Galileo's Paradox
in [Math]
|
From: Tony Orlow on 9 Dec 2006 16:49 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. 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? 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. Tony
From: Tony Orlow on 9 Dec 2006 16:55 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. 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. >> I prefer the first route, where IFR and N=S^L grow out of >> infinite-case induction and provide a means for finely ordering all >> sorts of infinite sets, countable and uncountable. > > IFR only applies, if at all, only for order preserving functions from > ordered sets to ordered sets, which leaves out all unordered sets and > multiply ordered sets. It cannot even deal with subsets of the Cartesian > plane. Yes any order-preserving bijection with the hypernaturals can produce a formulaic relative size of the set. > > N = S^L is only demonstrably valid for finite sets, for which there are > no problems with cardinality anyway, so is entirely redundant. When applied to the infinite case, it answers some questions about relationships between languages and measures very nicely.
From: David Marcus on 9 Dec 2006 17:01 Virgil wrote: > In article <457af70f(a)news2.lightlink.com>, > Tony Orlow <tony(a)lightlink.com> 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. > > For one thing, measure theory already has many such measures on the real > line and on much more complicated spaces that the IFR cannot even begin > to deal with. > > For another, The IFR essentially can only measure certain special > subsets of the real line, but fails for many subsets which other > measures deal with easily. > > Before trying to invent something entirely new, TO should try to find > out a bit about what already exists. Being the second person to invent > something does not earn any points. Actually, if Tony was able to independently invent something useful, or even just explain what he is talking about coherently and without exaggerating, he would increase his score quite a bit. I once had what I thought was a very good idea. Eventually, I found out that it was already known under the name "automatic differentiation". I was happy that I had thought of something that others found useful and that people were deriving benefit from the technique, even though the latter was not due to me. -- David Marcus
From: David Marcus on 9 Dec 2006 17:04 Virgil wrote: > In article <457af8dd(a)news2.lightlink.com>, > Tony Orlow <tony(a)lightlink.com> wrote: > > > David Marcus wrote: > > > Tony Orlow wrote: > > >> David Marcus wrote: > > >>> Tony Orlow wrote: > > > > > >>>> Isn't the purpose of math be to quantify? > > >>> No. > > >> What,in mathematics, has a solution which is neither a real measure, or > > >> the measure of truth of a statement, 0, 1, or somewhere in between? > > >> Measure=maths. > > > > > > You have a very limited view of what mathematics is. A better > > > description is, Mathematics is the study of patterns. I believe Saunders > > > MacLane presented this view in his book "Mathematics, Form and > > > Function". You might like this book. MacLane believed set theory is not > > > the best foundation for mathematics. > > > > > > > I have a simplistic view of it. There is measure, the language of > > measure, and the operations within the language. To me, logic is a form > > of mathematics where all values lie within the first unit interval. I > > try to stick to the basics - Occam's Razor, eh? Thanks for the > > reference, by the way. > > > > > > TOny > > Saunders Mac Lane preferred to have his family name spelled with a space > between "Mac" and "Lane". > E.g., See http://en.wikipedia.org/wiki/Saunders_Mac_Lane > > It is a preference often ignored. My apologies. I looked at his and Birkhoff's book "Algebra" to check the spelling. The title page spells it "SAUNDERS MacLANE". On the other hand, the copyright page does spell it "Mac Lane". -- David Marcus
From: Tony Orlow on 9 Dec 2006 17:06
Virgil wrote: > In article <457aff26(a)news2.lightlink.com>, > Tony Orlow <tony(a)lightlink.com> wrote: > >> cbrown(a)cbrownsystems.com wrote: >>> Tony Orlow wrote: >>> >>>> What,in mathematics, has a solution which is neither a real measure, or >>>> the measure of truth of a statement, 0, 1, or somewhere in between? >>> "Find all pairs of distinct naturals (x,y) such that x^y = y^x". The >>> solution to which is the set {(2,4), (4,2)}, which doesn't appear to be >>> a "real measure", nor a "measure of the truth of a statement" (as far >>> as I can understand your meaning of the terms). >> Those are specifications for two points in a 2D array of naturals, the >> values within each pair denoting the distance of each point in each of >> the two directions, from the origin. Are you saying x and y and 2 and 4 >> are not taken to be quantities? That's a rather strange position to take. > > Where does TO get such peculiar misinterpretations? > > CBrown says 'the set {(2,4), (4,2)}, which doesn't appear to be > a "real measure", nor a "measure of the truth of a statement"'. > > 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. 2 and 4 are the integral real quantities which satisfy the requirements, as expressed in the language including those quantities, and the operators '=' and '^' and 'e'. >>> I assume that you offer more than the trivial observation that all >>> mathematical statements P, including the statement "2^4=4^2", are >>> examples of the (boolean) truth valued statement "it can be proved that >>> P". If so, I would claim that the mathematical question is "Find a >>> proof of P, or proof of not P". And the solution is not "0" or "1"; the >>> solution is either a proof of P, or a proof of not P. >> First of all, that trivial observation is plenty. All logic is subsumed >> under math as a calculation of truth values between 0 and 1. > > But one "embed" logic within mathematics without having logic a priori > with which to build that model of logic. > > As a practical matter, logic is a priori to every axiom system. One can > model logic within mathematics, but that is not at all the same thing. > I think you should read Boole's "An Investigation into the Laws of Thought." He seemed to have a pretty mathematical view of logic when he invented the modern form of it. >> That's fine. I am still of the opinion that math boils down to measure, >> the language of measure, and the operations allowed on that language. > > Fortunately, TO's opinion is of no weight. But it has a great value range. |