'When Are Relations Neither True Nor False?
in [Math]
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From: Newberry on 26 Mar 2010 00:37 On Mar 25, 3:38 am, stevendaryl3...(a)yahoo.com (Daryl McCullough) wrote: > Newberry says... > > >If you take the position that there are truth value gaps then the Liar > >papradox is solvable in English. > > What does it mean to be "solvable" and why do you want it to be solvable? It measn that there is a plasible explantion why there is no inconsistency. I do not like inconsitencies. > > -- > Daryl McCullough > Ithaca, NY
From: Newberry on 26 Mar 2010 00:39 On Mar 25, 3:49 am, stevendaryl3...(a)yahoo.com (Daryl McCullough) wrote: > Newberry says... > > >Tarski's theorem does not apply to formal systems with gaps. I think > >it is preferable. > > If you the way you express Tarski's theorem is like this, then truth > gaps don't change anything: > > There is no formula T(x) such that if x is a Godel code of a true > sentence, then T(x) is true, and otherwise, ~T(x) is true. > > Anyway, *why* is it preferable to have a formal system for which Tarki's > theorem does not apply? Preferable for what purpose? If truth is expressible then truth can be equivalent to provabilty. > > -- > Daryl McCullough > Ithaca, NY
From: Nam Nguyen on 26 Mar 2010 00:47 Marshall wrote: > On Mar 25, 11:00 am, stevendaryl3...(a)yahoo.com (Daryl McCullough) > wrote: >> Ultimately, the people on this newsgroup who object to standard >> mathematics are really objecting to the idea that there can be >> such a thing as a counter-intuitive result. The ultimate logic >> would be one in which it is impossible to prove any result that >> you couldn't already guess was true. > > QFT QFF
From: Newberry on 26 Mar 2010 00:49 On Mar 25, 3:37 am, stevendaryl3...(a)yahoo.com (Daryl McCullough) wrote: > Newberry says... > > >In my logic the Liar paradox can be expressed as follows. > > > ~(Ex)(Ey)(Pxy & Qy) (L) > > >where Pxy means that x is a proof of y, Q is satisfied by only one y = > >m, and m is Goedel number of (L). > > That's not the Liar sentence. The sentence seemingly says about itself that it is not provable. SInce Tarski's theorem does not apply we can equate provability with truth. > > -- > Daryl McCullough > Ithaca, NY
From: Transfer Principle on 26 Mar 2010 00:57
On Mar 22, 5:29 pm, MoeBlee <jazzm...(a)hotmail.com> wrote: > On Mar 22, 7:00 pm, Transfer Principle <lwal...(a)lausd.net> wrote: > > So we must seek out a theory that's either > > at least as powerful as, or at worst as hard to use as, ZFC. > If those are your criteria, then they would govern your > investigations. They're not my criteria though. They're Spight's criteria. Earlier in this thread, he criticizes the so-called "crank" theories as being "less powerful" and "more work to use." Thus I want to find theories which are the opposite of how Spight describes the "crank" theories. > > So we know that in ZFC, if phi(x) is a one-place predicate of the form > > Ayex (psi(y)) for some one-place predicate psi, then phi(0) must hold > > by vacuous truth. There are two ways to avoid this. The first would be > > to change the laws of inference of FOL in order to avoid vacuous, > > and the other would be to change the axioms of ZFC in order to prevent > > the empty set 0 from existing. > But that in itself doesn't block all instantances of vacuous > implication. I stand corrected. Apparently, the proof of "the empty set is a relation" requires only the Axiom of Extensionality (in addition to the FOL rules, of course). > Why don't you scout around for some theory of physics that doesn't (at > least implicity) use mathematics that provides ~P -> (P -> Q). This is the Third Paradox of Material Implication, mentioned several times by others in this thread. It does appear to be the common denominator between Newberry's and Clarke's claims. The Wikipedia link mentioned by others discusses why this may be considered paradoxical, though I'm not sure whether there's anything that can be done in physics to avoid it. |