The Definition of Points
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From: MoeBlee on 18 Apr 2007 15:24 On Apr 18, 12:04 pm, Tony Orlow <t...(a)lightlink.com> wrote: > Mike Kelly wrote: > The successor relation is a relation not related to 'e' except in a > particular model of the naturals upo which the rest is built. In set theory such as Z set theory, EVERY relation among objects is based on the membership relation. > >> but in my opinion doesn't produce believable results. > > > Which "results" of cardinality are not believable? You don't think the > > evens and the naturals and the rationals are mutually bijectible? Or > > you do think the reals and the naturals are bijectible? Or did you > > mean you have a problem not with what cardinality actually is but with > > what you hallucinate cardinality to be? > > As long as transfinite "equivalence" classes are referred to as such, > and not said to denote "equal" size or numerosity, then I have no problem. In theories such as Z set theory and its variants the equivalence classes are NOT themselves cardinalities. > I suppose it's more the way it's spoken about. Leave out "size" and > "equinumerous" and talk about "equivalence classes" and not "equality", > and I have no problem with the consistency of standard theory. You've already been told a million times that the words 'size' and 'equinumerous' are dispensible. So, whenever someone says 'same size' or 'equinumerous', just take them to have said 'equipollent' instead. Anyway, consistency has nothing to do with whether we use the word 'size' and 'equinumerous'. And we do NOT say that two sets are equal just on account of being equipolllent, but rather that the cardinality (and here you can use another word if 'cardinality' does not suit you; maybe a word such as 'pollent-ordinal') of the sets is equal, as indeed, if x and y are equipollent, then the pollent-ordinal of x (the least ordinal that is bijectable with x) IS IDENTICAL with the pollent- ordinal of y. MoeBlee
From: Virgil on 18 Apr 2007 15:25 In article <46265bac(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > Virgil wrote: > > In article <4625a9df(a)news2.lightlink.com>, > > Tony Orlow <tony(a)lightlink.com> wrote: > > > >> Mike Kelly wrote: > >> < snippery > > >>> You've lost me again. A bad analogy is like a diagonal frog. > >>> > >>> -- > >>> mike. > >>> > >> Transfinite cardinality makes very nice equivalence classes based almost > >> solely on 'e', but in my opinion doesn't produce believable results. > > > > What's not to believe? > > > > Cardinality defines an equivalence relation based on whether two sets > > can be bijected, and a partial order based on injection of one set into > > another. > > > > Both the equivalence relation and the partial order behave as > > equivalence relatins and partial orders are expected to behave in > > mathematics, so what's not to believe? > > > > > > > > What I have trouble with is applying the results to infinite sets and > considering it a workable definition of "size". Mike's right. If you > don't insist it's the "size" of the set, you are free to do with > transfinite cardinalities whatever your heart desires. We don't insist that it is "THE" size but we do insist that it is "A" size. It is TO's error to assume that there can only be one kind of size. What I object to > are statements like, "there are AS MANY reals in [0,1] as in [0,2]", > and, "the naturals are EQUINUMEROUS with the even naturals." If you say > they are both members of an equivalence class defined by bijection, then > I have absolutely no objection. If you say in the same breath, "there > are infinitely many rationals for each natural and there are as many > naturals overall as there are rationals", without feeling a twinge of > inconsistency there, then that can only be the result of education which > has overridden natural intuition. That's my feeling. Since "natural intuition" often contradicts the requirements of logic, mathematicians and logicians have learned to be cautious about giving "natural intuition" too much weight. > > I'd rather acknowledge that omega is a phantom quantity, and preserve > basic notions like x>0 <-> x+y>y, and extend measure to the infinite scale. > > >> So, > >> I'm working on a better theory, bit by bit. I think trying to base > >> everything on 'e' is a mistake, since no infinite set can be defined > >> without some form of '<'. I think the two need to be introduced together. > > > > Since any set theory definition of '<' is ultimately defined in terms of > > 'e', why multiply root causes? > > It's based on the subset relation, which is a form of '<'. 'Subset' is based on 'e', at least in all the set theories I am aware of. Does TO have a reference to some set theory in which 'subset' is NOT based on 'e'? It so he should post it immediately, as it will revolutionize mathematics.
From: Virgil on 18 Apr 2007 15:37 In article <4626606d(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > Virgil wrote: > > In article <462507cf(a)news2.lightlink.com>, > > Tony Orlow <tony(a)lightlink.com> wrote: > >>> It is one form of size for all sets? One might use the physical analogy > >>> that volume, surface area, and maximum linear dimension are all measures > >>> of the size of a solid. So implying that one "size" fits all is false. > >>> > >>> > >> Each of those is derivative of the last, given the proper unit of > >> measure. > > > > So does one use distance, area, volume? > > > > What is the maximum length of a solid in cubic meters, or in square > > meters? > > What is its surface area in cubic meters or in linear meters? > > What drug are you on? Since TO claims that one object can only have one measure of size, I am trying to show him what his claim implies. The TO chooses not to understand is better evidence that TO is drugging up than that anyone else doing so.
