The End of an Era - That of The Natural Numbers
in [Math]
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From: Aatu Koskensilta on 16 Jun 2010 08:53 "Jesse F. Hughes" <jesse(a)phiwumbda.org> writes: > 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. Shoenfield requires the domain of a structure to be nonempty. Of course, nothing in the definition of truth in a structure actually depends on this, and if we relax the requirement we do indeed find that every M-instance of x=x is true in a structure with empty domain. -- Aatu Koskensilta (aatu.koskensilta(a)uta.fi) "Wovon man nicht sprechan kann, dar�ber muss man schweigen" - Ludwig Wittgenstein, Tractatus Logico-Philosophicus
From: Jesse F. Hughes on 16 Jun 2010 09:11 Aatu Koskensilta <aatu.koskensilta(a)uta.fi> writes: > "Jesse F. Hughes" <jesse(a)phiwumbda.org> writes: > >> 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. > > Shoenfield requires the domain of a structure to be nonempty. Of course, > nothing in the definition of truth in a structure actually depends on > this, and if we relax the requirement we do indeed find that every > M-instance of x=x is true in a structure with empty domain. Yes, of course. It's a shame that Nam, an authority on FOL and the first person to declare the end of the natural numbers, still doesn't get vacuous universals. But we surely can't blame this on Shoenfield. -- "So now, The Hammer is here, and with it, the end of days. The world will be destroyed, and then remade, as foretold. You will be lost, with your children, and then there will be others, and one day they will be tested, and will pass, but that is another story." --James S Harris gets a bit excited.
From: Marshall on 16 Jun 2010 11:08 On Jun 16, 5:33 am, "Jesse F. Hughes" <je...(a)phiwumbda.org> wrote: > Nam Nguyen <namducngu...(a)shaw.ca> writes: > > Jesse F. Hughes wrote: > >> stevendaryl3...(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. No! Really? That's hilarious. Marshall
From: Marshall on 16 Jun 2010 11:10 On Jun 15, 11:41 pm, Nam Nguyen <namducngu...(a)shaw.ca> wrote: > 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 "Whatever" I'm saying here? You mean you don't know what it is? I guess that explain why you continue not to understand me. > 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!) Being called incompetent by a talentless buffoon is no insult. Marshall
From: MoeBlee on 16 Jun 2010 12:08
On Jun 15, 11:13 pm, Nam Nguyen <namducngu...(a)shaw.ca> wrote: > If you incorrectly assume a structure with an empty U isn't a structure, > then of course you'd arrive such a wrong conclusion. > > A structure for a language is a way of consistently assigning "true" > > or "false" to each closed formula in the language. > But such consistency could exist only in a structure with an U that's > NOT empty. (As long as you don't acknowledge the case where U is empty > your argument couldn't be successful here. Seriously: you got to _confront_ > that case, whether or not you win or loose the debate.) I'd like to be completely clear on a certain point. Nam, do you agree or disagree with these statements: In Shoenfield, by definition, a structure for a language has for its universe (domain of discourse) a non-empty set. If one wishes to consider a notion of a structure for a language in which the universe for the stucture is empty, then one must revise Shoenfield's definition and recognize that one is then working with a different definition from Shoenfield's. MoeBlee |