>>10155823

I think this formulation of the property is superior to Eudoxus' formulation:

>The set of real number is unbounded above.

However, I can almost accommodate Eudoxus' formulation if I change it to:

>for any positive numbers x and y, with x < y, you can find a rational number q such that q x > y

Indeed, although

inf - 1

is as large as "integer-looking numbers" get in the neighborhood of infinity, you can multiply it by 1/2 to get

0.5 (inf - 1 ) = inf - 0.5

where

inf - 0.5 > inf -1

Likewise, I could adapt Eudoxus' version of the property to say

> for any positive numbers x and y, with x < y, you can find an integer n such that n x > y or x/n > y

In that case I can take

inf - 1

and divide by two to get

(inf - 1) / 2 = inf - 0.5

where, again,

inf - 0.5 > inf -1.

I think the main thing in the principle is that the reals are not bounded above, and that is something I have preserved.