The End of an Era - That of The Natural Numbers
in [Math]
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From: Marshall on 18 Jun 2010 02:44 On Jun 17, 11:32 pm, Nam Nguyen <namducngu...(a)shaw.ca> wrote: > > Since there are _no n-tuples_ _in_ an empty > predicate and since there's no non-empty predicate in an empty U, > no formula can be true in the degenerated structure where U is empty. Here is the heart of your mistake. Without justification (and in fact incorrectly,) you make the jump from speaking of predicates to speaking of all formulas. You are absolutely right that in a structure with an empty carrier set, every predicate must be empty. You are absolutely wrong in thinking that in a structure in which every predicate must evaluate to false, that every *formula* must also evaluate to false. Formulas are more than just predicates. Marshall
From: Alan Smaill on 18 Jun 2010 05:55 Nam Nguyen <namducnguyen(a)shaw.ca> writes: > Jesse F. Hughes wrote: >> If a theory has constants, then its >> structures must be non-empty. > > No (model) structure within FOL= can be the empty _set_. It has at least > these 2 elements: > > <'A',U> and <'=',{all 2-tuples of some form} | empty set> > > _by definition_. The issue is whether the *universe* of the model can be non-empty. Since constants correspond to elements of the universe (0-ary functions in Shoenfiled's terminology), if the language has constants, the *universe* cannot be empty. -- Alan Smaill
From: Jesse F. Hughes on 18 Jun 2010 07:31 Nam Nguyen <namducnguyen(a)shaw.ca> writes: > So the closed formula A is in L(fancyA). So x=x is not in L(fancyA), > an extension of L, in the sense that x=x is already in L. L is a subset of L(fancyA), and so every formula of L is also a formula of L(fancyA). Lordy, but you're incapable of reading a relatively clear and simple text. -- "These mathematicians are worse than communists, as how do you explain their behavior? I *am* the American Dream, fighting for what should be mine, having to get past weak-minded academics who are fighting to block my success. But I shall prevail!!!" -- James S. Harris
From: Jesse F. Hughes on 18 Jun 2010 07:29 Nam Nguyen <namducnguyen(a)shaw.ca> writes: > Jesse F. Hughes wrote: >> 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. > > In reasoning framework, in the end we'll go only by _definition_ > and permitted inferences. Feelings such "Of course that's right" > or how thing "should/shouldn't" be would be at best just intuitions > which more often than not turn out to be wrong. > >> If a theory has constants, then its >> structures must be non-empty. > > No (model) structure within FOL= can be the empty _set_. It has at least > these 2 elements: > > <'A',U> and <'=',{all 2-tuples of some form} | empty set> > > _by definition_. Your capability of misreading is peerless. By "empty structure", of course I meant a structure with empty universe (or support or whatever Shoenfield calls it). -- "And I'll reinforce the point that you are an enemy of humanity, that my predecessors are people like Gauss, Euler, Newton, Archimedes and others who you are spitting upon as you do it to me by trying to keep their discipline trashed as it is now." -- James S. Harris
From: Marshall on 18 Jun 2010 10:56
On Jun 18, 4:29 am, "Jesse F. Hughes" <je...(a)phiwumbda.org> wrote: > > Your capability of misreading is peerless. Do you think he does it on purpose, at least some of the time? I mean, he even gets really simple and obvious things wrong. (Which is not to say that all of the things he gets wrong are simple and obvious.) I'm beginning to wonder if he isn't more of a troll than a crank. Marshall |