Re: Orlow cardinality question



Tony Orlow (aeo6) wrote:
> Martin Shobe said:
> > I don't simply declare omega to be the smallest infinity. In ZFC, it
> > is a theorem that a smallest infinite ordinal exists. Once we have
> > proven this, we name that smallest infinite ordinal, omega.
> A theorem proven by what axioms? The smallest infinity is as nonsensical as the
> largest finite.

Here's where your intuition is correct:
If we were to declare that there were a smallest infinite
SET, then removing one element from that set would make
a finite set. Removing an element creates a smaller subset.

Hence we must conclude that there is no such thing as
a smallest infinite set.

Where you are wrong is assuming that the cardinality of
an infinite set is changed by removing one element. It
is not. Your bigulosity has that property, so therefore
we must conclude that there is no such thing as a
smallest infinite bigulosity.

But that does not impose a similar requirement on
cardinality.

- Randy

.

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