New essay on Goedel's Incompleteness Theorem
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From: Newberry on 22 Sep 2009 01:11 On Sep 21, 7:33 am, "Scott H" <nospam> wrote: > I give a concise, formal proof and a short intuitive description of ZFC + > ~G. > > http://www.hoge-essays.com/incompleteness.html > > Any constructive feedback is welcome. This looks like a good article. I have not completely digested it yet. I will come back to you later. But here are some preliminary comments: I do not like conflating geometry with mathematics proper. Mathematics proper is analytic, gemometry is synthetic. If you believe Euclid and Kant it is synthetic a priori, if you believe Einstein it is synthetic a posteriori. I do not know if 7 circles on the surface of a sphere do touch. Stating that axioms are self-evident does not cut it. They ought to be implicit definitions of the primitive terms. Then you have a truly analytic system. [But then it is counter-intuitive that there should be unprovable truths. But that is a different issue.]
From: Scott H on 22 Sep 2009 07:54 On Sep 21, 11:06 pm, Newberry <newberr...(a)gmail.com> wrote: > On Sep 21, 7:03 pm, Scott H <zinites_p...(a)yahoo.com> wrote: > > The following is unprovable: The following is unprovable: The > > following is unprovable: ... > > Where is the self-reference in this? There is no actual self-reference in modern mathematics. When people talk about the statement, "This statement is unprovable," they are really giving a metaphorical simplification of the real statement: the one given above. Now, in order to make this endless reference *act like* self- reference, we need omega-consistency and recursive axiomatizability, as described.
From: Scott H on 22 Sep 2009 07:56 On Sep 22, 1:11 am, Newberry <newberr...(a)gmail.com> wrote: > I do not like conflating geometry with mathematics proper. Mathematics > proper is analytic, gemometry is synthetic. If you believe Euclid and > Kant it is synthetic a priori ... Are you sure? Kant believed that even the statement 7 + 5 = 12 is synthetic.
From: Daryl McCullough on 22 Sep 2009 09:41 Scott H says... >The substitution or arithmoquine function, which replaces the free >variable of a property with the symbol for that property, allows an >infinite substitution process to take place. Ultimately, we obtain a >statement similar to: > >The following is unprovable: The following is unprovable: The >following is unprovable: ... I'm not sure why you say that it is an "infinite" substitution. There are two sentences involved in Godel's proof. First, we construct the sentence G. Then we prove (using the construction) that G <-> not Provable(#G) where #G means the Godel code of G. If you want to give not Provable(#G) a new name, G', then you have two sentences: G <-> G' I don't see that there is an infinite sequence of sentences. -- Daryl McCullough Ithaca, NY
From: Newberry on 22 Sep 2009 12:08
On Sep 22, 4:56 am, Scott H <zinites_p...(a)yahoo.com> wrote: > On Sep 22, 1:11 am, Newberry <newberr...(a)gmail.com> wrote: > > > I do not like conflating geometry with mathematics proper. Mathematics > > proper is analytic, gemometry is synthetic. If you believe Euclid and > > Kant it is synthetic a priori ... > > Are you sure? Kant believed that even the statement 7 + 5 = 12 is > synthetic. Yes, but it does not contradict what I said. |