The End of an Era - That of The Natural Numbers
in [Math]
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From: Nam Nguyen on 16 Jun 2010 00:18 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)?
From: Nam Nguyen on 16 Jun 2010 00:51 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? > > > Marshall > > PS. Or "Ax:x=x" if you want to get pedantic, which pretty much > no one here ever does. Why "or", given being an open formula and being a closed one are so drastically different in semantic that they don't mean the same thing?
From: Marshall on 16 Jun 2010 02:00 On Jun 15, 9:13 pm, Nam Nguyen <namducngu...(a)shaw.ca> wrote: > Daryl McCullough wrote: > > Nam Nguyen says... > >> Marshall wrote: > > > 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, > > Right. For instance, ~(1+1=0) isn't true in all structures. > > > which is to claim that there is a structure for L in which > > the negation is true. > > Not necessarily. Yes, necessarily. Marshall
From: Marshall on 16 Jun 2010 02:08 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. 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). 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. Failing that, I shall consider the claim to still stand. Marshall
From: Nam Nguyen on 16 Jun 2010 02:12
Marshall wrote: > On Jun 15, 9:13 pm, Nam Nguyen <namducngu...(a)shaw.ca> wrote: >> Daryl McCullough wrote: >>> Nam Nguyen says... >>>> Marshall wrote: >>> 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, >> Right. For instance, ~(1+1=0) isn't true in all structures. >> >>> which is to claim that there is a structure for L in which >>> the negation is true. >> Not necessarily. > > Yes, necessarily. An your explanation is ...? Btw, have you found the answer to the below question put forward to you in our previous conversation? Marshall: > x=x > > A lovely little statement that happens to be true in every model. Nam: > So what's your definition of an open formula being true in a model, > to begin with? |