An uncountable countable set
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From: MoeBlee on 20 Oct 2006 16:26 Tony Orlow wrote: > I'm reading Non-Standard Analysis instead. What book? You really would be better served by having your basics in set theory and mathematical logic in order first and then taking on non-standard analysis. > Robinson agrees there's no > smallest infinity, Then that is not the same as the ordering of the ordinals we're talking about. I very much doubt that Robinson claims that there is not a least infinite ordinal. Please tell me exactly what passages or theorems you are referring to in Robinson's work so that I can see exactly what it is you are talking about. MoeBlee
From: Lester Zick on 20 Oct 2006 16:26 On 20 Oct 2006 07:35:13 -0700, imaginatorium(a)despammed.com wrote: >Ross A. Finlayson wrote: >> Ross A. Finlayson wrote: >> > MoeBlee wrote: >> > > Han de Bruijn wrote: > ><chop> > >> Well, this thread is getting pretty long. I think I can sum it up by >> saying nothing. > >Hmm. If you said nothing, Ross, we would all miss your witticism. I >think you mean: > >"I think I can sum it up by posting a note commenting that I can sum it >up by posting a note commenting that I can sum it up by posting a note >commenting that I can sum it up by posting a note commenting that I can >sum it up by posting a note commenting that I can sum it up by posting >a note commenting that I can sum it up by posting a note commenting >that I can sum it up by posting a note commenting that I can sum it up >by posting a note commenting that I can sum it up by posting a note >commenting that I can sum it up by posting a note commenting that I can >sum it up by posting a note commenting that I can sum it up by posting >a note commenting that I can sum it up by posting a note commenting >that I can sum it up by posting a note commenting that I can sum it up >by posting a note commenting that I can sum it up by posting a note >commenting that I can sum it up by posting a note commenting that I can >sum it up by posting a note commenting that I can sum it up by posting >a note commenting that I can [message length exceeded] Ah, a typical neomathematical regression. ~v~~
From: MoeBlee on 20 Oct 2006 16:39 MoeBlee wrote: > Tony Orlow wrote: > > I'm reading Non-Standard Analysis instead. > > What book? > > You really would be better served by having your basics in set theory > and mathematical logic in order first and then taking on non-standard > analysis. > > > Robinson agrees there's no > > smallest infinity, > > Then that is not the same as the ordering of the ordinals we're talking > about. I very much doubt that Robinson claims that there is not a least > infinite ordinal. > > Please tell me exactly what passages or theorems you are referring to > in Robinson's work so that I can see exactly what it is you are talking > about. And this reminds me that you never did come to understand the difference between cardinality and ordering. I and others have pointed out to you that you conflate these. One doesn't even need non-standard analysis to provide an ordering in which there is a set S with no least member yet with every member of S greater than some set with no greatest member. But that ordering is not an ordering by cardinality. Yes, PA has models in which there are different orderings so that objects are called 'infinite' per these orderings, but this is NOT the same sense of 'infinite' as that of the cardinality sense. And we can define a division operation on these "infinite" objects to get infinitesimals, but again, this is not the same as cardinality. Moreover, you must be very careful to distinguish between the proof of the existence of certain models and a RECURSIVE axiomatization for a theory of which the model is a model of. MoeBlee
From: Lester Zick on 21 Oct 2006 13:21 On 19 Oct 2006 23:41:11 -0700, imaginatorium(a)despammed.com wrote: >MoeBlee wrote: >> You wrote too many confused and uninformed things for me to even care >> to sort through. > >You don't say who "you" is... let me spend a little while guessing. Oh, >but I'll read on while I'm guessing. > >> I'll take you up on your last line, though: >> >> > You >> > say you set theory texts define "cardinality" in a certain way which >> > is pretty much circular if relying on cardinality for equinumerosity. >> >> I don't say that. And the definition of 'cardinality of' does use >> 'equinumerosity', but the definition of 'equinumerosity' does not use >> 'cardinality of', so there is not the circluarity you just arbitarily >> claim there to be. I've already been over the subject of mathematical >> definitions with you in other threads. But please do consider all your >> points, objections, and conceptions to be vindicated by my increasing >> apathy to try to bring you to reason about anything at all. > >Good news, Lester! I think you're going to win!! Jesus, Brian, winning hasn't been the issue for the last couple of years. ~v~~
From: Lester Zick on 21 Oct 2006 13:34
On Fri, 20 Oct 2006 14:12:27 -0400, Tony Orlow <tony(a)lightlink.com> wrote: >MoeBlee wrote: >> Tony Orlow wrote: >>> Also, upon which axioms is the definition of cardinality based? >> >> The usual definition is: >> >> card(x) = the least ordinal equinumerous with x >> >> The definition ultimately reverts to the 1-place predicate symbol 'e' >> (and the 1-place predicate symbol '=', if equality is taken as >> primitive). For the definition to "work out" ('work out' is informal >> here) in Z set theory, we usually suppose the axioims of Z set theory >> plus the axiom of schema of replacement (thus we're in ZF) and the >> axiom of choice (thus we're in ZFC). However, there is a way to avoid >> the axiom of choice by using the axiom of regularity instead with a >> somewhat different definition from just 'least ordinal equinumerous >> with'. Also, we could adopt a "midpoint" between the axiom schema of >> replacement and the axiom of choice by adopting the numeration theorem >> (AxEy y is an ordinal equinumerous with x) instead, which would be a >> method stronger than adopting the axiom of choice, but weaker than >> adopting both the axiom of choice and the axiom schema of replacement. >> As to the more basic axioms of Z, for the definition to "work out", I'm >> pretty sure we need extensionality, schema of separation (or schema of >> replacement if we go that way), union, and pairing (pairing is not >> needed if we have the schema of replacement). I'm not 100% sure, but my >> strong guess is that we don't need the power set axiom for this >> purpose. And we don't need the axiom of infinity. >> >> Why don't you just a set theory textbook? >> >> MoeBlee >> > >I'm reading Non-Standard Analysis instead. Robinson agrees there's no >smallest infinity, Just out of curiosity, Tony, what's his rationale? > and that there are an uncountable number of countable >neighborhoods with what he calls the '~' relation. I'm very encouraged >to see essentially the same ideas as mine, put in technical terms. Maybe >when I'm through with that, although I'm also reading Boole's treatise >on logic where he eventually talks about probabilistic logic. ~v~~ |