In a Consistent System, Can a True Sentence Imply a False One?
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From: Daryl McCullough on 6 Aug 2010 10:40 George Greene says... >A lot of this is arising out of cases where there is some intended >model or language in advance. That is NOT appropriate EITHER. I think as long as all those involved are consenting adults, we shouldn't make judgments. >The AXIOMS come first and LOGIC IS ABOUT what follows from the axioms >and/or the rules of inference. The axioms don't *historically* come first. Axioms are always an attempt to capture important facts about some already understood subject area. If you just start with axioms, it's not likely that you'll get anything interesting. I would say that logic today includes both deductive theories and models of languages. -- Daryl McCullough Ithaca, NY
From: MoeBlee on 6 Aug 2010 10:53 On Aug 6, 9:34 am, stevendaryl3...(a)yahoo.com (Daryl McCullough) wrote: > George Greene says... > >On Aug 4, 5:04=A0pm, MoeBlee <jazzm...(a)hotmail.com> wrote: > >> A couple of the other arguments that happen to stand out in my mind > >> were: > > >> (1) He wouldn't agree that there is such a thing as the theory of a > >> model (the set of sentences true in a given model), though it was > >> explained to him over a dozen times and is found in virtually any > >> textbook in mathematical logic. > > >Well, I don't agree with that either. > >Syntax is one thing. Semantics is another. > >The theory is one thing. The model is another. > >When I first brought this literally umpteen years ago, > >the smackdown was "Oh, YOU mean a FORMAL theory". > >The point is that the theory is inherently investigative. It is an > >attempt TO CHARACTERIZE the set of truths of the model IN A FINITE way. > > The model theoretic characterization of what it means for a sentence to > be true in a structure gives a finite characterization of the complete > theory associated with a model. Thank you, Daryl, for making it unnecessary for me to get into another senseless discussion with Greene. MoeBlee
From: MoeBlee on 6 Aug 2010 11:00 On Aug 6, 6:08 am, George Greene <gree...(a)email.unc.edu> wrote: > On Aug 4, 5:04 pm, MoeBlee <jazzm...(a)hotmail.com> wrote: > > > (2) The more general matter that a formula with symbols that > > do not occur in a given set of axioms may still be derivable from that > > set of axioms. > > NO, you do NOT get to derive Pv~P in a language that does not have P > in it. I didn't claim otherwise. I'll leave the rest of your message alone as there is not enough time in the universe to spend in rounds of senseless discussion with you. In the meantime, what I wrote is correct, and as understood by virtually every informed poster back during the original discussion. The language may have '0' as a primitive symbol, but '0' might not be mentioned in the axioms. Still, if 'Ax x=x' is an axiom, then 0=0 is a theorem. It's as simple as that, and not worth an eternity of wasted time arguing with you about it. MoeBlee
From: George Greene on 6 Aug 2010 13:36 On Aug 6, 10:34 am, stevendaryl3...(a)yahoo.com (Daryl McCullough) wrote: > The model theoretic characterization of what it means for a sentence to > be true in a structure gives a finite characterization of the complete > theory associated with a model. A finite "characterization" IS NOT a finite description. Why do people have to invent WHOLE NEW WORDS OUT OF THIN AIR to JUSTIFY THEIR *IGNORANT*BULLSHIT*??? N is infinite and there IS NO finitary characterization of first-order arithmetic truth. You have NOTHING to add to this.
From: George Greene on 6 Aug 2010 13:40
On Aug 6, 10:40 am, stevendaryl3...(a)yahoo.com (Daryl McCullough) wrote: > The axioms don't *historically* come first. Yes, actually, they did. > Axioms are always an attempt > to capture important facts about some already understood subject area. Except IT WASN'T understood! That's THE WHOLE point! It STILL IS NOT understood! It STILL IS NOT known whether Goldbach's conjecture is true! > If you just start with axioms, it's not likely that you'll get anything > interesting. This is IDIOTICALLY false. In point of actual fact, NOBODY HAS EVER started WITH ANYthing OTHER than axioms!!! ALL starts HAVE ALWAYS BEEN from axioms! They just weren't properly understood AS such. You act as though there were, out there, somewhere, some NON- axiomatic description of the true first-order arithmetic of the naturals. THERE ISN'T. ALL math HAS ALWAYS had RULES AND FIRST PRINCIPLES DESCRIBING the things that the investigation is intended to be about! THOSE HAVE ALWAYS BEEN the relevant axioms! |