The End of an Era - That of The Natural Numbers
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From: Aatu Koskensilta on 9 Jun 2010 01:56 Nam Nguyen <namducnguyen(a)shaw.ca> writes: > And didn't you also say mathematical logic is a branch of mathematics? Yes. > How could the whole thing has _nothing_ to do with something when part > of it has a lot to do with that something? Mathematical logic is not a part of the usual definition of "branch of mathematics". Rather, it is a branch of mathematics. > Didn't you say 'from my explanation that the usual meaning of "branch of > mathematics"'? You seem to be dishonest here. No? I mentioned "my explanation that the usual meaning of 'branch of mathematics' has nothing to do with formal theories". I didn't claim to have offered any explanation of what is usually meant by a branch of mathematics. That you can easily find out for yourself. -- Aatu Koskensilta (aatu.koskensilta(a)uta.fi) "Wovon man nicht sprechan kann, dar�ber muss man schweigen" - Ludwig Wittgenstein, Tractatus Logico-Philosophicus
From: WM on 9 Jun 2010 06:10 On 22 Apr., 05:50, Nam Nguyen <namducngu...(a)shaw.ca> wrote: > It's widely believed our intuition of the natural numbers > has led to foundational understandings of mathematical > reasoning and not the least of which is the validity > of GIT proof. Goedels incompleteness proof presupposes and is based upon the infinite hierarchy of infinities. "Der wahre Grund für die Unvollständigkeit, welche allen formalen Systemen der Mathematik anhaftet, liegt, wie im lI. Teil dieser Abhandlung gezeigt werden wird, darin, daß die Bildung immer höherer Typen sich ins Transfinite fortsetzen läßt [...] während in jedem formalen System höchstens abzählbar viele vorhanden sind. [Kurt Gödel: "Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme I", Monatshefte für Mathematik und Physik 38 (1931) S.173198.] This assumption is wrong. The axiom of power set allows for the construction of set P(M) with larger cardinal number than M, but this is only a must if some counting function phi(omega) is not missing in M. In case phi is missing, there must be the possibility that objective external (though not internal) cardinalities remain the same for M and P(M) and P(P(M)) and P(P(P(M))) and so on in infinity. This is obviously nonsense. Regards, WM
From: Marshall on 9 Jun 2010 09:07 On Jun 8, 8:00 pm, Nam Nguyen <namducngu...(a)shaw.ca> wrote: > > But what are the naturals? A model of PA? I'm sure you know what > circularity means! And I am sure that you don't. Marshall
From: Marshall on 9 Jun 2010 09:12 On Jun 8, 9:55 pm, Nam Nguyen <namducngu...(a)shaw.ca> wrote: > > > That's what I just explained to you. > > Wrong kind of explanation though. At this point, the only kind of explanation that might stand a chance of working with Nam is one that involves whacking him on the head with some sort of large migratory fish. I propose the stately salmon as the best candidate. However I do not hold out much hope even for this method. Marshall
From: Nam Nguyen on 9 Jun 2010 14:48
Aatu Koskensilta wrote: > Nam Nguyen <namducnguyen(a)shaw.ca> writes: > >> Well then he has yet to demonstrate the formula is true in a false >> model (where U is empty). > > To your satisfaction? I doubt that's possible. > No. Demonstrate using set-membership and 2 complementary predicates in an empty U. That's purely technical requirements and whether or one is satisfied is an entirely different matter from the demonstration. If he or you can't, I have the counter demonstration: would he or you like to hear? |