The Definition of Points
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From: Tony Orlow on 18 Apr 2007 14:55 Lester Zick wrote: > 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~~ Uh, what? 01oo
From: Tony Orlow on 18 Apr 2007 14:57 Lester Zick wrote: > On Tue, 17 Apr 2007 13:29:44 -0400, Tony Orlow <tony(a)lightlink.com> > wrote: > >>> Let me ask you something, Tony. When you send off for some truth value >>> according to "true(x)" and it returns a 1 or 0 or whatever, how is the >>> determination of that "truth value" made? >> From the truth values of the posited assumptions, of course, just like >> yours. > > So you just posit truth values and wing it whereas I'm more inclined > to demonstrate the truth of what I posit instead? > > ~v~~ What truth have you demonstrated without positing first? 01oo
From: Tony Orlow on 18 Apr 2007 15:04 Mike Kelly wrote: > On 18 Apr, 06:17, Tony Orlow <t...(a)lightlink.com> wrote: >> Mike Kelly wrote: >> >> < snippery > >> >>> You've lost me again. A bad analogy is like a diagonal frog. >>> -- >>> mike. >> Transfinite cardinality makes very nice equivalence classes based almost >> solely on 'e', > > Why "almost solely"? > The successor relation is a relation not related to 'e' except in a particular model of the naturals upo which the rest is built. >> but in my opinion doesn't produce believable results. > > Which "results" of cardinality are not believable? You don't think the > evens and the naturals and the rationals are mutually bijectible? Or > you do think the reals and the naturals are bijectible? Or did you > mean you have a problem not with what cardinality actually is but with > what you hallucinate cardinality to be? > As long as transfinite "equivalence" classes are referred to as such, and not said to denote "equal" size or numerosity, then I have no problem. >> I'm working on a better theory, bit by bit. > > You have never managed to articulate a problem with what cardinality > actually says (all it says is which sets are bijectbile). You've never > managed to explain the motivation behind your "better theory". I don't > even know whether your ideas are supposed to be formulated in ZF or in > something else, possibly a new foundation of your own devising(or, > perhaps, never formalised at all?). FWIW I don't think it would be > very hard to define, say, "density in the naturals" in ZF. I just > don't see the motivation behind doing so. > The idea is to create a method that works for comparing countable and uncountable sets alike, and properly defining the relationship between the two. >> I think trying to base >> everything on 'e' is a mistake, since no infinite set can be defined >> without some form of '<'. I think the two need to be introduced together. > > But you are, in fact, completely mistaken here. Several people have > told you this. You have some inability to acknowledge this. Oh well. > > PS you snipped a lot of my post; maybe you found it boring. I find > your evasiveness quite interesting, though. What does cardinality > claim to tell that it can't? > > -- > mike. > I suppose it's more the way it's spoken about. Leave out "size" and "equinumerous" and talk about "equivalence classes" and not "equality", and I have no problem with the consistency of standard theory. tony.
From: Tony Orlow on 18 Apr 2007 15:13 MoeBlee wrote: > On Apr 17, 10:02 am, Tony Orlow <t...(a)lightlink.com> wrote: > >> I am saying it's obvious that any countable set has a well ordering. It >> is not obvious for uncountable sets. > > It's not only obvious, but it's trivial to prove that every countable > set has a well ordering. > > So, just to be sure you understand, what is at stake with an axiom of > countable choice is not at all that every countable set has a well > ordering (since we don't need any choice principle to prove that) but > rather that for any countable set there is a function that chooses > exactly one member from every nonempty subset of that countable set > (and actually, we need concern ourselves only with denumerable sets, > since we don't need a choice principle to prove that for every finite > set there is a function that chooses exactly one member from every > nonempty subset of the finite set). > > MoeBlee > It's not trivial to prove any such thing for an uncountable set. That requires full Choice, does it not? In fact, it was Virgil, I believe, who said that was the MOTIVATION behind Zermelo's formulation of Choice, so that ALL sets could be considered well orderable. However, while I see that a choice function produces a well ordering for any countable set, I don't see that it does for an uncountable set. Maybe I am confused about something. Can a well ordering include an uncountable number of limit elements? If so, then I can see how such a well ordering could occur. Are limit elements which are an infinite number of limit elements beyond the first what are called "inaccessible" limit ordinals? If this is the case, then I guess I can see that in this sense, one could concoct a well ordering. But, no, then there would exist an infinite descending sequence of limit ordinals withoin the well order. So, that wouldn't work, right. I just don't see that a well order can be accomplished in principle on an uncountable set, whether one declares an axiom to that effect of not. TOEknee
From: MoeBlee on 18 Apr 2007 15:24
On Apr 18, 12:04 pm, Tony Orlow <t...(a)lightlink.com> wrote: > Mike Kelly wrote: > The successor relation is a relation not related to 'e' except in a > particular model of the naturals upo which the rest is built. In set theory such as Z set theory, EVERY relation among objects is based on the membership relation. > >> but in my opinion doesn't produce believable results. > > > Which "results" of cardinality are not believable? You don't think the > > evens and the naturals and the rationals are mutually bijectible? Or > > you do think the reals and the naturals are bijectible? Or did you > > mean you have a problem not with what cardinality actually is but with > > what you hallucinate cardinality to be? > > As long as transfinite "equivalence" classes are referred to as such, > and not said to denote "equal" size or numerosity, then I have no problem. In theories such as Z set theory and its variants the equivalence classes are NOT themselves cardinalities. > I suppose it's more the way it's spoken about. Leave out "size" and > "equinumerous" and talk about "equivalence classes" and not "equality", > and I have no problem with the consistency of standard theory. You've already been told a million times that the words 'size' and 'equinumerous' are dispensible. So, whenever someone says 'same size' or 'equinumerous', just take them to have said 'equipollent' instead. Anyway, consistency has nothing to do with whether we use the word 'size' and 'equinumerous'. And we do NOT say that two sets are equal just on account of being equipolllent, but rather that the cardinality (and here you can use another word if 'cardinality' does not suit you; maybe a word such as 'pollent-ordinal') of the sets is equal, as indeed, if x and y are equipollent, then the pollent-ordinal of x (the least ordinal that is bijectable with x) IS IDENTICAL with the pollent- ordinal of y. MoeBlee |