Re: Galileo's Paradox

Virgil wrote:
In article <457b8ccf@xxxxxxxxxxxxxxxxxxx>,
Tony Orlow <tony@xxxxxxxxxxxxx> wrote:

Virgil wrote:
In article <457b5606@xxxxxxxxxxxxxxxxxxx>,
Tony Orlow <tony@xxxxxxxxxxxxx> 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.

What, now mathematics has declared what the single letter n means? It's a variable.

Every variable has a domain of definition. The variable 'n' is quite commonly required to have the set of finite naturals as its domain.

I merely remarked that when that is the case, there are no values for n other than finite ones.

Why do you have a problem with the mere suggestion of an infinite value?
It is arithmetical operations with infinite values absent any definition of what those operations mean or what properties they have, to which we make legitimate objections.
Not really.

Really.


No. Reread the following:

If the expressions used can themselves be ordered using infinite-case induction, then we can say that one is greater than the other, even if we may not be able to add or multiply them. Of course, most such arithmetic expressions can be very easily added or multiplied with most others. Can you think of two expressions on n which cannot be added or multiplied?

I can think of legitimate operations for integer operations that cannot be performed for infinites, such as omega - 1.

Omega is illegitimate schlock. Read Robinson and see what happens when omega-1<omega.

Surely you must have guessed enough what I meant to follow the paragraph? (sigh)
Guessing is not a reliable way of finding things out.
Of course, reading is a little more reliable, but when one gets stuck on such words as "positive" and "infinite", maybe one needs to do a little guessing.

If TO's descriptive powers are so insubstantial as to leave us perpetually guessing what he means, he will never be able to convey anything of mathematical interest or significance.

If you don't know what "positive" or "infinite" mean, that's not my fault.
.

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