An uncountable countable set
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From: Lester Zick on 18 Oct 2006 15:56 On 18 Oct 2006 11:02:28 -0700, "MoeBlee" <jazzmobe(a)hotmail.com> 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 As noted on a collateral thread, Moe, this kind of definition doesn't tell us what cardinality or ordinality is. It only tells us when x is or isn't a cardinal in non general restrictive mathematical terms at best. As a matter of fact this kind of definition is circular because it uses x both as subject and predicate. >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? Are you kidding? Why would anyone want to use a set theory textbook for general mathematical definition and analysis? ~v~~
From: imaginatorium on 18 Oct 2006 16:09 Ross A. Finlayson wrote: > imaginatorium(a)despammed.com wrote: <snip> > > You've said most, if not all, of the above before. The question here is > > whether any of the "counterexamples to standard real analysis" that you > > speak of can be expressed in mathematical language, or even in any > > language, in a way that can be understood - or even parsed - by anyone > > other than yourself. You said a day or so ago that you thought you were > > not liked - I think that's a misunderstanding. I can't see any reason > > not to like you - you do not resort to abuse, let alone to threats to > > contact employers etc. The reason most people ignore you is, I'm sure, > > that you have never really said anything that makes enough sense not to > > ignore. > > > > If you think this is wrong, or unfair, please explain to me one of your > > counterexamples to standard real > > analysis. > Hi Brian, > > I actually recommend that you read the book "Counterexamples in Real > Analysis." That contains scores of what are called "counterexamples in > standard real analysis." Sorry, Ross, perhaps I misread. Um, well, on looking closely, perhaps I didn't. Here's what you refer to: "counterexamples to standard real analysis" I read that as meaning that there are counterexamples that show that standard real analysis is inconsistent, wrong, or otherwise defective. But the title of the book you mention is subtly different: "Counterexamples in Real Analysis" I went to Amazon.com, and found a book with that title, and looked in the front. I saw a list of "counterexamples", and noticed that one of them was (from memory) "There exists a set with an infimum greater than its supremum". Sounds very odd - intuitively, it seems obvious that, assuming the Latin words mean greatest lower bound and least upper bound, the glb must be less than the lub, *until* you think about proving it, and think about the definitions. Aha! Consider the reals in [0, 1] with the usual ordering, then these have a subset whose glb is 1 and whose lub is 0. (At least if I've guessed the standard definitions correctly.) Can you see what it is? Thus, it appears that this book is about "counterexamples", in the sense of counterexamples to statements that might seem intuitively obvious, but which are not true. I don't see how this has any connection to any of your stuff. > .... iota, the least positive real, ... Can you either: provide the basis for your construction of the Rreals such that there is a least positive one, or point us at a textbook that does? The proof that in normal reals there can be no least positive is trivial, and has been posted here many times. It goes without saying that I can make no sense of anything else you said either. Brian Chandler http://imaginatorium.org
From: MoeBlee on 18 Oct 2006 17:10 Lester Zick wrote: > On 18 Oct 2006 11:46:47 -0700, "MoeBlee" <jazzmobe(a)hotmail.com> wrote: > > >Han de Bruijn wrote: > >> But why are the finite ordinals not equivalent with the naturals (I mean > >> in mainstream mathematics)? > > > >The set of finite ordinals IS the set of natural numbers. > > Natural numbers are ordinals? Yes, in virtually any standard treatment of set theory, the following holds, whether as a definition or as a theorem: x is a natural number <-> x is a finite ordinal. > I've yet to see textbooks call naturals ordinals. What set theory textbooks are you reading? Virtually every prominent textbook in set theory mentions that the set of finite ordinals is the set of natural numbers. > They're cardinals > not ordinals. All cardinals are ordinals (while not all ordinals are cardinals). The set of natural numbers is the set of finite ordinals which is the set of finite cardinals. This is standard set theory. >Totally different concepts. In set theory, cardinals are a special kind of ordinal. However, in the finite case, the set of cardinals is the set of ordinals. That is, the set of finite ordinals is the set of finite cardinals. > If textbooks do call > naturals ordinals then that's a pretty good reason to ignore textbook > definitions. Ignore away, then. > Just because 1 is first, 2 second, . . .etc. is no reason > to say 1, 2, . . . etc. are ordinals. First, second, . . . etc. are > ordinals. 1, 2, . . . etc. are cardinals. The terms 'ordinal' and 'cardinal' have special definitions in set theory. It is not guaranteed that these definitions agree with your notions (nor even common grade school mathematics) regarding 'first, etc.' Though, you will find that set theoretic ordinals do capture the notion of ordering while the set theoretic cardinals do capture the notion of cardinality. But I'm sure not eager to argue with you about your own notions about these things. If you don't want to find out how set theory actually works, so be it. MoeBlee
From: MoeBlee on 18 Oct 2006 17:20 Lester Zick wrote: > On 18 Oct 2006 11:02:28 -0700, "MoeBlee" <jazzmobe(a)hotmail.com> 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 > > As noted on a collateral thread, Moe, this kind of definition doesn't > tell us what cardinality or ordinality is. It only tells us when x is > or isn't a cardinal in non general restrictive mathematical terms at > best. It's a mathematical defintion in a certain theory (theories). Otherwise I really don't care about what the cardinality operation is in whatever Zickian universe you commune with. > As a matter of fact this kind of definition is circular because > it uses x both as subject and predicate. That is completely incorrect. Such definitions of the form I used are PROVEN NOT to be circular. 'x' is a VARIABLE in such definitions, it is not 'x' that is being defined but rather 'card', which appears on the left side of the equation and not on the right side of the equation, just as is compeltely correct and standard for the definition of an operation symbol. > Are you kidding? Why would anyone want to use a set theory textbook > for general mathematical definition and analysis? The discussion was about set theoretic definitions of certain terminology of set theory. The best place to find that is in a textbook on set theory. As to mathematical definition, the best place to find explication is in textbooks on mathmatical logic. MoeBlee
From: MoeBlee on 18 Oct 2006 17:29
Ross A. Finlayson wrote: > I actually recommend that you read the book "Counterexamples in Real > Analysis." That contains scores of what are called "counterexamples in > standard real analysis." I have browsed that book. It does not give counterexamples that shown an inconsistency in analysis. What it gives are counterexamples to generalizations that are NOT theorems. The counterexamples themselves are theorems, not contradictions of theorems. It's of the form (though of course, this is not an actual example): A counterexample to the statement that "all prime numbers are odd" is the prime number 2. > Did you know model theory posits the existence of a maximal ordinal of > which there is none in ZF? Of course no one knows that, since it is not true. MoeBlee |