Re: An uncountable countable set

In article <ecb0gm$rie$1@xxxxxxxxxxxxxxxxxxxx>,
Tony Orlow <aeo6@xxxxxxxxxxx> wrote:

Virgil wrote:
In article <ec9s3c$8qv$1@xxxxxxxxxxxxxxxxxxxx>,
Tony Orlow <aeo6@xxxxxxxxxxx> wrote:

Then these infinite numbers would not deserve the name "natural number".
Why is that? If they are whole numbers, each with successor, does that
not fit the bill?

What is your definition of "whole number", TO, that allows it to be
applied to objects not members of the minimal of the inductive sets
required by the axiom of infinity?

A point on the real line exactly reachable with an endpoint by placing
unit intervals end-to-end starting from 0 (using either a "countable" or
"uncountable" number of intervals).

No wonder TO has so much trouble with arithmetic.


One of TO's faults is that he will not allow any definition to mean
only what it states, but always has to suppose it to mean something else.
This is a very anti-mathematical anti-logical attitude, and explains
much of TO's difficulty in comprehending things logical and things
mathematical.


More vacuous dismissal. Whatever.

Those aspiring to be mathematicians are supposed to dismiss the vacuous.

In ZF, ZFC andr NBG, a set, S, is called an inductive set if and only if
(1) {} is a member of S, and
(2) Whenever x is a member of S, then union {x,{x}} is a member of S.
The axioms of infinity in those systems declare that there exists
inductive sets, and other axioms guarantee that there must be a minimal
inductive set which is called the set of natural numbers and its
members, and only its members, are natural numbers in those systems.

Very untrue. Those are the von Neumann ordinals, but that is not the
general definition of an inductive set.

They are the only sort of inductive set whose existence is guaranteed by
the axiom of infinity(also called the inductive axiom).
http://en.wikipedia.org/wiki/Axiom_of_infinity
There is a set N, such that the empty set is in N and such that whenever
x is a member of N, the set formed by taking the union of x with its
singleton {x} is also a member of N. Such a set is sometimes called an
inductive set.


If TO wants to call anything else natural numbers, he has put himself
outside the pale.

Into the tan, indeed. Out from the cave. Over the rainbow, if you will.
Why not? Why object to infinite induction as an alternative to
tranfinitology when it's no less reasonable? Why hide behind Georg?

There is a perfectly legitimate version of transfinite induction, see:
http://en.wikipedia.org/wiki/Transfinite_induction
or
http://en.wikipedia.org/wiki/Three_forms_of_mathematical_induction
but it does not support TO's "extended induction" or "infinite
induction" claims.
.

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