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

*** T. Winter wrote:
> In article <1132553483.030538.90530@xxxxxxxxxxxxxxxxxxxxxxxxxxxx> "Ross A. Finlayson" <raf@xxxxxxxxxxxxxxx> writes:
> > Well-ordering leads to problems, for example in well-ordering the
> > reals.
> >
> > A well-order is a binary relation, that is in computer talk a function
> > with two arguments, that returns less than, or greater than or equal.
> > It might instead admit greater than, or less than or equal. Then, also
> > there exists an element for the set that is less than, for any other
> > element of the set.
>
> You clearly do not know what a well-ordering is, that is the definition
> Tony Orlow uses, who also does not know it.
> --
> *** t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131
> home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~***/

I'll dispute that. Not that I don't mind Tony, I find Tony refreshing,
but that's a property of a well-ordering, as applied a well-ordered
set, it has a least element. That applies to non-empty sets, which was
an omission.

The existence of a well-ordering on a set implies the existence of a
choice function that given the set as an argument returns the unique
least element of the set by the well-ordering.

Why do you not agree with that? I have not been following Tony's
definition of what a well-ordering is. "Maybe you are thinking of a
total ordering". - Tapio Hurme

The existence of a well-ordering is convenient for quantifying over the
elements of a set, for example for purposes of induction.

Ross

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