An uncountable countable set
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From: MoeBlee on 19 Sep 2006 15:00 Tony Orlow wrote: > Now, the general rule that > adding more unique elements makes a set larger, that is, increases its > measure of "size", is clearly violated by the "general" rules of set > theory. If it serves as a foundation for all of mathematics, then how > can it contradict any portion thereof? Yes, by all means, when you've devised a recursively axiomatized theory sufficient for ordinary calculus but such that there are no sets that can be put into 1-1 correpsondence with a proper subset, then do let us know. MoeBlee
From: Tony Orlow on 19 Sep 2006 16:14 MoeBlee wrote: > Tony Orlow wrote: >> If you are so well-acquainted with these notions, could you be troubled >> to define a distinction between them? > > Oh, please. They are well known concepts. I don't offer any explication > beyond what you would find in any general presentation of the subject, > just as I don't offer any special explication of such concepts as > 'objective'/'subjective', or 'a priori'/'a posteriori', or hundreds of > others. > > MoeBlee > Oh. Well, then, allow me to un-snip the context: MoeBlee wrote: > Han de Bruijn wrote: >> MoeBlee wrote: >> >>> Tony Orlow wrote: >>> >>>> Unbounded but finite may >>>> be considered potentially, but not actually, infinite. >>> That will be jiffy, once you give axioms and/or our definitions for >>> 'potentially infinite' and 'actual infinite'. Until then, it's pure >>> handwaving. >> Come on, Moeblee, don't be silly! Let Google be your friend! > > I'm well aware of the notions of 'potential infinity' and 'actual > infinity' that go back through at least a couple thousand years of > philosophy and philosophy of mathematics. So, you ask for axioms defining the terms, then cannot gives any "explication" yourself? I gave you axioms a little while ago. ToeKnee
From: Virgil on 19 Sep 2006 16:38 In article <45102757(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > Transfinitology draws conclusions well beyond the purview of any finite > arithmetic. TO draws conclusions fare beyond the purview of any arithemetic, finite or otherwise.
From: Virgil on 19 Sep 2006 16:41 In article <45102b5d(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > MoeBlee wrote: > > Tony Orlow wrote: > >> I am well aware my position is "provably false", but I am > >> also aware that that depends on the axioms assumed and the rules > >> regarding logical inference. > > > > No, your "position" is not coherent enough to be addressed as either > > true or false. A bunch of gibberish doesn't admit of examination for > > truth or falsehood. > > You are wrong. If I assert that S = proper_subset(T) -> |S|<|T|, then > this is a basic axiom that one can accept which makes all of > transfinitology untenable. Depends on how one defines |S| and |T|. Given TO's assertion above, Card(S) and |S| are not the same, but that in no way interferes with the validity of cardinality.
From: Virgil on 19 Sep 2006 16:47
In article <45102c08$1(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > MoeBlee wrote: > > I'm well aware of the notions of 'potential infinity' and 'actual > > infinity' that go back through at least a couple thousand years of > > philosophy and philosophy of mathematics. > > If you are so well-acquainted with these notions, could you be troubled > to define a distinction between them? > > > But to use those notions in > > an axiomatic mathematics requires either defining the terms > > 'potentially infinite' and 'actually infinite' in the axiomatic theory > > or taking them as primitive and giving axioms for them in the axiomatic > > theory. So far, that has not been done in this thread or in any other > > thread I've happened to read. > > It depends on the definition of an index for each element in an ordered set. Absent any details about the nature of that alleged dependency, Moblee's criticism stands. |