The End of an Era - That of The Natural Numbers
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From: MoeBlee on 17 Jun 2010 14:45 On Jun 17, 8:00 am, "Jesse F. Hughes" <je...(a)phiwumbda.org> wrote: > x=x is > neither true nor false in Shoenfield's terminology, but rather valid or > invalid. I agree that that's the way in Shoenfield. And he says that formula is valid. But where does he define 'valid'? MoeBlee
From: Jesse F. Hughes on 17 Jun 2010 15:43 MoeBlee <jazzmobe(a)hotmail.com> writes: > On Jun 17, 8:00 am, "Jesse F. Hughes" <je...(a)phiwumbda.org> wrote: > >> x=x is >> neither true nor false in Shoenfield's terminology, but rather valid or >> invalid. > > I agree that that's the way in Shoenfield. And he says that formula is > valid. > > But where does he define 'valid'? Page 19, near the middle or lower half, as I recall, he defines "valid in M" (or "valid in fancyA", if you prefer). On the following page, he defines valid (simpliciter). -- Jesse F. Hughes "She moaned, in pain and pleasure, as, in a confused whirlwind, she glimpsed an image of Saint Sebastian riddled with arrows, crucified and impaled." --Mario Vargas Llosa on category theory
From: Jesse F. Hughes on 17 Jun 2010 15:41 MoeBlee <jazzmobe(a)hotmail.com> writes: > But if the universe is empty while the set of constants is non-empty > then there is no such function. > > So, I don't see how we can have an empty universe for a structure for > a language that has constants. Of course that's right. And that's how it *should* be. If a theory has constants, then its structures must be non-empty. -- Jesse F. Hughes Jesse: Quincy, you should trust me more. Quincy (age 4): Baba, I never trust you. And I've got good reasons.
From: MoeBlee on 17 Jun 2010 18:00 On Jun 17, 2:43 pm, "Jesse F. Hughes" <je...(a)phiwumbda.org> wrote: > MoeBlee <jazzm...(a)hotmail.com> writes: > > On Jun 17, 8:00 am, "Jesse F. Hughes" <je...(a)phiwumbda.org> wrote: > > But where does he define 'valid'? > > Page 19, near the middle or lower half, as I recall, he defines "valid > in M" (or "valid in fancyA", if you prefer). On the following page, he > defines valid (simpliciter). Thanks. To be exact with his terminology. He defines "valid in fancyA" and then "logically valid" (what you mean by 'valid simpliciter'). MoeBlee
From: MoeBlee on 17 Jun 2010 18:01
On Jun 17, 2:41 pm, "Jesse F. Hughes" <je...(a)phiwumbda.org> wrote: > MoeBlee <jazzm...(a)hotmail.com> writes: > > But if the universe is empty while the set of constants is non-empty > > then there is no such function. > > > So, I don't see how we can have an empty universe for a structure for > > a language that has constants. > > Of course that's right. > > And that's how it *should* be. If a theory has constants, then its > structures must be non-empty. Then (as long as Aatu agrees with you) I have nothing further for the court. All other details may be found in my amicus brief. MoeBlee |