A Constructive Renewal Theorem

On the Maximum Theorem: A Constructive Analysis

We examine the maximum theorem by Berge from the point of view of Bishop style constructive mathematics. We will show an approximate version of the maximum theorem and the maximum theorem for functions with sequentially locally at most one maximum. Keywords—Maximum theorem, Constructive mathematics, Sequentially locally at most one maximum.

A Constructive Proof of Gleason's Theorem

A Constructive Study of Landau's Summability Theorem

A summability theorem of Landau, which classically is a simple consequence of the uniform boundedness theorem, is examined within Bishop-style constructive mathematics. It is shown that the original theorem is nonconstructive, and that a natural weakening of the theorem is constructively equivalent to Ishihara’s principle BD-N. The paper ends with a number of results that, while not as strong a...

A Constructive Proof of Vizing's Theorem

We consider finite graphs with no self-loops and no multiple edges. A graph is valid if all edges incident on a vertex have different colors. We prove Vizing’s Theorem. All the edges of a graph of maximum degree less than N can be colored using N colors so that the graph is valid. We call a color incident on a vertex if an edge incident on that vertex has that color; otherwise, the color is fre...

Frege's Theorem in A Constructive Setting

By Frege's Theorem is meant the result, implicit in Frege's Grundlagen, that, for any set E, if there exists a map ν from the power set of E to E satisfying the condition ∀XY[ ν(X) = ν(Y) ⇔ X ≈ Y] , then E has a subset which is the domain of a model of Peano's axioms for the natural numbers. (This result is proved explicitly, using classical reasoning, in section 3 of [1].) My purpose in this n...