An uncountable countable set
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From: MoeBlee on 19 Oct 2006 14:23 Han de Bruijn wrote: > MoeBlee wrote: > > > 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. > > Ah! Therefore we can limit ourselves to the naturals and forget all the > fuzz about ordinals and cardinals. Because the infinite counterparts of > these beasties do not exist in _my_ universe anyway. Do what you want in _your_ universe. Meanwhile, if you ever get around to formulating a mathematical theory, do let us know. MoeBlee
From: Lester Zick on 19 Oct 2006 14:26 On 18 Oct 2006 14:20:40 -0700, "MoeBlee" <jazzmobe(a)hotmail.com> wrote: >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. But do you care whether 2 is the second in the Moebius strip of the universe? >> 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. Weeell, Moe, you may or may not have any expertise in transgendered arithmetic but you certainly seem to have little or none when it comes to definition. Your definition for card(x) assumes we know what x is to begin with. Certainly there's no evidence in your definition for what x may be that wouldn't affect the definition of cardinality. So you wind up with no definition for "card" that doesn't rely on x. You go on to claim that this means you're defining "card" but I don't see any definition for "card". Maybe it's a counter intuitive definition? >> 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. And not mathematics? Then I repeat my objection. > 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. So what is it exactly that "set" theory allows us to do in mathematics that we couldn't already do without it? Define infinity? Define regularity? Define choice? Define ordinals? Define cardinals? You seem to be of a psychological frame of reference prevalent among modern mathematikers that arithmetic in the form of set theory represents some kind of TOE. ~v~~
From: MoeBlee on 19 Oct 2006 14:28 Han de Bruijn wrote: > The confusion stems from the fact that I cannot and shall not understand > the _infinite_ counterparts of the finite cardinals and ordinals. How can you understand if you won't read a book that explains it? (By the way, Halmos is a good book, but it's just an overview; it doesn't give you the full explanations that you need.) So you seem to think it is better to spout nonsense on the Internet about a subject you cannot possibly understand since you insist that you won't. MoeBlee
From: MoeBlee on 19 Oct 2006 15:16 Lester Zick wrote: > to definition. Your definition for card(x) assumes we know what x is > to begin with. x is a variable. It's just ridiculous that you don't understand the basic form of a mathematical definition. If you want it in intuitive terms, 'x' is a pronoun, such as 'it' or an expression such as 'that thing'. Then the definition is understood as saying that if we have a given thing, call it 'x', then the cardinality of that thing (called 'x') is the least oridinal equinumerous with that thing (called 'x'). We're not defining the variable. We're USING the variable as a "place holder" that you plug in any object of the domain of discourse so that the defintion tells you what the cardinality of that object is. That's longwinded for simply: card(x) = the least ordinal equinumerous with x It's exactly analogous to such mathematical defintions as: square(n) = n times n > Certainly there's no evidence in your definition for > what x may be that wouldn't affect the definition of cardinality. 'x' is a variable that ranges over the entire domain of disourse. x could be instantiated to ANY object of the theory. > So > you wind up with no definition for "card" that doesn't rely on x. Of course it relies on x when you instantiate 'x' to any object of the theory. This is ridiculous that you don't understand what a variable is in mathematics. > You > go on to claim that this means you're defining "card" but I don't see > any definition for "card". Maybe it's a counter intuitive definition? The definition is there. Give me an object, ANY object, which is instantitated from the variable 'x', then the cardinality of that object is the least ordinal equinumerous with that object. Again, it's a simple as: card(x) = the least ordinal equinumerous with x > So what is it exactly that "set" theory allows us to do in mathematics > that we couldn't already do without it? Any axiomatization (such set theories have axioms) allows you to show proofs of theorems from those axioms. And definitions of operation symbols added to the language of the theory are from primitives of the theory as long as the axioms prove the existence of a unique object apropros the definition. For example, ZFC proves that for any object, there exists a unique least ordinal equinumerous with that object. (Conditional definitions are the subject of another lessson.) Answering each of your objections without you knowing even the most basic things about the subject is not an efficient way for me to teach you about basic mathematical logic. You just need to do some of this basic work for yourself with a good textbook. > You seem > to be of a psychological frame of reference prevalent among modern > mathematikers that arithmetic in the form of set theory represents > some kind of TOE. What in the world are you talking about? I never said anything like that set theory "represents a theory of everything" (let alone that arithmetic in the form of set theory does that). On the contrary, I've posted that I do not make such a claim. Please do not put words in my mouth. MoeBlee
From: Lester Zick on 19 Oct 2006 17:26
On 19 Oct 2006 11:23:26 -0700, "MoeBlee" <jazzmobe(a)hotmail.com> wrote: >Han de Bruijn wrote: >> MoeBlee wrote: >> >> > 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. >> >> Ah! Therefore we can limit ourselves to the naturals and forget all the >> fuzz about ordinals and cardinals. Because the infinite counterparts of >> these beasties do not exist in _my_ universe anyway. > >Do what you want in _your_ universe. Meanwhile, if you ever get around >to formulating a mathematical theory, do let us know. And if set theorists ever get around to formulating a mathematical theory do let us know. ~v~~ |