'When Are Relations Neither True Nor False?
in [Math]
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From: Newberry on 3 Apr 2010 12:49 On Apr 3, 6:54 am, stevendaryl3...(a)yahoo.com (Daryl McCullough) wrote: > Newberry says... > > >If it absolutely certain that PA is consistent why don't we formalize > >the reasoning? > > It has been. It's easily formalized in ZFC. I do not know why we are going through this circle again. Look it is very simple. All you have to do is to divorce ~(Ex)(Ey)(Pxy & Qy) (1) from ~(Ex)Pxm (2) [No need to repeat that m is the Goedel number of (1).] Then there is no reason why (2) could not be proven. > -- > Daryl McCullough > Ithaca, NY
From: Daryl McCullough on 3 Apr 2010 12:51 Newberry says... > >On Apr 3, 6:54=A0am, stevendaryl3...(a)yahoo.com (Daryl McCullough) wrote: >> Newberry says... >> >> >If it absolutely certain that PA is consistent why don't we formalize >> >the reasoning? >> >> It has been. It's easily formalized in ZFC. > >I do not know why we are going through this circle again. > >Look it is very simple. All you have to do is to divorce > >~(Ex)(Ey)(Pxy & Qy) (1) > >from > >~(Ex)Pxm (2) > >[No need to repeat that m is the Goedel number of (1).] Then there is >no reason why (2) could not be proven. It can be proven. Just not in PA. -- Daryl McCullough Ithaca, NY
From: Nam Nguyen on 3 Apr 2010 12:51 Nam Nguyen wrote: > Nam Nguyen wrote: > >> >> Let me put to rest the idea we know enough about the natural numbers, >> to prove important thing such as the consistency of PA. I'll do that >> by pointing out the existence of a specific unknown natural number. >> >> Let N be the set of natural numbers and R the set of standard reals. > > Please disregard my mentioning about the reals (R) here. Thanks. > >> Let a natural number n be expressed as n = d0d1d2...dn, where d's are >> the decimal digits. Let's also define the following functions: >> >> f1: N -> N, f1(n=d0d1d2...dn) = dn...d2d1d0 >> f2: N -> N, pE(n) = p, where p is the greatest prime <= the even n >> [assuming n >= 0]. Of course it was meant: "[assuming n > 0]". Also since we'd apply f1() only to primes here, the issue of dn = 0 would be a moot point for f1(). >> >> Let S1 = {n | n is an example of GC} >> Let S2 = {n' | n' is a counter example of GC} >> >> Note that at least one of S1, S2 must be infinite. Now if S1 is >> finite of length l > 0, then there is an infinite sequence: >> >> Seq1: n1, n2, ..., nl, 0, 0, 0, ... >> >> where all terms are either in S1 or 0. (If S1 were empty, then all terms >> are defined equal to 0). Similarly, an _infinite_ sequence Seq2 would >> exist, where all terms are either in S2 or defined to be 0. >> >> Let's define the set S as: >> >> Let S = { m | max(f1(f2(nth-term-of-Seq2)),f1(f2(nth-term-of-Seq1))) } >> >> By Well Ordering Principle, S has a minimal number which would be >> the called Un: the desired "unknown" natural number. >> >> To know the natural numbers then is to know the value of Un, which we >> can not know.
From: Daryl McCullough on 3 Apr 2010 12:53 Newberry says... > >On Apr 2, 3:19=A0am, stevendaryl3...(a)yahoo.com (Daryl McCullough) wrote: >> Then, as I said, it's a true sentence that Gaifman's procedure does >> not return true for. > >Gaifman's procedure is not applicable. That's exactly right. That's why it makes no sense to identify truth with provability (or any other procedure). -- Daryl McCullough Ithaca, NY
From: Marshall on 3 Apr 2010 19:16
On Apr 2, 6:38 pm, Transfer Principle <lwal...(a)lausd.net> wrote: > > To repeat, what may be intuitive to one poster may be counterintuitive > to another. And I see no reason to favor one poster's intution over > another's, no matter what the standard theorists try to say. I don't see any reason to pay much attention to anyone's intuition, my own included. "Intuition" is just a fancy word for "hunch." It might be something that suggests areas to investigate, but as far as these sorts of discussions go, it's the beginning of the process, not the end. Marshall |