The End of an Era - That of The Natural Numbers
in [Math]
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From: Nam Nguyen on 16 May 2010 02:50 J. Clarke wrote: > > Have you ever taken an abstract algebra course? If not you might want > to. After you have completed it you should understand how vacuous your > whole line of argument is. Like you could technically demonstrate how abstract algebra would lead to the conclusion our knowledge of the naturals is not just an intuition! (Iow, if you really knew what you were talking about, you utterance above doesn't show it!) [Btw, I took 2 semesters of Abstract Algebra as an undergraduate math major.]
From: Nam Nguyen on 16 May 2010 04:38 Nam Nguyen wrote: > Alan Smaill wrote: >> Nam Nguyen <namducnguyen(a)shaw.ca> writes: >> >>> Marshall wrote: >>>> On May 13, 7:13 pm, Nam Nguyen <namducngu...(a)shaw.ca> wrote: >>>>> ... the fact that nobody could have a single example of a _FOL_ >>>>> absolute (formula) truth. >>>> x=x >>>> >>> Is that formula true in the theory T = {(x=x) /\ ~(x=x)}? >> >> What does "true in a theory" mean? > > How about true in all models of a consistent theory (formal system). > >> >> Please be precise. >> It's unfortunate my opponents don't see that there's no absolute truth for FOL formulas and that the knowledges of the naturals is of intuitive nature. It's unfortunate because if they adhere to fundamental definitions in FOL reasoning such as that of formulas, of inference rules, or of being true, then they'd clearly see why they've been quite incorrect. Instead labeling their opponents with all sort of names, I wish they just give, as you mentioned, _precise_ technical definition of what "being true" is. Because in doing so, they'd understand why they have taken FOL reasoning for granted.
From: Marshall on 16 May 2010 08:44 On May 15, 11:42 pm, Nam Nguyen <namducngu...(a)shaw.ca> wrote: > Marshall wrote: > > > It's true in all models. > > The question was whether or not x=x true or false in the > inconsistent theory T = {(x=x) /\ ~(x=x)}? Your utterance > above is NOT an answer (i.e. irrelevant) to the question. By your own definition of "true in a theory", this is not the question. Your definition was "true in all models of a consistent theory." Since T is not consistent, the meaning (by *your* definition) of "true in T" is undefined. Of course, all the while you've been babbling incoherently about absolute truth, you've never said what you mean by that. Marshall
From: Nam Nguyen on 16 May 2010 09:57 Marshall wrote: > On May 15, 11:42 pm, Nam Nguyen <namducngu...(a)shaw.ca> wrote: >> Marshall wrote: >> >>> It's true in all models. >> The question was whether or not x=x true or false in the >> inconsistent theory T = {(x=x) /\ ~(x=x)}? Your utterance >> above is NOT an answer (i.e. irrelevant) to the question. > > By your own definition of "true in a theory", this is not > the question. Your definition was "true in all models of > a consistent theory." Since T is not consistent, the > meaning (by *your* definition) of "true in T" is undefined. We live in a binary logic world remember? By _default_, if a formula doesn't meet the definition of being true in a context then it is defined to be false in that context. In technical terms, which I did mention in a conversation with Aatu before, a formula F is true in PA: > F is true <-> (PA isn't inconsistent) and (PA |- F) Just replace PA by my specific inconsistent T and note my "and" above. Do you see now x=x is false in T? > > Of course, all the while you've been babbling incoherently > about absolute truth, you've never said what you mean > by that. I did: you're just too quick engaging in flame war to notice. A formula is absolutely true (in the context of FOL reasoning) if there's no other FOL context that it's false. Do you now understand that x=x is not an absolute truth, as you originally thought?
From: Nam Nguyen on 16 May 2010 10:31
Nam Nguyen wrote: > Marshall wrote: >> By your own definition of "true in a theory", this is not >> the question. Your definition was "true in all models of >> a consistent theory." Since T is not consistent, the >> meaning (by *your* definition) of "true in T" is undefined. > > We live in a binary logic world remember? By _default_, if a formula > doesn't meet the definition of being true in a context then it is > defined to be false in that context. In technical terms, which > I did mention in a conversation with Aatu before, a formula F is > true in PA: > > > F is true <-> (PA isn't inconsistent) and (PA |- F) > > Just replace PA by my specific inconsistent T and note my > "and" above. Do you see now x=x is false in T? Btw, that's not my definition as you mentioned above. Although he used a slightly different word "valid", the definition could be found in Shoenfield's book: "A formula is valid in T if it is valid in every model of T" (Pg. 22) Note also what I had stipulated above with Aatu and what I said to Alan are equivalent. Again, hopefully by now you understand x=x is false in my inconsistent T. |