The End of an Era - That of The Natural Numbers
in [Math]
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From: Jesse F. Hughes on 5 Jun 2010 14:54 "Jesse F. Hughes" <jesse(a)phiwumbda.org> writes: > But you don't seem to understand it. According to that clause, a > (closed) formula ~B is false in M if and only if B is true in M. > Thus, *even in the empty structure*, it cannot be the case that > every formula is false. If a formula B is false in M, then ~M is ^^ I meant ~B. > true in M. -- Jesse F. Hughes "It is not as satisfying to disagree with a book." -- Russell Easterly, on why he argues against set theory without reading a book on set theory.
From: Nam Nguyen on 6 Jun 2010 14:12 Jesse F. Hughes wrote: > Nam Nguyen <namducnguyen(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? >>> Unfortunately, Nam thinks >>> that the clause for non-equation atomic formulas (A is of the form >>> pa_1,...,a_n) applies for arbitrary formulas. >> Unfortunately Jesse just contradicted the fact that "among the binary >> predicate symbols must be the equality symbol =." (Shoenfield, pg. 14) >> and so x=x is just =(x,x) which is of the form pa_1,...,a_n Jesse >> just mentioned. > > Yes, = is a binary predicate symbol. So what? > > The clause I mentioned *specifically* applies for atomic predicates > *other than* =. Why are you saying that? [I know condition iii on previous page does stipulate "other than =", but how would you go from that stipulation to the conclusion the clause applies only "for atomic predicates *other than* ="? Note: on the definition of a structure (pg. 18) he only stated "consists of the following things" but he didn't state "consists of _only_ the following things" and note he had stipulated earlier = must be present in _all_ formal systems.] >> Don't know much about me admitting anything here but I'll never >> understand why he border-lined dishonesty when he said "He _completely_ >> ignores the clause for, say, negation" when he and I had _actually_ >> talked about it! > > But you don't seem to understand it. So there was only _your perception_ of the matter but there wasn't any _fact_ to the matter. I wouldn't spend much time to "debate" this silly thing; I just hope in general one should be more careful in asserting such things when one doesn't have facts to support the assertions. > According to that clause, a > (closed) formula ~B is false in M if and only if B is true in M. Yes in high level that's how it was phrased. Note, as we both seem to have noticed before, this is the typical case where the U of the structure M is _non-empty_. > Thus, *even in the empty structure*, it cannot be the case that every > formula is false. If a formula B is false in M, then ~M is true in M. But how do you support this conclusion if you use the argument that applies only for the typical (non-degenerated) cases where the U is non-empty? [Remember I alluded to something like in the argument H implies C, we'd try to attack C by attacking H or the methods of reasoning, but not attacking C by brining in an irrelevant H'.] > > That is what the clause for negation says. Right. But didn't he assume U be non-empty and a formula be closed, here? > (But I bet a quarter that you still don't get it.) > You still have not shown why x=x is false is wrong in the one case, where U is empty! (I wouldn't bet before knowing the likelihood of losing, if I were you.) If I were you, among the few key issues in his book related to model truth, I'd reflect hard on: - Why he _only_ had a definition of a formula being "valid" in all models of T, and not a formula being "true" in all models of T, on pg. 22? Would that mean in his book, the concept of a formula being true/false in a theory isn't even defined? - Why he chose the open formula x=x as a logical axiom of FOL= (pg 21.), yet on pg. 19 he defined structure truth _only_ for closed formulas? - On pg. 18 he had: We want to define a formula A to be valid in M if all the meanings of A are true in M. Would what he said here apply to _both_ open and closed formulas? And how would what he said here impact your conclusion x=x is true in all cases and my conclusion x=x is false in one particular case where U = {}? [After all, as you've noticed, x=x isn't a closed formula and hence shouldn't even qualify the definition of structure truth on pg. 19!] Once you get over the fact that by a formula's being "valid" in a model of a T he actually meant a formula's being "true" in a model, which is a meta level _quantification_ of all the being "true"'s of the meanings of the formula, then you'd get a hang on why: - both x=x and Ax[x=x] are "true" in a model where U is _not_ empty - both x=x and Ax[x=x] are "true" in a model where U is _empty_. Typically authors might choose to ignore the case where U is empty, but that doesn't mean using set membership, Tarski's concept of "factual truth", Shoendfield's meta level truth "quantification", etc... we can't conclude x=x is false when U is empty, and still stay within reasoning framework of FOL=.
From: Nam Nguyen on 6 Jun 2010 14:15 Nam Nguyen wrote: > Once you get over the fact that by a formula's being "valid" in a model > of a T he actually meant a formula's being "true" in a model, which is > a meta level _quantification_ of all the being "true"'s of the meanings of > the formula, then you'd get a hang on why: > > - both x=x and Ax[x=x] are "true" in a model where U is _not_ empty > - both x=x and Ax[x=x] are "true" in a model where U is _empty_. There was a typo; I meant: > - both x=x and Ax[x=x] are "false" in a model where U is _empty_.
From: Jesse F. Hughes on 6 Jun 2010 20:16 Nam Nguyen <namducnguyen(a)shaw.ca> writes: > Jesse F. Hughes wrote: >> Nam Nguyen <namducnguyen(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.) Again: p. 19 does not define truth for open formulas. I'm skipping the rest because I'm tired of this senseless back-and-forth. -- "You got more out of it than I put into it last night. Who were you thinking of when we were loving last night?" -- Texas Tornadoes
From: Marshall on 6 Jun 2010 22:14
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. Marshall |