Re: Galileo's Paradox

Tony Orlow wrote:
David Marcus wrote:
Tony Orlow wrote:
David Marcus wrote:
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.

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?

No. See above.

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?

--
David Marcus
.

Relevant Pages

  • Re: Galileos Paradox
    ... I don't have a problem with infinite values. ... construction of these positive infinite things or you need to specify ... Then list the axioms for them. ... such as that the size of the even naturals is half that of the naturals. ...
    (sci.math)
  • Re: Galileos Paradox
    ... I don't have a problem with infinite values. ... construction of these positive infinite things or you need to specify ... Then list the axioms for them. ...
    (sci.math)
  • Re: Galileos Paradox
    ... I don't have a problem with infinite values. ... construction of these positive infinite things or you need to specify ... then state your first theorem about them ... such as that the size of the even naturals is half that of the naturals. ...
    (sci.math)
  • Re: Galileos Paradox
    ... 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. ... Big'un is the number of reals per unit interval. ...
    (sci.math)
  • Re: Galileos Paradox
    ... I don't have a problem with infinite values. ... construction of these positive infinite things or you need to specify ... Then list the axioms for them. ...
    (sci.math)