Abstract
There are some fundamental mathematical theories, such as the Fraenkel set-theory and the Bernays-Gödel set-theory, in which, I believe, all the actually important formal theories of mathematics can be embedded. Formal theories come into existence by being shown their consistency. As far as this is admitted, not all the axioms of set theory are necessary for a fundamental mathematical theory. The fundierung axiom is proved consistent by v. Neumann, the axiom of extensionality is proved consistent by Gandy, and even the axiom of choice is proved consistent by Göldel. Although it is not evident that a set-theory does not cease from being a fundamental theory of mathematics after abandoning these axioms all at once, the theory must be enough for being a fundamental theory of mathematics without some of them.
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