Re: Galileo's Paradox

Virgil wrote:
In article <457c658c$1@xxxxxxxxxxxxxxxxxxx>,
Tony Orlow <tony@xxxxxxxxxxxxx> wrote:

David Marcus wrote:
Tony Orlow wrote:
David Marcus wrote:
Tony Orlow wrote:
I don't need to construct these numbers. Consider them axiomatically declared.
Then list the axioms for them.
(sigh)

infinite(x) <-> A yeR x>y
Is that your only axiom? If so, then state your first theorem about them and give the proof.

That's the only one necessary for what defining a positive infinite n.

It is a definition, not an axiom. And as it does not imply existence of anything, one has no more that other axioms will supply.




Since it is a statement of implication, it's a rule, and an axiom. That it serves as the definition of infinite doesn't make it not a rule for determining what values are infinite, once finites are defined.

A whole array of theorems pop forth from infinite-case induction and IFR, such as that the size of the even naturals is half that of the naturals.

Not unless you declare them as axioms, as neither holds without being assumed.

If n>k -> f(n) (inductively proven), and infinite(n) -> n>k, then infinite(n) -> f(n), and the property can be said to hold in the infinite case. I suppose what you want is an exact statement of IFR so that we can determine which inductive proofs are allowable? Or, what?
.

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