Re: Galileo's Paradox

cbrown@xxxxxxxxxxxxxxxxx wrote:
Tony Orlow wrote:
David Marcus wrote:
Tony Orlow wrote:
David Marcus wrote:
Tony Orlow wrote:
David Marcus wrote:
Tony Orlow wrote:
Why do you have a problem with the mere suggestion of an infinite value?
I don't have a problem with infinite values. However, you have to do
more than merely say "positive infinite n". Assuming your positive
infinite things aren't the same as something that we already know about
(in which case, you should just say so), you either need to give a
construction of these positive infinite things or you need to specify
their properties. You haven't done either. For example, how many of
these things are there? How do they relate to each other? How do they
interact with the natural numbers? Are the operations of addition and
multiplication defined for them?
Well, I have been through much of that regarding such specific language
approaches as the T-riffic digital numbers, but that's not necessary for
this purpose. It suffices to say that, if a statement is proved true for
all n greater than some finite k, that that also includes any postulated
infinite values of n, since they are greater than any finite k. I don't
need to construct these numbers. Consider them axiomatically declared.
Then list the axioms for them.
(sigh)


(T1) infinite(x) <-> A yeR x>y

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. A
whole array of theorems pop forth...

Before going there, you might want to start by adding the axiom:

(T2) exists B such that infinite(B)

Otherwise, who cares if you can prove a whole bunch of theorems about
something that doesn't exist?

Cheers - Chas


What do you mean by "exist"? We can postulate the existence of such a set, and derive conclusions about it. It seems to me that it exists by virtue of being referenced.
.

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