The End of an Era - That of The Natural Numbers
in [Math]
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From: Jesse F. Hughes on 6 Jun 2010 23:30 Marshall <marshall.spight(a)gmail.com> writes: > On Jun 6, 5:16 pm, "Jesse F. Hughes" <je...(a)phiwumbda.org> wrote: >> Nam Nguyen <namducngu...(a)shaw.ca> writes: >> > Jesse F. Hughes wrote: >> >> Nam Nguyen <namducngu...(a)shaw.ca> writes: >> >> >>> Jesse F. Hughes wrote: >> >> >>>> x=x is not a closed formula and so nothing from p. 19 of Shoenfield >> >>>> applies to it. >> >>> So Ax[x=x] isn't a logical axiom? >> >> >> x=x is not a closed formula. >> >> > Right. And I've never said otherwise (I've never stated x is bound in x=x >> > or free in Ax[x=x]). >> >> >> That fact has [nothing] to do with >> >> whether Ax[x=x] --- which is a *different* formula --- is an axiom or >> >> not. >> >> > Then, are you saying that, unlike Ax[x=x], x=x isn't true in a model >> > of a consistent T where U is non-empty, simply because x is free in >> > x=x? >> >> I said that p.19 from Shoenfield defines truth in a structure only for >> closed formulas. (Someone else -- Daryl? -- says that according to >> Shoenfield, x=x is in fact *not* true or false, but rather valid, >> because true/false applies only to closed formulas. I assume he's >> correct, but I haven't checked.) > > Apropos of nothing, when I say "x=x" I am referring to the > implicitly universally quantified x=x. That is, Ax.x=x. Just > like when I describe commutativity as "x^y=y^x". > > Prolly a bad habit in a forum that's so oriented on FOL. I don't know whether it's a bad habit or not, but Nam claims to be sticking closely to Shoenfield's presentation, and in that text, the author treats the two formulas differently. -- "Being who I am, I know that's a solution that will run in polynomial time, but for the rest of you, it will take a while to figure that out and know why [...But] it's the same principle that makes n! such a rapidly growing number." James S. Harris solves Traveling Salesman
From: Marshall on 7 Jun 2010 09:50 On Jun 6, 8:30 pm, "Jesse F. Hughes" <je...(a)phiwumbda.org> wrote: > Marshall <marshall.spi...(a)gmail.com> writes: > > > Prolly a bad habit in a forum that's so oriented on FOL. > > I don't know whether it's a bad habit or not, but Nam claims to be > sticking closely to Shoenfield's presentation, and in that text, the > author treats the two formulas differently. Right. Marshall
From: Nam Nguyen on 7 Jun 2010 22:36 Marshall wrote: > On Jun 6, 8:30 pm, "Jesse F. Hughes" <je...(a)phiwumbda.org> wrote: >> Marshall <marshall.spi...(a)gmail.com> writes: >> >>> Prolly a bad habit in a forum that's so oriented on FOL. >> I don't know whether it's a bad habit or not, but Nam claims to be >> sticking closely to Shoenfield's presentation, and in that text, the >> author treats the two formulas differently. > > Right. Of course Shoenfield did treat the 2 formulas differently: he mentioned 1 kind on pg. 19 but not the other kind. But how does that have anything to bear on the arguments here? Specifically, how does such observation help you to prove there's no context in FOL in which x=x (or even Ax[x=x]) is false?
From: Marshall on 7 Jun 2010 22:50 On Jun 7, 7:36 pm, Nam Nguyen <namducngu...(a)shaw.ca> wrote: > Marshall wrote: > > On Jun 6, 8:30 pm, "Jesse F. Hughes" <je...(a)phiwumbda.org> wrote: > >> Marshall <marshall.spi...(a)gmail.com> writes: > > >>> Prolly a bad habit in a forum that's so oriented on FOL. > >> I don't know whether it's a bad habit or not, but Nam claims to be > >> sticking closely to Shoenfield's presentation, and in that text, the > >> author treats the two formulas differently. > > > Right. > > Of course Shoenfield did treat the 2 formulas differently: he mentioned > 1 kind on pg. 19 but not the other kind. But how does that have anything > to bear on the arguments here? Specifically, how does such observation > help you to prove there's no context in FOL in which x=x (or even Ax[x=x]) > is false? See any of the previous hundreds of messages pointing out how completely untenable your position is. My favorite was Phil Carmody's: "I always try skip past your long drawn-out threads, and have never spend enough time to ascertain whether you are a loon or not. I think finally I have the evidence that you really don't have a clue what you're talking about.... *PLONK*." Marshall
From: Nam Nguyen on 7 Jun 2010 22:58
Marshall wrote: > On Jun 7, 7:36 pm, Nam Nguyen <namducngu...(a)shaw.ca> wrote: >> Marshall wrote: >>> On Jun 6, 8:30 pm, "Jesse F. Hughes" <je...(a)phiwumbda.org> wrote: >>>> Marshall <marshall.spi...(a)gmail.com> writes: >>>>> Prolly a bad habit in a forum that's so oriented on FOL. >>>> I don't know whether it's a bad habit or not, but Nam claims to be >>>> sticking closely to Shoenfield's presentation, and in that text, the >>>> author treats the two formulas differently. >>> Right. >> Of course Shoenfield did treat the 2 formulas differently: he mentioned >> 1 kind on pg. 19 but not the other kind. But how does that have anything >> to bear on the arguments here? Specifically, how does such observation >> help you to prove there's no context in FOL in which x=x (or even Ax[x=x]) >> is false? > > See any of the previous hundreds of messages pointing out how > completely untenable your position is. My favorite was Phil Carmody's: > > "I always try skip past your long drawn-out threads, and have never > spend enough time to ascertain whether you are a loon or not. I think > finally I have the evidence that you really don't have a clue what > you're talking about.... *PLONK*." And you call that _your_ _technical_ refutes? |