The End of an Era - That of The Natural Numbers
in [Math]
|
From: Alan Smaill on 16 May 2010 15:58 Nam Nguyen <namducnguyen(a)shaw.ca> writes: > A formula is absolutely true (in the context of FOL reasoning) > if there's no other FOL context that it's false. What is an "FOL context"? > Do you now understand that x=x is not an absolute truth, as you > originally thought? -- Alan Smaill
From: Alan Smaill on 16 May 2010 16:09 Nam Nguyen <namducnguyen(a)shaw.ca> writes: > Alan Smaill wrote: >> Nam Nguyen <namducnguyen(a)shaw.ca> writes: >> >>> William Hughes wrote: >>>> On May 4, 9:30 am, Nam Nguyen <namducngu...(a)shaw.ca> wrote: >>>>> William Hughes wrote: >>>>>> On May 4, 2:21 am, Nam Nguyen <namducngu...(a)shaw.ca> wrote: >>>>>> <snip> >>>>>>>> It is interesting to note that while you have been presented with >>>>>>>> many putative "intuitions" given which the truth or falsehood >>>>>>>> of (1) is knowable (you have accepted none and explicitly rejected >>>>>>>> one), you have not presented a single "intuition" under which the >>>>>>>> truth or falsehood of (1) is not knowable. >>>>>>> That's correct: I have not - yet. That doesn't mean I'm not going to. >>>>>> I'm not holding my breath. >>>>> If you don't have a good faith on that then that's your issue and >>>>> isn't my concern. [Btw, the post about imprecision in reasoning and >>>>> the recent T post are part of the explanation. So in effect I've been >>>>> doing the explanation, whether or not you're listening to.] >>>> Can give an intuition without using multiple >>>> long posts? Forget about Observation 1. >>>> Just give an example of an intuition >>>> Any intuition will do. >>> OK. If you just want a short description intuition about (1) then here >>> it is. >>> >>> Intuitively, to see _either_ cGC or ~cGC as true or false, you have to >>> do the same impossible thing: transverse the entire set of natural numbers >>> to figure it out, hence (again intuitively) it's impossible to know the >>> truth value of cGC, hence of (1). >>> >>> [In contrast, intuitively it's not impossible to see ~GC as true since >>> a counter example is still a distinct possibility. So in principle, >>> we can't say it's impossible to know the truth value of GC, though >>> *IF* GC is genuinely true then intuitively it's impossible to know so.] >>> >>> And that is as short as I could put it. >> >> Do you also have the intuition that it is impossible to see that >> the associativity of addition holds for natural numbers, >> where addition is defined as usual for the recursive definition >> (0 case and successor case)? > > I don't happen to have that intuition. But what would you mean by the > "natural numbers"? Those in which cGC is true? Or ~cGC is true? I don't know if cGC is true or not; that doesn't stop me referring to the natural numbers. The idea that unless we know *everything" about X, then we know *nothing* about X is just weird. Saying you have no intuition for whether addition is associative is also weird, of course. Do you add up your bills in all possible ways, in case one answer that is cheaper than the others? Would you be surprised if one way was cheaper? Do you in fact have an intuition that addition is *not* associative? -- Alan Smaill
From: Alan Smaill on 16 May 2010 16:23 Nam Nguyen <namducnguyen(a)shaw.ca> writes: > 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 Once you supply a definition of "true in a theory" (terminology not generally used) then yes you have a formula not "true in all theories". This doesn't tell us anything about the coherence of the usual usage of "true", "provable" etc, though. if by "no absolute truth for FOL formulas", you mean that the truths of arithmetic are not truths of of pure logic, then that is the standard view these days. Do you see anyone saying that the truths of arithmetic can be derived from logic alone? > and that the knowledges of the naturals is > of intuitive nature. If all you mean is that it goes beyond pure logic, then that is also the normal view these days. I really can't tell what you mean by "intuitive" beyond that, if anything. (When you say that anyone who has any intuition about arithmetic, they must also have some other intuition, then you look like you must have something else in mind.) > 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. Well, Shoenfield *is* precise about these matters. Where do you think he goes wrong? After all, he accepts that truth and provability are distinct notions. -- Alan Smaill
From: Marshall on 16 May 2010 16:29 On May 16, 6:57 am, Nam Nguyen <namducngu...(a)shaw.ca> wrote: > 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? Oh my yes, absolutely. That's why every sentence about the natural numbers is either true or else it's false. You agree with that, don't you? Binary logic world, right? > 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? I see that you have your own personal definition of false in a theory, in which every sentence in an inconsistent theory is false. Sure. > > 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? It's true in every model, and provable in every theory; that's all that matters. Note that I won't be adopting your personal definition of true/false in a theory. Amusing side note: by Nam's definition, all the axioms in an inconsistent theory are false. All the axioms are false by definition, is what he wants. Marshall
From: Alan Smaill on 16 May 2010 16:34
Nam Nguyen <namducnguyen(a)shaw.ca> writes: > 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. It looks like you are claiming Shoenfield's formulation and yours are *equivalent* (after replacing "valid" with "true"). They are not: Shoenfield's version allows a formula to be valid even for an inconsistent T, and yours does not. -- Alan Smaill |