MoeBlee calls "hopeless cranks" across the board. > > No it doesn't. Suppose I say the Riemann hypothesis is true because, you > see, if we look very carefully at the zeta function, and squint our eyes > a bit, we find it's in reality made out of paper clips, hotdog buns and > naval lint. Am I vindicated if someone proves the Riemann hypothesis? Interesting that in your argument (pre-Perelman) against the Poincare conjecture you used boomerangs, tongue sandwiches, and navel lint, but here you use naval lint. Personally, I think a more elegant solution is found with air force crumbs. MoeBlee |