The End of an Era - That of The Natural Numbers
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From: Nam Nguyen on 12 Jun 2010 20:24 Nam Nguyen wrote: > Daryl McCullough wrote: >> >> I cannot imagine any possible use for your definition of >> truth in the empty model. > > Actually, that's Tarski's definition of _falsehood_ in the empty > model. It also actually has some use in meta reasoning: it proves > that relativity of mathematical reasoning runs deep in the > foundation and that even the truth value of the logical axiom x=x > isn't absolute, as Marshall would believe. (And no, changing it > to Ax[x=x] doesn't change the fact neither truth value is absolute.) > > (If nothing else, there's alway an inconsistent theory that'd be happy > to have this model!) If the Big Bang had been symmetrical then e.g. "sci.logic" couldn't have existed since, say, all the protons would have been uniformly dispersed in all directions and matters couldn't have coalesced into galaxies, planets, etc... In fact if God hadn't desired to create the universe then he would have already applied the most perfect symmetry into the Big Bang: the Emptiness, as perfectly devoid of content as the abstract ZF empty set, so to speak. And in this case there would have been no physics _facts_ hence no _truths_ to make any true physics statements about non-existing facts. There's a similar situation with FOL, in term of model truth and falsehood. T0 = {x=x} is a formal system from which all formal systems must extend. But there's absolutely no model of T0 _without_ a non- logical (contingent) formula formula being true in it. In a nutshell: if there's no contingent truth, there's no logical truth, but when there's contingent truth there's a contingent falsehood, and in meta level the 2 are symmetrically interchangeable as far as FOL reasoning is concerned. For example, Axy[x=y] and ~Axy[x=y] symmetrically can be either true or false, and either of them could be considered as "the standard" truth about, say, "cardinality". This has supported the Principle of Symmetry and has signaled the end of the so called the standard-ness of the naturals. So, if we uphold GIT to the highest level of absoluteness, we'd also uphold the below symmetrical counterpart along side, by renaming "the naturals" "the un-naturals", and vice versa: For any formal system capable of expressing the arithmetic of the naturals, there are statements that are false and provable. Kind of interesting to to observe the similarity between this and some QM theories speculating there are different universes, different from our own in some ways. Reality is hard to fathom isn't it? Well, so is the realm of mathematical abstraction.
From: Nam Nguyen on 12 Jun 2010 20:38 Daryl McCullough wrote: > > No, it doesn't, but I don't actually care what Shoenfield or Tarski > said. What I care about is having a non-stupid definition of "truth > in a model" that applies to models with empty domain. But you shouldn't have worried about that: because that "non-stupid definition" could only render falsehood. Why would you get so concerned about "truths" of the emptiness, anyway? I mean would you be able to discern something in the emptiness that, say, I for one couldn't possibly be able to? > The definition > that you are insisting follows Tarski or Shoenfield is a pointless > definition, and I don't accept it. > > You want to propose that we use some particular definition of truth > in the model with empty domain, say *why* you want to use that definition. > Is there any point? Arguing that Tarski did it is no good, unless you > can reproduce Tarski's reasoning for doing it that way. I _already_ did mention one reason for the "why": the relativity nature of reasoning in FOL.
From: Marshall on 12 Jun 2010 21:27 On Jun 12, 4:02 pm, Nam Nguyen <namducngu...(a)shaw.ca> wrote: > [nothing whatsoever] That's the most intelligent post you've made yet! Marshall
From: Nam Nguyen on 12 Jun 2010 21:36 Nam Nguyen wrote: > Daryl McCullough wrote: >> >> No, it doesn't, but I don't actually care what Shoenfield or Tarski >> said. What I care about is having a non-stupid definition of "truth >> in a model" that applies to models with empty domain. > > But you shouldn't have worried about that: because that "non-stupid > definition" could only render falsehood. Why would you get so concerned > about "truths" of the emptiness, anyway? I mean would you be able to > discern something in the emptiness that, say, I for one couldn't possibly > be able to? > >> The definition >> that you are insisting follows Tarski or Shoenfield is a pointless >> definition, and I don't accept it. >> >> You want to propose that we use some particular definition of truth >> in the model with empty domain, say *why* you want to use that >> definition. >> Is there any point? Arguing that Tarski did it is no good, unless you >> can reproduce Tarski's reasoning for doing it that way. > > I _already_ did mention one reason for the "why": the relativity nature of > reasoning in FOL. There's also another immediate reason related to the weakness of FOL as we accept today. But let me do an IOU an this and would come back at it later.
From: Marshall on 12 Jun 2010 21:37
On Jun 12, 5:38 pm, Nam Nguyen <namducngu...(a)shaw.ca> wrote: > Daryl McCullough wrote: > > > You want to propose that we use some particular definition of truth > > in the model with empty domain, say *why* you want to use that definition. > > Is there any point? Arguing that Tarski did it is no good, unless you > > can reproduce Tarski's reasoning for doing it that way. > > I _already_ did mention one reason for the "why": the relativity nature of > reasoning in FOL. Nam's stupid definition is motivated by his goal of justifying the "relativity nature" of reasoning in FOL. Nam proves the relativity nature of FOL reasoning by using his definition to show it. All the while, Nam accuses others of circular reasoning! Marshall |