An uncountable countable set
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From: Virgil on 25 Aug 2006 21:00 In article <44ef344b(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > For every two new edges, one new path is produced But all such paths, by having to have the terminal edges which created them, are finite, so irrelevant.
From: Tony Orlow on 25 Aug 2006 21:07 MoeBlee wrote: > Tony Orlow wrote: >>> Indeed. Not step by step. And so? I have said this before. I am quite >>> sure others have said that before. You can not get the set of naturals >>> by adding the one by one. It is the axiom of infinity that asserts that >>> that set does exist. >> And that it's infinite. Why do you need an axiom for that? WHy is it not >> derivable logically? > > What do you mean "why"? It is not derivable since there are finite > domains of discourse that satisfy Z-I (Z set theory without the axiom > of infinity). > > There is NO system that will give you any kind of even minimal > mathematics without adopting axioms that are not true in all domains of > discourse. I am not convinced of that. > >> If Ross and I each have our linearizations of the reals, then they >> exist, independent of the axiom of choice (which might as well be called >> the axiom of free will). > > We prove existence of objects in a theory from axioms. You posit the > existence of objectw without any system of logic, primitives, or > axioms. Logic is the primitive of argument. Logical truth is founded upon quantity, and quantity upon geometry. Space is a priori. :) > >>> Yes, but in that case you are not using standard mathematical terminology. >>> And I have yet to see a *definition* of you about *infinite*. >> Larger than any finite. The set of naturals is as large as, but no >> larger than, every natural. > > Define 'larger'. m>n <=> A x (x>m ^ x>n) v (m>x ^ x>n) v (m>x ^ n>x) Define 'finite'. That is a little more difficult - it's relative. Actually, nevermind, since you have > no logicistic system, primitives, or axioms from which to make > defintions. Moreover, you have no idea what are correct forms for > defintions, why such forms are correct and others are incorrect. You > even resort to circular definitions. Comment on 'larger'. > >>> You are using words without definition, using words from standard terminology >>> but with (apparently) not the standard definition. That is what makes you >>> on occasion not understandable. >> I thought it was clear that I was using a notion of infinite, like WM, >> from a quantitative standpoint, rather than set-theoretic. > > Oh, yes, the "quantitative standpoint". Have you read > Zerbernieskoskywoskyozerlichmanosty's paper on the standard > quantitative geometric induction? It's right up your alley. I'll have to google it. Thanks for the reference. :| > > MoeBlee >
From: Tony Orlow on 25 Aug 2006 21:10 MoeBlee wrote: > Tony Orlow wrote: >> Given a set x, can we always determine card(x)? > > I'm guessing, but I'm pretty sure that there is not an algorithm that > will tell you the ordinal index of the aleph that is the cardinality of > any given set. Good guess, except I have no idea what your driving at. > >>>> However, the set of bit positions >>>> required to list the naturals in binary defies classification in this >>>> system. >>> Whatever you mean by that, based on remark that you know of no proof of >>> a contradiction in set theory, it is a problem of your own >>> misconception and not a contradiction in set theory. Fortunately, we >>> don't obligate ourselves to the burdens of your own misconceptions. >> No, but you are obligated to define a cardinality for this set which is >> consistent, if you claim the theory is consistent. You can't. > > First you need to define the set in set theory and prove that it exists > in set theory. Then, set theory would not be inconsistent for there not > being an algorithm to determine the ordinal index of the aleph that is > the cardinality of any given set. So, you're saying that identifying a set with no cardinality doesn't make set theory inconsistent? I think I saw a different opinion yesterday. > > Again, you're just yapping without regard for the specific definitions > of such things as 'consistent'. Consistent: Adj. Without contradiction. > > MoeBlee >
From: Tony Orlow on 25 Aug 2006 21:16 Virgil wrote: > In article <1156501377.098380.50700(a)m79g2000cwm.googlegroups.com>, > mueckenh(a)rz.fh-augsburg.de wrote: > >> Virgil schrieb: >> >> >>>> But as we just investigate consistency, you cannot presuppose it. With >>>> your attitude it is impossible to find any inconsistency even in an >>>> inconsistent theory. Deplorably you are too simple to recognize that. >>> If "Mueckenh" can deduce from any axiom system both a statement within >>> the system and its negation, "Mueckenh" will have found his >>> inconsistency. >> Which part of my proof concerning the binary tree is not in accordance >> with the ZFC axioms in your opinion? > > The part that says a bijective image of the naturals bijects with a > bijective image of the power set of the naturals. But, you claim that a bijective image of the naturals bijects with a bijective image of a set whose power set is the naturals, so who are you to complain? The naturals in binary are the power set of the bit positions in those strings. How can the set biject with its own power set?
