Re: Galileo's Paradox

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
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Relevant Pages

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