In the terminology of that program, the least upper bound principle is equivalent to ACA 0 over RCA 0.
12.
Instead of just adding the least upper bounds for pairs, add the least upper bounds for every finite subset.
13.
Instead of just adding the least upper bounds for pairs, add the least upper bounds for every finite subset.
14.
More formally, real numbers have the two basic properties of being an ordered field, and having the least upper bound.
15.
More specifically, among the models of the formula, the excluded model is the least upper bound of the two selected models.
16.
By definition, the arc length of a curve is the least upper bound of the lengths of all polygonal approximations of the curve.
17.
In the formalization of order theory, this is just the "'least upper bound "'of the directed set.
18.
The motivation for considering completeness properties derives from the great importance of suprema ( least upper bounds, union of a collection of sets ).
19.
Has an upper bound in, but does not have a least upper bound in ( since the square root of two is irrational ).
20.
It should be noted, though, that this proof of the completeness of the real numbers implicitly makes use of the least upper bound axiom.
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