Abstract
In what follows, essentially two things will be accomplished: Firstly, it will be proven that a version of the Arzel\`a--Ascoli theorem and the Fr\'echet--Kolmogorov theorem are equivalent to the axiom of countable choice for subsets of real numbers. Secondly, some progress is made towards determining the amount of axioms that have to be added to the Zermelo--Fraenkel system so that the uniform boundedness principle holds.
Highlights
OutlineIn Subsection 1.2, we briefly describe the greater goal of this paper. In Section 2, we treat the theorems of Arzela–Ascoli and Frechet–Kolmogorov in the following manner: In Subsection 2.1, we state precisely the theorems which we will investigate and briefly comment on them, in Subsection 2.2 we prove modified versions of both theorems that hold without any choice axiom, and using these, in Subsection 2.3 we prove that both theorems under consideration are equivalent to the axiom of countable choice for subsets of real numbers
For the axiom of choice, the situation is somewhat different because it is logically independent of the Zermelo–Fraenkel system; from this follows that ZF plus the axiom of choice can only lead to a contradiction if ZF already leads to a contradiction
We will prove that Theorem 2.1.1 is equivalent to the axiom of countable choice for subsets of the real numbers. (Note that other versions of the Arzela–Ascoli theorem have been studied with regard to their axiomatic strength, eg in Herrlich [11].) The Frechet–Kolmogorov theorem concerns compactness in certain Lp spaces; it is contained within several introductory functional analysis textbooks
Summary
In Subsection 1.2, we briefly describe the greater goal of this paper. In Section 2, we treat the theorems of Arzela–Ascoli and Frechet–Kolmogorov in the following manner: In Subsection 2.1, we state precisely the theorems which we will investigate and briefly comment on them, in Subsection 2.2 we prove modified versions of both theorems that hold without any choice axiom, and using these, in Subsection 2.3 we prove that both theorems under consideration are equivalent to the axiom of countable choice for subsets of real numbers. A F D Fellhauer we give a weak version of the uniform boundedness principle which is equivalent to the axiom of countable multiple choice, and in Subsection 3.5 we elaborate on how our results are incomplete and what seem promising directions for further investigation
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