Uncountability of the Rationals?
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From: K_h on 4 Jun 2010 17:31 "Tony Orlow" <tony(a)lightlink.com> wrote in message news:dc7fa188-5dc1-4171-bd46-50a5ae84de27(a)c11g2000vbe.googlegroups.com... On Jun 4, 2:37 pm, MoeBlee <jazzm...(a)hotmail.com> wrote: > On Jun 4, 11:16 am, Tony Orlow <t...(a)lightlink.com> wrote: > > > Do you not remember the thread not > long ago where you methamaticians > started arguing amonst yourselves? > What is the "ordinary" definition, > again? Never mind. It's a sidebar, as > always with you. The term sequence has a mathematical and non-mathematical usage. In mathematics, an infinite sequence is defined as follows: A set S is a sequence of the elements of a set T if S is a function with domain N and codomain T. For a finite sequence the definition is the same except the domain is some n in N and not all of N. Remember, in the ZF approach to set theory, functions are defined as sets. _
From: Virgil on 4 Jun 2010 17:47 In article <613255cf-d67c-4c71-ad3d-f388a0329848(a)c22g2000vbb.googlegroups.com>, Tony Orlow <tony(a)lightlink.com> wrote: > Sequences are sets with order. But sets with order often are not sequences. Consider the standard reals. In terms of order, a set must be well-ordered with one and only one non-successor element to be a sequence. But there are sequences which are not well-ordered sets.
From: Pollux on 4 Jun 2010 13:42 That is, those infinite sets "potentially" exist, but maybe not in "actuality", right? You could also deny the status of "mathematical entity" to such an infinite construction, just because it is "infinite" (looks like I'm in good mathematical company on that line of argument). Pollux
From: MoeBlee on 4 Jun 2010 18:16 On Jun 4, 3:24 pm, Tony Orlow <t...(a)lightlink.com> wrote: > On Jun 4, 2:32 pm, MoeBlee <jazzm...(a)hotmail.com> wrote: > > > On Jun 3, 7:34 am, Tony Orlow <t...(a)lightlink.com> wrote: > > > > mathematicians misrepresent the theory, > > > claiming that every conclusion they draw "follows logicallly from the > > > axioms." This claim is simply not true, as cardinality is not > > > mentioned in the axioms, much less anything about omega or the alephs, > > > except for a declaration that something homomorphic to the naturals > > > exists, which is ultimately not a set, but a sequence. > > > No, all of those terms are given by DEFINITIONS, which may take the > > form of definitional AXIOMS, which provide a conservative extension of > > the theory in its primnitive form. This has been explained to you at > > least a hundred times already, but you choose just to ignore. > > Can you please provide a complete list of your definitional axioms for > my inspection? They are not part of ZFC, are they? The answer to your question is embedded in about a hundred explanations I've already given you over the years, as well as in the very paragraph you just quoted. The definitional axioms are not themselves part of ZFC in its primitive form. Each definitional axiom is a sentence added to ZFC and introduces a new defined symbol. However, by the method of proper definitions, definitional axioms do not add any theorems to the primitive theory in the primitive language but rather add only theorems that have defined symbols in them as such theorems can be reduced to equivalent theorems with only primitive notation. All of this is covered in many a textbook in mathematical logic, and is a formalization of basic common sense notions in mathematics. It would be impractical for me to post a list of all the definitions I've entered just in my own compendium. And such a list would vary from author to author anyway, as each author will adopt definitions as he or she needs for his or her own purposes of exposition. In any case, ordinarily the list starts with such symbols as '0', 'P' (for power set), 'U' (for union), etc. > > Moreover, we don't just show that there exists a set homomorphic to > > the naturals (and their customary ordering) but rather that we define > > 'w' (omega) to be the least set that has 0 as a member and is closed > > under successsorship. Moreover, in set theory, sequences are sets. > > Sequences are sets with order. That is not the set theoretic definition of a sequence. A set with an order is of the form: <S R> where S is a set and R is an ordering on S. A sequence (in a general sense) is a function whose domain is an ordinal > Sets in general have no order. Every set can be partially ordered. With the axiom of choice, every set can be well ordered. And with the axiom of choice (but weaker than the axiom of choice), every set has a linear ordering. > If set > theory would like to hide the recursive nature of infinite sets I have no idea what you mean by the "recursive nature". If there is some specific mathematical statement or statement about mathematics you wish to make, would you please find out how to couch in some sensible way? > > > By demanding to > > > know with which axioms a non-transfinitologist disagrees, > > > mathematicians obscure the question. > > > WHAT question? > > MoeBlee eat some meat or take vitamins. The question is, "Why do you > object to set theory, O Crackpot?" Isn't that what you've been asking > all along? I had a top sirloin last night and later a big bottle of Pomegranate vitamin water. Don't worry about my nutrition. If you