From: Tony Orlow on 25 Aug 2006 21:23
MoeBlee wrote: > Tony Orlow wrote: >> MoeBlee wrote: >>> Tony Orlow wrote: >>>> If the naturals are not a subset of the reals in ZFC and NBG, then those >>>> theories are even more screwed up than they already seemed. >>> With either of the usual constructions of the reals (Dedekind cuts or >>> equivalence classes of Cauchy sequences), both the system of naturals >>> and the system of rationals are isomorphically embedded in the system >>> of reals. Mathematicians usually speak of the system of natural numbers >>> and the system of rational numbers as subsystems of the the system of >>> reals. Strictly speaking, that is incorrect, but it is harmless given >>> the isomorphism. >> Harmless, even though incorrect? What makes it incorrect, if not >> inconsistency? Doesn't inconsistency cause a problem? > > Now you're being captious. The informality is harmless, as I said, > because we are within ISOMORPHISM. It has nothing to do with the > consistency of the theory. It's only a matter of INFORMAL convenience > to not have to say each time "the system that is isomorphically > embedded" but instead speak directly of the system as if it were a > subsystem, since, to WITHIN ISOMORPHISM, it is a subsystem. This kind > of informality is common throughout mathematics and is harmless. So, is it technically incorrect to say 1 = 3/3 = 1.000... = 0.999...? I never thought so. > >> I used y in the equation (or statement of inequality) and then specified >> in parentheses at the end, with a space, which condition for y made that >> statement true. I didn't think that was tooooo confusing. > > It was unclear enough that I had to check with you what you meant. It's > not a big deal, but you could acheive clarity by being more explicit. > >>> 'number of any real sort' is not a defined predicate of set theory. w >>> is not in the standard ordering of the reals. That does not make w an >>> undefined object. >> My experience is that asking amthemticians for a definition of "number" >> results in.....nothing. > > Because it's more a philosophical issue or an issue of terminology > outside the system. The purpose of set theory is not to address the > question of what is and is not a number. Rather, among the purposes of > set theory is to axiomatize and construct the various number systems > that are of interest. So, how do you know when you're doing math, and when you've strayed into other territory? > >>>> you don't see scientists accepting transfinitology either. >>> Of course, you have a survey of scientists to support your claim. >> Uh, yes, right here. Why don't you survey thise that object, regarding >> what they do? > > That's quite an unscientific survey method of yours. Yo, man, it's empirical evidence. Ask around. Life's an experiment. > >>> You have no coherent system of definitions at all. It's not the job of >>> set theory to define objects that obey the whims of your informal >>> notions. >> It's the job of mathematicians to work with numbers. > > Numbers are among the primary concern of mathematics. Set theory > axiomatizes and constructs number systems. Which are the other primary concerns, if any? (how did we define "number" again?) > >>>> The selection of any unit is done by simply choosing a point separate >>>> from the origin. When it comes to division using infinite values, one >>>> translates the geometric definition into symbolic form and applies >>>> induction formulaically. >>> "Translates the geometric definition into symolic form and applies >>> induction formulaically" is no less doubletalk than "Coordinates the >>> numeric form into geometric postulates and applies the recursive >>> definition metrically." >>> >> Is that a question? > > Is that a rhetorical question? I'll take as an "I dunno". > > MoeBlee > |