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On the Foundations of Finitely Supported Sets
Andrei Alexandru and Gabriel Ciobanu

In this paper we analyze the validity of various fundamental results of set theory in a newly developed framework of finitely supported sets having important applications in both mathematics and computer science. We focus on the validity of maximal principles, cardinalities and Dedekind-finiteness. We prove that Hausdorff maximal principle, first and second trichotomy principles and Tarski theorem about choice are all inconsistent in the framework of invariant sets, while Schröder- Bernstein theorem is still valid in this new framework. We also provide a characterization of Dedekind-finite sets which allows us to provide several surprising equivalences between injectivity and surjectivity.

Keywords: Finitely supported, maximality, cardinality, Dedekind-finiteness, logical foundations, choice principles

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