Galileo's Paradox
in [Math]
|
From: David Marcus on 9 Dec 2006 17:09 Tony Orlow wrote: > Lester Zick wrote: > > On Fri, 08 Dec 2006 14:20:32 -0500, Tony Orlow <tony(a)lightlink.com> > > wrote: > > > >>> Now we're all aware of phase transition rules for imaginary numbers. > >>> c+ni is no greater or less than c alone unless n is even in which case > >>> there is a transition from the i phase to the c phase and c+ni becomes > >>> larger than c alone. > >> I am not aware of that. Please give me a good link that explains it. It > >> might prove quite important. Thanks. > > > > Just complex algebra, Tony, where powers of i contribute to cardinal > > magnitudes according to whether they're even or not. > > > > Uh, why don't I see any powers of i in c+ni? Do you mean c+i^n? Yes, > there's a funny interaction between the real and imaginary components of > complex numbers which maps to angular and linear operations on the > radius of a circle. What does this have to do with infinitesimals? They > are not produced by calculus, but used to form the basis of it. A number > infinitesimally close to x is considered equal to x in standard math. I think Lester would like to avoid having a real number plus an infinitesimal be a real number. So, he wants to keep them separate, like we do when we add a real number and an imaginary number. -- David Marcus
From: David Marcus on 9 Dec 2006 17:12 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"? 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. -- David Marcus
From: David Marcus on 9 Dec 2006 17:15 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"? > 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? No. See above. > Now, where we are bijecting the naturals with a subset of the reals > through a mapping formula, if that function is monotonically increasing > or decreasing, then there is a formulaic relation between the count and > the value range of each set. For instance, we map the naturals to the > evens using e=2n. For every set of naturals up to n, there is a > corresponding set of evens up to 2n. For every set of evens up to e, > there is a corresponding set of naturals up to e/2. In other words, the > COUNT of the set up through the value e is e/2, because that's the upper > bound on the naturals which map to it. So, the inverse of the formula > describes the size of the set. That is the Inverse Function Rule. > > This works for finite sets mapped from the naturals as well as infinite, > but in order to accommodate any value range that one might plug in, > whether the values map to naturals or not, we have to employ the floor > function. Where N maps to S using f(n), and f(g(n))=g(f(n))=n, within > the value range [x,y] we have floor(|g(y)-g(x)|+1) elements. For > infinite sets we can dispense with the floor function, and consider the > interval [0,n], for some assumed infinite range of n, such that the > count if g(n)-g(0). > > I think this might be appealing to Six Letters. Let's hope not. -- David Marcus
From: Virgil on 9 Dec 2006 17:23 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. > 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.
From: Virgil on 9 Dec 2006 17:31
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? >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. |