Does Such a Relation Exist?
in [Logic]
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From: Herman Rubin on 3 Dec 2009 12:43 In article <2cc9ee5d-1f0f-4c8b-adc5-8e70223e61a7(a)u20g2000vbq.googlegroups.com>, Charlie-Boo <shymathguy(a)gmail.com> wrote: >On Dec 2, 1:39=A0pm, hru...(a)odds.stat.purdue.edu (Herman Rubin) wrote: >> In article <77840a3f-bde8-4f61-a35f-31554ffea...(a)h10g2000vbm.googlegroups= >.com>, >> Charlie-Boo =A0<shymath...(a)gmail.com> wrote: >> >On Nov 16, 8:25am, "Jesse F. Hughes" <je...(a)phiwumbda.org> wrote: >> >> Charlie-Boo <shymath...(a)gmail.com> writes: >> >> > On Nov 10, 1:32pm, "Jesse F. Hughes" <je...(a)phiwumbda.org> wrot= >e: >> >> >> I'm not sure what you're going on about. The axioms of ZF do n= >ot >> >> >> define function, but it is easy enough to introduce such a >> >> >> definition. Here it is: >> >> >> Let f, X and Y be sets. Then f is a function with domain= > X and >> >> >> codomain Y (written f:X -> Y) iff the following hold: >> >> >> (1) f c X x Y (f is a subset of X x Y) >> >> >> (2) (Ax in X)(Ey in Y)( <x,y> in f ) >> >> >> (3) (Ax in X)(Ay in Y)(Ay' in Y)( ( <x,y> in f & <x,y'> in f = >) -> y = y' ) >> >> >> What's the issue? >> >> > Set Theory is all about what can be formally proven using a specific >> >> > set of axioms. If you add more (axioms, definitions) then you a= >re not >> >> > addressing the same question. >> >> The introduction of definitions for function (and ordered pair and so >> >> on) are simple niceties, done in order to shorten presentations and >> >> make theorems and their proofs more readable. You seem to think t= >hat >> >> something more substantial is going on. >> >But the "definition" is subjective. Might not two different >> >definitions of function using relations lead to different results? >> >That's why an axiomatic system requires you to specify everything, >> >which ZF unfortunately does not. >> The definition of ordered pair is irrelevant. >Some definitions have been shown to be inconsistent. Some authors >have writen "if this form of definition is legitimate" (I believe >Kleene was one.) People debated for years whether definitions can >refer to themselves. Whether to allow certain axioms has never been >settled. >Something is missing from ZF: a definition of function. Yet you say >it's irrelevant??? >You cannot dismiss an omission in such a cavalier manner. I can and do. In von Neumann's thesis, function is primitive, and a class is defined as a function which takes on two designated values only. Yours assumes that X x Y exists; this is not needed if you replace (1) and (2) by (Au in f)(Ex in X)(Ey in Y)(u = <x,y>). Now in this form, <x,y> can be replaced by <<1,x><2,y>>, or other formulations. The essence is preserved. There have been many "definitions" of ordered pair; the one now used is the simplest. What is usually called a function should really be called a representation of that function, and this can be useful. If a different representation was used, one could still refer to this representation if needed to prove theorems, but nothing important would be changed. >> Whatever >> definition is used can be shown equivalent to the usual >> one, >How do you know that? You cannot systematically consider every >possible definition that anyone might propose. That makes no sense. >"It doesn't matter. They're all the same." Is there any logic that >supports such a broad statement as that? No, but you could introduce the current on and show that the current representation could be produced from whatever other one was used, by showing that a class with the desired properties can be produced therefrom. >> and this one is not the original one. >> In fact, the von Neumann axiomatization of set theory, >> the first one with a finite number of axioms (not >> axiom schemata), has function as a primitive; >That's right - they were smarter. But ZF did not. Since NBG is a conservative extension of ZF allowing proper classes, this is not the case. In NBG, one usually uses the current definition, which was the result of several papers by Kuratowski and Wiener. Bernays replaced the von Neumann approach with the current use of ordered pairs, and it was easily seen to be equivalent. Godel equated classes and sets with the same elements. -- This address is for information only. I do not claim that these views are those of the Statistics Department or of Purdue University. Herman Rubin, Department of Statistics, Purdue University hrubin(a)stat.purdue.edu Phone: (765)494-6054 FAX: (765)494-0558 |