Re: An uncountable countable set


Tony Orlow wrote:
*** T. Winter wrote:
In article <44ef3da9@xxxxxxxxxxxxxxxxxxx> Tony Orlow <tony@xxxxxxxxxxxxx> writes:
> *** T. Winter wrote:
....
> > > Why do you need an axiom for that? Why is it
> > > not derivable logically?
> >
> > Because without the axiom of infinity the set of naturals need not exist,
> > and indeed, you can build a completely logical system with the negation
> > of the axiom of infinity and with all other axioms remaining. It is
> > similar to the parallel axiom in geometry.
>
> But without an axiom of infinity, it is demonstrable that, given the
> axiom of internal infinity (continuity), x<z -> x<y<z, that any finite
> interval includes an infinite number of points. Start with the line, and
> identify points. There's infinity.

Your axiom uses things that are not defined. What is the *meaning* of
"x<z"?

Geometrically it means that x is left of z on the number line.

And what does this mean? What is "the number line"? What is "left"?

--
mike.

.

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