The End of an Era - That of The Natural Numbers
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From: J. Clarke on 16 May 2010 10:40 On 5/16/2010 2:50 AM, Nam Nguyen wrote: > 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.] And yet you continue to blather about how "our knowledge of the naturals is just an intuition". Define the naturals and the operations on them in terms of what you learned in those classes (assuming you learned anything) and then get back to us on how it's "just an intuition".
From: Jim Burns on 16 May 2010 11:32 Nam Nguyen wrote: > James Burns wrote: >> Nam Nguyen wrote: >>> Nam Nguyen wrote: >>>> herbzet wrote: >>>>> Nam Nguyen wrote: >>>>> >>>>>> In other word there's no absolute truth. >>>>> >>>>> That's absolutely true. >>>> >>>> Relatively speaking. >>> >>> All truths are relative, including this truth. >>> (Kind of rings the bell, doesn't it?) >> >> Please put this statement of yours in context. >> Otherwise, I may not understand it. > > Sure. > >> And then, please put the context in context. > > Sure, 1 more time. > >> And then, continue contextualizing context >> until you're done, of course. > > If I'm sure at any given context, I'd be sure 1 more time. > All of which means I'd be done - by induction reasoning. > >> Thanks in advance. > > You're welcomed. > > *** > > On a more serious note, my > > >>>>> In other word there's no absolute truth. > > only meant _within the context of FOL reasoning_ there's > no such thing as an absolute truth of a formula! My mistake. The last time I exchanged posts with you, you refused to agree that "2+2=5" was false, citing missing context. You neither agreed nor disagreed with my last post on the topic, so I assumed that you still believed "no absolute truth" for a broader range of statements. Even if it does come a bit late, and in a different thread, thank you for conceding my point. Jim Burns > That's all I ever meant and herbzet should have realized that > and not jumped on the bandwagon Oh-Nam-is-making-philosophical- > comment-again, which the "standard theorists" tend to jump, to > hide the fact that nobody could have a single example of a _FOL_ > absolute (formula) truth. > > I hope you understand the context and the situation now.
From: Daryl McCullough on 16 May 2010 11:33 Nam Nguyen says... >>> 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. A "theory" is just a collection of sentences. It doesn't make any sense to ask whether a statement is "true" in a theory. You can ask whether it logically follows from other sentences in a theory, or you can ask whether a statement is true in every model of a theory (which turns out to be the same thing, for first-order logic). I'm sure you can come up with a screwy definition of "true in a theory" such that "x=x" is not true in an inconsistent theory. But why do so? By the *usual* meaning of "true", "x=x" is always true, under every interpretation. -- Daryl McCullough Ithaca, NY
From: Nam Nguyen on 16 May 2010 11:35 J. Clarke wrote: > On 5/16/2010 2:50 AM, Nam Nguyen wrote: >> 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.] > > And yet you continue to blather about how "our knowledge of the naturals > is just an intuition". > > Define the naturals and the operations on them in terms of what you > learned in those classes (assuming you learned anything) and then get > back to us on how it's "just an intuition". > You're clueless in the matter of abstract algebra: the first few chapters and the key subject of discussion in abstract algebra are about the concept of a group. And the naturals as you yourself intuit it is NOT a group. Jesuz! You're lecturing people about the knowledge of natural numbers and abstract algebra?
From: Daryl McCullough on 16 May 2010 11:40
Nam Nguyen says... >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) I think you misquoted. Isn't it "A formula is valid in T if it is *true* in every model of T (or "is satisfied by every model of T"). Anyway, in light of the completeness theorem, that definition is equivalent to "A formula is valid in T if it is provable from formulas in T". So any formula provable from pure first-order logic is valid in every theory. >Again, hopefully by now you understand x=x is false in my inconsistent >T. No, that's wrong. -- Daryl McCullough Ithaca, NY |