Re: abundance of irrationals!)

Virgil said:
> In article <MPG.1ce421053d62a336989bc4@xxxxxxxxxxxxxxxxxxxxxxxxx>,
> Tony Orlow (aeo6) <aeo6@xxxxxxxxxxx> wrote:
>
>
> > The natural numbers are defined using the iterative process of counting, to
> > begin with. "By definition" we declare the set infinite and the members
> > finite,
> > but if that is an illogical definition, then it's not useful.
>
> It is not possible to build a physical model of the natural numbers, so
> that those whose imagination is limited to what they can see see illogic.
>
> Those who, like mathematicians, are used to dealing with axiom systems
> in their mental world, require actual internal conflicts in those axiom
> systems before rejecting them.
This is exactly my point. What is "internal"? Is math a field in itself, and if
so, should it contradict itself? Many different kinds of math have developed
from two sides at once independently, and this is a beautiful thing, when we
discover two apparently different systems are actually equivalent in some
manner, when they agree. What agrees with cardinality, within the world of
math? Does it actually contradict other areas of math and logic? If so, does
the mistake belong to cardinality or to the other maths? These are valid
questions.
>
> > What goo does
> > it
> > do us to have it this way? What does it solve, and what would it break to
> > change it?
>
> You can change to any axiom system you like, but you cannot persuade
> others give up an axiom system except by showing that it is internally
> inconsistent. Which you have not done.
I have shown that one needs to take into account the nature of the elements in
one's set, and make sure the conclusions drawn don't violate the rules for
those elements. For instance, a set of strings of length l constructed of
symbols from a set with size s is known to have a total number of possible
strings of s^l. So, we know the finiteness of such a set is dependent on the
finiteness of the length of its strings and vice versa, and so a set of strings
cannot be said to have an infinite number of distinct finite strings
constructed from a finite alphabet. If these strings are digital numbers as we
all know they are defined, then to say an element has an infinite number of
non-zero digits necessarily implies it has an infinite value. Therefore an
infinite set of distinct finite natural numbers represented as digital numbers
necessarily includes infinite values. To have an infinite set of distince
finite natural numbers is mathematically impossible, and if Cantor disagrees,
there's a problem there. Cantor is at odds with so much of mathematics, that
it's hard to imagine ANY real utility coming from the bizarre conclusions it
draws.
>
> Your continued assertions that our axiom system does not satisfy what
> you insist on as axioms is irrelevant. You have not, and cannot show
> that, for example, the Zermelo-Frankel axiom system, is internally
> inconsistent. This does not assert that there is no such inconsistency,
> only that neither Orlow nor WM are sufficiently competent to be able to
> find one.
>
These are the points of which Lester speaks, the withdrawal not only into
specialized fields like physics, psychology or math, but into tiny
departmentalized sets of axioms or systems, and a focus on a very few points at
a time. The lines I would like to draw at this time are between various areas
of math and the concept of infinity, but whenever these things come up,
somebody whips out set theory and dismisses any differing conclusions. I am
sick to death of set theory as it stands, so I am afraid I am going to have to
struggle with pulling it down, like the Berlin wall. Don't worry, I already
have beautiful shrubberies to replace it, and a rather nice set of obelisks and
fountains lined up. :D

--
Smiles,

Tony
.

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