Re: Logarithm of transfinite numbers

Tony Orlow wrote:
There are not a quantitatively infinite number of finite natural numbers. There
CAN be an infinite number of finite numbers in a set, IF the set is dense at at
least one point. The naturals are everywhere sparse.

What do you mean by 'dense at a point'? Do you mean dense within an
upper and lower bound?

Density is per an ordering. The rationals in [0 1] are dense with the
standard linear ordering of the rationals, and there is a bijection f
between omega and the set of rationals in [0 1]. Let R be the ordering
of omega as: Rnk iff f(n) < f(k), where '<' stands for the standard
linear ordering of the rationals. Unless I'm mistaken, that provides
that the set of natural numbers is dense and bounded with the ordering
R.

MoeBlee

.

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