are to nag me, I wish you would nag me about something actually in need of attention, such as I need to renew my bus pass this month rather than waste money each day on individual fares. > You may call N a set, but it is really a sequence that is > never complete at any point. I don't begrudge that you have that notion of N and of sequence and of 'complete', whatever that may mean. Let me know when you have such notions evinced in a formal theory. > the Axiom of infinite is not about a > set, really, but a sequence without end. You may have your own notions about things, but you're off-base when you presume to command the import of notions quite independent of you. The axiom of infinity, as a sentence in a formal language, "reads off" in an ordinary way as asserting that there exists an successor inductive set. > > > and the simplistic assumption of equivalence of > > > "size" based on bijection alone. > > > I've presented to you at least a hundred times the answer (in > > variation) that instead of the word 'size' we could use the word > > 'zize'. Then your protest disappears. You've never dealt with that > > point. > > I used "Bigulosity", remember? Surely you do, you rascal. You could use "jiggywiggywosity" or whatever you want. It doesn't address the point I've made. That YOU have some neologism concerning your own mathematical thought-cloud doesn't address that, as far as mere formal ZFC is concerned, it is not substantive whether we say 'size' or 'zize'. > > > If bijection determines equal > > > cardinality that's fine, but to equate cardinality with set size > > > simply does not work for infinite sets to the satisfaction of most > > > people's intuitions. > > > Most people haven't studied the subject. And, again, everywhere in > > discussions about set theory you may substitute 'size' with 'zize'. No > > substantive matter to the FORMAL theory. > > "The" formal theory. I like that. You should really become a Hare > Krishna devotee. Oh please, Swami Sri Nondevotee, of course, I mean whatever particular theory is under discussion at the time. That doesn't imply that I don't recognize that there are INFINITELY many theories. > > > To claim that they are being illogical by > > > objecting to the theory without identifying an axiom at fault is > > > disingenuous, since the axioms are clearly not the problem. > > > The problem is that you demand that the word 'size' be used in a way > > that does not conflict with your ordinary notions about size. But set > > theory does not represent that such words are used in a way that > > conforms in every sense to such everyday notions. > > Well, perhaps it satisfies your "everyday" experiences with infinity. No, of course not. I HAVE NO everyday experiences with infinity (at least not in the sense of sensory experience). Don't twist me. I said 'notions' not 'experiences'. And I didn't even say that anyone has everyday notions about infinity. I simply said that set theory does not (in itself, I'll add) represent that certain words are used in an everyday sense; indeed, since the everyday sense of size pertains to finite. > Or, perhaps, you don't have any. If you don't call it "size" I won't. > You say "cardinality", and I sqay "bigulosity". So what? You persist to miss my point even after I've stated it point blank over a hundred times. > > > The von Neumann limit ordinals are > > > simply bunk. > > > Fine, then probably you want to throw out the axiom of infinity. > > It says nothing about the non-finite limit ordinals. See above. So what? But along with other axioms it entails the existence of infinite ordinals and of limit ordinals. You're wasting our time. > > > Everything Jesus said was true and wise, but the Trinity > > > and the Immaculate Conception are Christianity's von Neumann limit > > > ordinals, thrown in later for Constantine's political goal of state > > > religion incorporating pagan beliefs and christian. > > > So who is the Jesus of set theory before statist limit ordinals > > corrupted the whole thing? > > Good question. Perhaps Cantor and Dedekind in modern times, though > Galileo certainly explored bijection long before, and decided there > was no real answer, and Aristotle first made the distinction between > potential (countable) and actual (uncountable) infinities. This is > some of the ancestry of transfinitology. As far as I know, neither Galileo nor Aristotle proposed what could fairly be called a set theory. Anyway, so everything Galileo and/or Aristotle said was true and wise? > Good enough? Why do you ask? Curious to see how you'd finish your half-baked analogy. > Keep on truckin' Don't do your knockin' where your sheeps are aflockin'. MoeBlee > ToeKnee BoerEeng. MoeBlee
From: MoeBlee on 4 Jun 2010 18:18
On Jun 4, 3:25 pm, Aatu Koskensilta <aatu.koskensi...(a)uta.fi> wrote: > MoeBlee <jazzm...(a)hotmail.com> writes: > > It's not fascinating. Aatu is just reminding of the dry fact that in > > conversational mathematics, we do use the word 'more' in a sense not > > confined to cardinality. > > Well, I was also pointing out there are in the logical literature > numerous not at all nonsensical attempts at defining a general notion of > size on which e.g. the rationals are of a larger size than the naturals. Okay, I stand corrected. In any case, the more prosaic sense applies also, as you illustrated. MoeBlee |