From: Mike Kelly on 18 Apr 2007 15:39 On Apr 18, 6:55 pm, Tony Orlow <t...(a)lightlink.com> wrote: > Virgil wrote: > > In article <4625a...(a)news2.lightlink.com>, > > Tony Orlow <t...(a)lightlink.com> wrote: > > >> Mike Kelly wrote: > >> < snippery > > >>> You've lost me again. A bad analogy is like a diagonal frog. > > >>> -- > >>> mike. > > >> Transfinite cardinality makes very nice equivalence classes based almost > >> solely on 'e', but in my opinion doesn't produce believable results. > > > What's not to believe? > > > Cardinality defines an equivalence relation based on whether two sets > > can be bijected, and a partial order based on injection of one set into > > another. > > > Both the equivalence relation and the partial order behave as > > equivalence relatins and partial orders are expected to behave in > > mathematics, so what's not to believe? > > What I have trouble with is applying the results to infinite sets and > considering it a workable definition of "size". Mike's right. If you > don't insist it's the "size" of the set, you are free to do with > transfinite cardinalities whatever your heart desires. What I object to > are statements like, "there are AS MANY reals in [0,1] as in [0,2]", > and, "the naturals are EQUINUMEROUS with the even naturals." If you say > they are both members of an equivalence class defined by bijection, then > I have absolutely no objection. Then you have absolutely no objection. Good that you recognise it. > If you say in the same breath, "there > are infinitely many rationals for each natural and there are as many > naturals overall as there are rationals", And infinitely many naturals for each rational. > without feeling a twinge of > inconsistency there, then that can only be the result of education which > has overridden natural intuition. That's my feeling. Or, maybe, other people don't share the same intuitions as you!? Do you really find this so hard to believe? > I'd rather acknowledge that omega is a phantom quantity, By which you mean "does not behave like those finite numbers I am used to dealing with". >and preserve basic notions like x>0 <-> x+y>y, and extend measure to the infinite scale. Well, maybe you'd like to do that. But you have made no progress whatsoever in two years. Mainly, I think, because you have devoted rather too much time to very silly critiques of current stuff and rather too little to humbling yourself and actually learning something. > >> So, > >> I'm working on a better theory, bit by bit. I think trying to base > >> everything on 'e' is a mistake, since no infinite set can be defined > >> without some form of '<'. I think the two need to be introduced together. > > > Since any set theory definition of '<' is ultimately defined in terms of > > 'e', why multiply root causes? > > It's based on the subset relation, which is a form of '<'. Get this through your head : every relation between objects in set theory is based on 'e'. It's really pathetic to keep mindlessly denying this. Set theory doesn't just "try to base everything on 'e'". It succeeds. -- mike.
From: Virgil on 18 Apr 2007 15:40
In article <462660f1(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > Virgil wrote: > > In article <46250ad6(a)news2.lightlink.com>, > > Tony Orlow <tony(a)lightlink.com> wrote: > > > >> Virgil wrote: > > > >> Yes, it can be that x<y and y<x and y<>x. > > > > Then that is not an order relation. In any form of order relation "<" > > allowed in mathematics (x<y and y<x) requires x=y. > >>>>> For any in which "<" is to represent the mathematical notion of an > >>>>> order > >>>>> relation one will always have > >>>>> ((x<y) and (y<x)) implies (x = y) > >>>>> > >>>> Okay, I'm worried about you. You repeated the same erroneous statement. > >>>> You didn't cut and paste without reading, did you? Don't you mean "<=" > >>>> rather than "<". The statement "x<y and y<y" can only be true in two > >>>> unrelated meanings of "<", or else "=" doesn't have usable meaning. > >>> TO betrays his lack of understanding of material implication in logic. > >>> For "<" being any strict order relation, "(x<y) and (y<x)" must always > >>> be false so that any implication with "(x<y) and (y<x)" as antecedent > >>> for such a relation, regardless of conseqeunt, is always true. > >> Oh, yes, well. Any false statement implies any statement, true or false, > >> as long as you're not an intuitionist. If (x<y) -> ~ (y<x), then x<y ^ > >> y<x is of the form P ^ ~P, or ~(P v ~P) which is false in classical > >> logic, but not intuitionistically. There is debate on this topic. > > > > I very much doubt that any intuitionist would say that for an order > > relation,"<", on any set one could have x<y and y<x without having x=y. > > > >>>>> Is TO actually claiming that the irrationals form a subset of the > >>>>> countable set NxN. > >>>>> > >>>>> That is NOT how it works in any standard mathematics. > >>> Then why does TO claim it? > >> I am not. I am saying it is a set equal in magnitude to the redundancies > >> in NxN. > > > > The set of "redundancies in NxN" must be a subset of NxN itself, so must > > be countable, so that TO is saying that the set of irrationals is "equal > > in magnitude" to a countable set. > > Well, actually, I mean N*xN*, including infinite naturals in N*. Sorry, > I should have been more specific. At what point did N morph into N*? Everything up to now has been in terms of plain N. And isn't doesn't N* surject to N*xN*, making N*xN* countable in R*? |