In a Consistent System, Can a True Sentence Imply a False One?
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From: Daryl McCullough on 6 Aug 2010 14:32 George Greene says... > >On Aug 6, 11:00=A0am, MoeBlee <jazzm...(a)hotmail.com> wrote: >> The language may have '0' as a primitive symbol, but '0' might not be >> mentioned in the axioms. > >That is debatable to some and ridiculous to me. This is a silly thing to argue about. To specify a theory, one first specifies a *language* for that theory, which means constant symbols, function symbols, predicate symbols. Then once the language is specified, the rules of first-order syntax determine the set of formulas for the language. Then the axioms are a subset of those formulas. There is no particularly good reason to require that the axioms must actually mention every symbol appearing in the language. I suppose you could require that, but why would you? What's the point? -- Daryl McCullough Ithaca, NY
From: Daryl McCullough on 6 Aug 2010 14:35 George Greene says... > >On Aug 6, 11:00=A0am, MoeBlee <jazzm...(a)hotmail.com> wrote: >> 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=3D0 is = >a >> theorem. > >How exactly do you think 0 GOT INTO the language if not via an >axiom??? Somebody said something along the lines: Consider the language with one constant symbol, 0, one unary function symbol, S, two binary function symbols, plus and times, and one binary relation symbol, =. Let T be the theory in this language with the single axiom: "Ax x=x". -- Daryl McCullough Ithaca, NY
From: Daryl McCullough on 6 Aug 2010 14:46 George Greene says... > >On Aug 6, 10:40=A0am, stevendaryl3...(a)yahoo.com (Daryl McCullough) >wrote: >> The axioms don't *historically* come first. > >Yes, actually, they did. No, they don't. People did arithmetic a long time before Peano formalized it with axioms. >> 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! That has nothing to do with whether arithmetic is axiomatized or not. Goldbach introduced his conjecture way back in 1742. Peano wasn't born until 1835. Nobody needed his axioms in order to understand what GC meant, or to attempt to prove it or refute it. >> 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! That's just not true. When children learn arithmetic, they are not taught it as deductions from axioms. Maybe you want to say that it is somehow equivalent to deduction from axioms, if you look at it in the right way. That might be true, but it isn't so obviously true as to make the contrary opinion "IDIOTICALLY false". -- Daryl McCullough Ithaca, NY
From: Daryl McCullough on 6 Aug 2010 14:49 George Greene says... > >On Aug 6, 10:34=A0am, 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. It certainly is. >Why do people have to invent WHOLE NEW WORDS OUT OF THIN AIR >to JUSTIFY THEIR *IGNORANT*BULLSHIT*??? Okay, I've had enough of you George. I will give it a break of 4 weeks or so before I respond to any of your posts. -- Daryl McCullough Ithaca, NY
From: MoeBlee on 6 Aug 2010 15:12
On Aug 6, 1:32 pm, stevendaryl3...(a)yahoo.com (Daryl McCullough) wrote: > George Greene says... > >That is debatable to some and ridiculous to me. > > This is a silly thing to argue about. There is insanity and then there is INSANITY. Trying to reason with George Greene is INSANE. MoeBlee |