Galileo's Paradox
in [Math]
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From: Tony Orlow on 9 Dec 2006 19:36 Virgil wrote: > In article <457b30ed(a)news2.lightlink.com>, > Tony Orlow <tony(a)lightlink.com> wrote: > >> Virgil wrote: >>> In article <457af56a(a)news2.lightlink.com>, >>> Tony Orlow <tony(a)lightlink.com> wrote: >>> >>>> But, >>>> there are serious questions regarding the measure of infinite sets, as >>>> Six Letters points out. >>> Questions about mathematics considered serious by non-mathematicians >>> most often merely demonstrate that they are non-mathematicians. >>> >> Yes, well, math can use a little feedback sometimes. >> >>>> It's not unreasonable to consider the proper >>>> subset as always smaller than the superset, though that is clearly not >>>> how cardinality works for infinite sets. >>> It is not unreasonable to have different measures of set size depending >>> on what properties of those sets are being measured. >>> >>> And it is not unreasonable to note that different measures will behave >>> differently, that when for one measure of one property, two sets are >>> the same but for a measure of a different property the same two sets >>> differ, that the two measures will disagree as well. >>> >>> >> Sure, and it's not unreasonable to say that, if one measure says equal >> and another says unequal, that they are unequal in a way that the first >> measure does not detect, but that exists. >> >>>> So, there is an ongoing attempt >>>> in various directions to develop a better theory, or to reject it as an >>>> absurdity. >>> The absurdity is to suppose that sets have only one measurable quality. >>> >> Nobody says that. > > Then why is TO trying to exile cardinality? > > I'm not. I'm just trying to dethrone the court jester. >> Sets may have many measures. The question is how best >> to formulate and compare sizes for infinite sets, the size being the >> count of elements. > > For finite sets, "count" is unambiguous, but for infinite sets, it can > have more than one meaning, and therefore more than one measure, > depending on which "counting" process one is using. Yeah. some with better powers of distinction than others. Cardinality is woefully blind.
From: Tony Orlow on 9 Dec 2006 19:39 Virgil wrote: > In article <457b2f54(a)news2.lightlink.com>, > Tony Orlow <tony(a)lightlink.com> wrote: > >> David Marcus wrote: >>> Tony Orlow wrote: >>>> There is nothing wrong with saying E and N have the same cardinality. >>>> It's a fact. Six Letters is essentially suggesting IFR, my Inverse >>>> Function Rule, which indeed can parametrically compare sets mapped onto >>>> the real line, using real valued functions, as a generalization to set >>>> density. It works for finite and infinite sets. So, what is wrong with >>>> trying to form a more cohesive theory of infinite set size, which >>>> distinguishes set sizes that cardinality cannot? There certainly seems >>>> to be intuitive impetus for such a theory. >>> Fine. Please state your rule. Let's take a look. >>> >> I've been over this a lot, but hey, what's one more time. Practice makes >> perfect. >> >> We start with the notion of infinite-case induction, such that a >> equation proven true for all n greater than some finite k holds also for >> any positive infinite n. > > Whatever for? Math already has both finite and transfinite induction. > For the purpose of establishing an intuitively satisfying theory of infinite sets. Duh. >> Inequalities can also be proven true for >> infinite n, but only provided that the difference between expressions >> which forms the inequality does not have a limit of 0 as n grows without >> bound. If it does, the inequality holds only for finite n. Now, given >> this extension of classical inductive proof, we can easily prove such >> facts as, say, 2<n <-> 2 < n < 2n < n^2 < 2^n < n^n, and this ordering >> will be true for any infinite n. Thus we have a full spectrum of >> infinite expressions which can be ordered, provided we have some common >> infinite n with which to express them. Okay so far? > > Nope. Until one has unambiguous definitions of what is meant by the > various arithmetical operations with infinite "numbers" as arguments, > the whole thing is nonsense. Not so. If we can express infinite values in a language with rules that enable us to order the various possible expressions, then we have ordered the values expressed in that language. You can call it nonsense if you like, but all that means is that you can't make sense of even the simplest concepts that you haven't already been taught before your neurons got calcified. Tsk tsk.
