Axiomatic set theory
Axiomatic set theory grew in response to Russell’s paradox, which showed that the intuitive notions underlying naïve set theory were flawed, in particular the General Comprehension Principle.
Unter diesen Umständen bleibt gegenwärtig nichts anderes übrig, als den umgekehrten Weg einzuschlagen und, ausgehend von der historisch bestehenden „Mengenlehre“, die Prinzipien aufsuchen, welche zur Begründung dieser mathematischen Disziplin erforderlich sind. Diese Aufgabe muss in der Weise gelöst werden, dass man die Prinzipien einmal eng genug einschränkt, um alle Widersprüche auszuschließen, gleichzeitig aber auch weit genug ausdehnt, um alles Wertvolle dieser Lehre beizubehalten.1
Ernst Zermelo sought to find axioms that were both self-consistent and powerful enough to yield important results from naïve set theory, and his axioms form the basis of ZFC, the most commonly used set theory as a Foundation of mathematics.
Axiomatic set theories
There are two main approaches to axiomatic set theory: Those based on Substance, and those based on Form.
All of these are formulated in 1st-order logic, refer to Conventions of 1st-order logic in these notes.
Sets within other foundations
The alternative to axiomatic set theory is the study of sets as objects within some other foundations such as a Type theory, see e.g. 2013. Homotopy Type Theory: Univalent Foundations of Mathematics, §10.
Footnotes
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1908. Untersuchungen über die Grundlagen der Mengenlehre. I. “In this situation their remains presently no other option other than to embark upon the inverse path and, beginning with the historically conceived ‘set theory’, seek out the principles required for the foundations of this mathematical discipline. This task must be executed in such a way that one at once restricts the principles tightly enough to close out all contradictions, but also simultaneously extends these broadly enough to retain of this theory all which is valuable.” ↩