The Definition of Points
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From: David R Tribble on 17 Apr 2007 18:55 Tony Orlow wrote: >> Defining "=" depends, as far as I can tell, on defining "<". Is this >> wrong, in your "opinion"? > MoeBlee wrote: > In what context? In set theory, given an appropriate axiomatization, > we can define '=' from just 'e' (the membership symbol). Indeed, you can define a system with '<', '>', and '=' operators defined entirely in terms of set membership. To wit: 1. X < Y iff for all a in X, a in Y, and there exists b in Y such that b not in X. 2. X > Y iff Y < X. 3. X = Y iff for all a in X, a in Y, and for all b in Y, b in X. (These definitions resemble cardinal relations.) But note specifically that '=' is not defined in terms of '<'. Both are defined in terms of the more primitive 'in' relation/operator.
From: Lester Zick on 17 Apr 2007 18:58 On Tue, 17 Apr 2007 13:39:45 -0400, Tony Orlow <tony(a)lightlink.com> wrote: >>> The question about axioms is whether each one is justifiable and >>> sufficiently general enough to be accepted as "true" in some universal >>> sense. >> >> No the actual question is whether each and every axiom is actually >> true and demonstrably so in mechanically exhaustive terms. Otherwise >> there's not much point to the exhaustively rigorous demonstration of >> theorems in terms of axioms demanded of students if axioms themselves >> are only assumed true. >> >> ~v~~ > >I am saying that one can assume axioms for the sake of deduction, but >that the conclusions derived are only as reliable as the starting >axioms, and so there is an inductive process in deciding which axioms to >accept for the sake of one's "theory", expecially when looking for >universal truths that serve as axions in a TOE, depending on whether the >conclusions drawn fit the empirical evidence. Lots of axioms and conclusions fit the empirical evidence, Tony. That's the whole problem in determining which are actually true and why. Given various experimental circumstances the question is how to explain them all in terms of one another. Modern mathematikers just assume they can explain them one way to the exclusion of other ways and indulge in special pleading and excuses to justify their choices. It might help if we could just start from true assumptions to begin with for a change. You want to start by making certain assumptions for the sake of an argument and then argue the truth of the assumptions according to the apparent plausibility of the argument. Doesn't work that way. ~v~~
From: Lester Zick on 17 Apr 2007 19:01 On Tue, 17 Apr 2007 13:33:39 -0400, Tony Orlow <tony(a)lightlink.com> wrote: >>> Define "assumption". >> >> Any declarative judgment not demonstrated in mechanically exhaustive >> terms. >> >>> Do you "believe" that truth exists? >> >> Of course. >> > >Prove it in "mechanically exhaustive terms". Prove that I believe truth exists? How? I testify that I believe it. What difference does it make anyway whether I believe truth exists? It's what I can demonstrate in mechanically exhaustive terms that matters and if you haven't seen me do that any number of times already then there's nothing I can help you with. ~v~~
From: Lester Zick on 17 Apr 2007 19:10 On Tue, 17 Apr 2007 13:33:39 -0400, Tony Orlow <tony(a)lightlink.com> wrote: >>> Is there a set >>> of statements S such that forall seS s=true? >> >> No idea, Tony. There looks to be a typo above so I'm not sure exactly >> what you're asking. >I am asking, in English, whether there is a set of all true statements. No. There are predicates to which all true statements and all false statements are subject respectively but no otherwise exhaustively definable set of all true or false statements because the difference between predicates and predicate combinations in true or false statements is subject to indefinite subdivision. ~v~~
From: Lester Zick on 17 Apr 2007 19:14
On Tue, 17 Apr 2007 13:33:39 -0400, Tony Orlow <tony(a)lightlink.com> wrote: >>> Is there such a thing as >>> truth, or falsity? >> >> Of course. >"Prove" logic exists, in terms that precede logic. Can't be done. You can only show that everything doable, thinkable, and knowable can only be done, thought, and known by alternatives because there is no alternative to alternatives. That's the way we mechanize logic in tautological terms. ~v~~ |