From: Tony Orlow on 9 Dec 2006 19:41 Virgil wrote: > In article <457b3353(a)news2.lightlink.com>, > Tony Orlow <tony(a)lightlink.com> wrote: > >> Virgil wrote: > >>> I also, do not regard the set {(2,4), (4,2)} as a"real measure", nor a >>> "measure of the truth of a statement", but that does not mean that I >>> regard 2 or 4 as anything other than natural number quantities. >>> >> {x,y}| xeN ^ yeN ^ x^y=y^x = {(2,4),(4,2)} >> >> That's a logical statement with a truth value of 1. > > It is, logically speaking, not even a WFF, so cannot be either true or > false. > > { (x,y) | xeN ^ yeN ^ x^y=y^x} = {(2,4),(4,2)} is a logically WFF. Oh wow, you put brackets. It was so ambiguous before, and obviously had no truth value. Now it does, right? (sigh)
From: Virgil on 9 Dec 2006 20:03 In article <457b5573(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > David Marcus wrote: > > Tony Orlow wrote: > >> David Marcus wrote: > >>> Tony Orlow wrote: > >>>> Countably infinite means potentially infinite. > >>> If you wanted to say "potentially infinite" means countably infinite, > >>> why didn't you just say so? I don't think anyone else who has been using > >>> the phrase "potentially infinite" recently shares your meaning. > >> No? Why don't you ask them. It seems pretty clear to me that that's a > >> pretty common conception. > > > > Are you using the standard meaning for "countably infinite"? > > Yes, an unbounded set (no last element), with all elements finitely > close to the beginning of the set, in whatever order is chosen. In other > words, a set bijectible with the naturals. That requires that a TO-countable set be an ordered set, rather than merely being orderable, and be a metric space in some senserealted to that order, whereas the set of points in R^2 with both coordinates algebraic (or equivalently, the set of algebraic complex numbers), while being countable in the ordinary way, does not meet TO's requirements. > > > > > As for asking WM, EB, etc., they rarely give coherent answers when they > > are asked questions. I'm sure I've asked both WM and EB several times to > > explain what they mean by "potential infinity". I'm afraid I still have > > only the vaguest idea what they mean. > > > > I think "countable infinity" is what they mean. Endless, and yet, not > requiring infinite strings to index any of its elements, using a normal > finite digital system. I am reasonably sure that no such notions of indexing is involved in their "countably infinite". > I personally see a conflict between countable > infinity and what I consider actually infinite. That is your problem, TO.
From: Virgil on 9 Dec 2006 20:09
In article <457b5606(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > David Marcus wrote: > > Tony Orlow wrote: > >> David Marcus wrote: > >>> Tony Orlow wrote: > >>>> There is nothing wrong with saying E and N have the same cardinality. > >>>> It's a fact. Six Letters is essentially suggesting IFR, my Inverse > >>>> Function Rule, which indeed can parametrically compare sets mapped onto > >>>> the real line, using real valued functions, as a generalization to set > >>>> density. It works for finite and infinite sets. So, what is wrong with > >>>> trying to form a more cohesive theory of infinite set size, which > >>>> distinguishes set sizes that cardinality cannot? There certainly seems > >>>> to be intuitive impetus for such a theory. > >>> Fine. Please state your rule. Let's take a look. > >> I've been over this a lot, but hey, what's one more time. Practice makes > >> perfect. > >> > >> We start with the notion of infinite-case induction, such that a > >> equation proven true for all n greater than some finite k holds also for > >> any positive infinite n. > > > > What is a "positive infinite n"? > > > > A value greater than any finite value. But if "n", is, as usual, reserved for indicating natural numbers, those which are members of every inductive set, there is no such thing. > > Why do you have a problem with the mere suggestion of an infinite value? It is ar\ithmetical operations with infinite values absent any definition of what those operations mean or pwhat properties they have, to which we make legitimate objections. > Surely you must have guessed enough what I meant to follow the > paragraph? (sigh) Guessing is not a reliable way of finding things out. |