Re: when does induction fail? well-ordering property?

On Sun, 20 Nov 2005, comtech wrote:

> My understanding of induction proof goes as follows:
>
> 1. First prove P(1) true;
> 2. Prove P(n) implies P(n+1) for any n in set of Natural numbers ...
>
> I vaguely heard that it works for finite number of cases, and works for
> countably infinitely many of cases; and it does not work for uncountably
> infinitely many of cases, because the n is in the Natural numbers
> set...
>
> Is my understanding right?
>
Understanding of what? What works? Be clear and precise or nothing
mathematical will be understood.

Induction
Induction is (1) and (2) implies for all n in N, P(n)
where N is the natural numbers or positive integers.

> How does it related to the well-ordering property?
>
> What is well-ordering property exactly? To me, it sounds like just
> another version of "a set has a infnimum inside this set is called to
> have the well-ordering property"... or "a set has a minimum is defined
> to have the well-ordering property"... am I right?
>
A set is well ordered when every nonnul subset has a least element.
Traditionally, well ordered also included totally ordered.

If N is well ordered, then Induction holds.
Visa versa, if Induction holds, then N is well ordered.

> The well-ordering theorem says that any set can be made to possess the
> well-ordering property... is that true?
>
No, it's an axiom of much dispute. The Axiom of Choice, AxC or AC has
even become a paramount issue in politics. Should the Axiom of Choice be
taught in schools? Yea say prochoice mathematicians. Nay, say the
constructionist, viewed upon some as obstructionists, intelligent choice
should be taught. Others think intelligent choice should be taught in sex
education. Perhaps they're right, have you ever been bother or haunted by
an unintelligent choice? However the axiom of choice isn't just about you
or me making a choice or two, it's about us simultaneously making
infinitely many choices. This bothers some, as for example, with it you
can cut a sphere into a few pieces and rearrange them into another
sphere with a different surface area.
.

Relevant Pages
Loading