The End of an Era - That of The Natural Numbers
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From: Nam Nguyen on 16 Jun 2010 02:25 Marshall wrote: > On Jun 15, 9:51 pm, Nam Nguyen <namducngu...(a)shaw.ca> wrote: >> Marshall wrote: >>> On Jun 14, 9:42 pm, Nam Nguyen <namducngu...(a)shaw.ca> wrote: >>>> Marshall wrote: >>>>> On Jun 14, 9:25 pm, Nam Nguyen <namducngu...(a)shaw.ca> wrote: >>>>>> Where above did I _claim_ anything exist? >>>>> The claim that x=x isn't always true is a claim that there exists some >>>>> thing that is not equal to itself. >>>> What I asked you: >>>> > So, Marhsall, does the-thing-that-doesn't-equal-itself equal itself, >>>> > mathematically speaking? >>>> So, now you seem to have reversed your answer and claim "Yes" it's true >>>> the-thing-that-doesn't-equal-itself equals itself! >>>> I mean couldn't you give a clear cut answer "Yes" or "no" to my question? >>>> After all, you'd believe whatever the-thing-that-doesn't-equal-itself is >>>> it must equal to itself, right? >>> There is another possibility you haven't mentioned, and that >>> is that there is no thing which is not equal to itself, no matter >>> the model. This happens to be the possibility that is actually true. >>> In formal language, that is written as follows: >>> x=x >>> A lovely little statement that happens to be true in every model. >> So what's your definition of an open formula being true in a model, >> to begin with? > > I've already explained that. I've also already explained how > I don't follow Shoenfield. > > x=x is true in every model. You state before (above) that x=x > happens to be true in every model and now you've "explained": > x=x is true in every model. well then ... > If you want to refute it, the only > refutation I will accept is to be shown some value, call it > "x", in whatever model you choose, such that not(x=x). I'm sorry, Marhsall, in reasoning we call that a circularity and nobody in the right mind would care to consider that, let alone refuting it. > The FOL definition of "=" and "not" shall of course be > required. No making up your own logic, unless you want > to say your claim is specific to your logic. So Shoenfield made up his own FOL logic then? > > Failing that, I shall consider the claim to still stand. Iirc, in many years being here, I think I've seen a few times cranks came up with circularity reasoning argument and never backed down. Hopefully you wouldn't let your own reasoning fall into that category.
From: Marshall on 16 Jun 2010 02:33 On Jun 15, 11:12 pm, Nam Nguyen <namducngu...(a)shaw.ca> wrote: > So what's your definition of an open formula being true in a model, > to begin with? How many times do you need me to say I don't follow Shoenfield? How many times do you need me to say that I'm using the convention that unbound variables are implicitly universally quantified? I guess the answer to these and many other questions about your comprehension is that no finite number is sufficient. My hat is off to Daryl. Marshall
From: Nam Nguyen on 16 Jun 2010 02:36 Marshall wrote: > On Jun 15, 9:51 pm, Nam Nguyen <namducngu...(a)shaw.ca> wrote: >> So what's your definition of an open formula being true in a model, >> to begin with? > > I've already explained that. I've also already explained how > I don't follow Shoenfield. > > x=x is true in every model. You should also try to understand the _very basic_ notion that there's a difference between a definition and an explanation.
From: Nam Nguyen on 16 Jun 2010 02:41 Marshall wrote: > On Jun 15, 11:12 pm, Nam Nguyen <namducngu...(a)shaw.ca> wrote: > > So what's your definition of an open formula being true in a model, > > to begin with? > > How many times do you need me to say I don't follow Shoenfield? > How many times do you need me to say that I'm using the convention > that unbound variables are implicitly universally quantified? > > I guess the answer to these and many other questions about > your comprehension is that no finite number is sufficient. > My hat is off to Daryl. You're mistaken Marshall: whatever you're saying here is just a poor smokescreen for you incompetence to demonstrate x=x is true in a structure with U being non-empty. (There are people who can, you know!)
From: Jesse F. Hughes on 16 Jun 2010 08:33
Nam Nguyen <namducnguyen(a)shaw.ca> writes: > Jesse F. Hughes wrote: >> stevendaryl3016(a)yahoo.com (Daryl McCullough) writes: >> >>> Nam Nguyen says... >>>> Marshall wrote: >>>>> I thought that *you* were the one claiming that x=x is not true in >>>>> all contexts. >>>> I'm still claiming that. What have I just said that made you think >>>> otherwise? >>> To claim that a formula in a language L is not true in all contexts >>> is to claim that there is a structure for L in which the formula is >>> false, which is to claim that there is a structure for L in which >>> the negation is true. There is no such structure. >>> >>> A structure for a language is a way of consistently assigning "true" >>> or "false" to each closed formula in the language. >> >> Given that Nam (allegedly) uses Shoenfield, I think you ought to stick >> to Shoenfield's terminology. An open formula is neither true nor false, >> but is instead either valid or invalid. > > So obviously you implied: > > (a) x=x is "neither true nor false". > >> >> The point remains, of course: x=x is valid, since it is true in every >> interpretation of every structure. > > In "since it is true" it looked like by "it" you meant x=x. So apparently > you meant: > > (b) x=x "is true". > > Why such a contradiction between (a) and (b)? > No contradiction. As a formula, x=x has no truth value. But each interpretation (or M-instance, in Shoenfield's terms) of x=x has a truth value. That is, for every structure and every assignment of x to an element in the structure, the result is true. In Shoenfield's terms, for every structure M, every M-instance of x=x has a truth value. A formula F is valid in M if every M-instance of F is true. It is valid (simpliciter) if it is valid in every such M. Shoenfield even gives an example of a valid formula, right there on p. 20. Know what it is? Yep. It's x=x. The formula x=x is valid for every structure (of its language). That is, for every structure M, every M-instance of x=x is true. You have read the first twenty pages of Shoenfield before declaring that the era of natural numbers has ended, right? -- "Rob Enderle [predicts] that Longhorn will provide 'vast improvements in security.' We can cheer this happy prospect, but at the same time we must ignore the snide laughs of Macintosh users who have yet to encounter a virus..." -- New York Times |