In a Consistent System, Can a True Sentence Imply a False One?
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From: Marshall on 6 Aug 2010 17:33 On Aug 6, 11:35 am, stevendaryl3...(a)yahoo.com (Daryl McCullough) wrote: > 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=0 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". As a minor historical point, the axiom was "Axy x=y". I vaguely recall there was some challenge made to axiomatize a condition that implied the model had only a single element. Various sets of three or more axioms were proposed; I then suggested "x=y" as I am generally an impatient person. Proving various terms are equivalent to various other terms is generally quite straightforward using this axiom. Marshall
From: Nam Nguyen on 6 Aug 2010 22:02 Daryl McCullough wrote: > 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". How would you know the everybody didn't actually mishear that somebody because it was too noisy in the room (and what was said was actually 0' and S', instead of 0, and S) ? -- ----------------------------------------------------------- Normally, we do not so much look at things as overlook them. Zen Quotes by Alan Watt -----------------------------------------------------------
From: Nam Nguyen on 6 Aug 2010 22:10 Daryl McCullough wrote: > 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. Is it? > To specify a theory, > one first specifies a *language* for that theory, which means > constant symbols, function symbols, predicate symbols. So what would the standard textbooks suggest as the standard-that-every -one-has-to-conform way of "specifying" this language: in an MS Word file, or in an PDF document? -- ----------------------------------------------------------- Normally, we do not so much look at things as overlook them. Zen Quotes by Alan Watt -----------------------------------------------------------
From: Nam Nguyen on 6 Aug 2010 22:43 Daryl McCullough wrote: > In article <BOE5o.47972$xZ2.27206(a)newsfe07.iad>, Nam Nguyen says... > >> Secondly, not all mathematical truths about the natural numbers are >> definable and hence in general if a particular formula's truth value >> hasn't already been known, there's the possibility it couldn't be >> unknown - in the impossibility sense. > > First of all, it's not the case that the truths of arithmetic are > undefinable. It's just that they cannot be defined using a formula > of arithmetic. You can certainly define the truths of arithmetic using > very weak set theory. So is it impossible to know the truth value of cGC in this definition via the "very weak set theory"? > > Second, it is hard to know how to make sense of the concept of > a statement being *impossible* to know. Does my question Q1 ask about "impossible statement", whatever that might mean? > For any particular formal > system, there are sentences that cannot be proved in that system, > but those sentences can be proved in some other formal system. Right. For instance, if F isn't provable in T then it's provable in T + F. But what does that have to do with anything I said about Q1 and Q2? > > I suppose what you might mean is that there might be a sentence > of arithmetic that can never be proved in any formal system with > "self-evident" axioms. Fwiw, I've actually never cared much for the concept of "self-evidence", and probably never will. What I asked in Q1 is actually very simple: "Is it impossible ... to know the [arithmetic] truth value of cGC". If you actually don't know what "possible to know", or "impossible to know" in the context of mathematical reasoning in FOL, I could find some examples to illustrate the meaning, but really these concepts are as simple as a priori. -- ----------------------------------------------------------- Normally, we do not so much look at things as overlook them. Zen Quotes by Alan Watt -----------------------------------------------------------
From: Nam Nguyen on 7 Aug 2010 12:15
Aatu Koskensilta wrote: > Nam Nguyen <namducnguyen(a)shaw.ca> writes: > The notion of the truth of a formula being impossible to know in some > absolute sense is not unproblematic, and requires elucidation. But what kind of problem would you think there be? If you could give an example of such problem, I'll try to elucidate. > A necessary first step to this line of investigation is to clarify for > ourselves just what sort of absoluteness and relativity is in > question. We don't need -- and indeed can't in general expect -- any > formal explication, but we need a clear informal explanation, detailed > and precise enough that we can be sure we know what we're talking > about. Fair enough. (Though I think I explained before at least once, using SR as an example). Let me make another attempt to illustration of the concept of relativity in mathematical reasoning. In the basic group theory G of the language L(e,*), the truth of the formula F1 = Ax[x*e=x] is an absolute truth, while the truth of of F2 = Axy[x*y = y*x] is a relative truth. The reason being is F1 is true in _all_ models of G, while the truth of F2 is _relative_ to which particular underlying model of G. So in general, if there's a concept C, e.g. the concept of being true in a model of a formal system T, in which a statement S about C is true in _all_ instances of C, then the truth of S would be defined to be "absolute"; if it is not so, the truth of S would be defined as "relative". The *general* idea here is the concept C would comprise of 2 parts: a constant part in which C wouldn't be a concept as such without this part being constant, and a variant part in which the concept C would still be a concept as such, despite the different variances of this part. Similarly, what I've been saying in one way or the other is that the concept of the natural numbers overall _seems_ to comprise of 2 parts: the statements [formulas in L(PA)] that we know for certain the truth values, e.g. 0=0, Ax[~(x=0) -> x > 0], etc..., while the truth value of cGC appears to be assumable without contradicting anything as far as FOL is concerned. At least that's what I'm proposing as a thesis to be accepted as valid in reasoning in FOL. -- ----------------------------------------------------------- Normally, we do not so much look at things as overlook them. Zen Quotes by Alan Watt ----------------------------------------------------------- |