Proof — Axioms of strong infinity in axiomatic set theory (PSPUM XIII.1)
Statement
Foundational survey of “axioms of strong infinity”: extensions of ZFC that assert the existence of large cardinals (e.g. inaccessible, Mahlo, and related strengthenings), their motivation inside the cumulative hierarchy, mutual consistency implications, and the idea that such axioms codify progressively stronger idealizations of completed infinite structures. The paper is expository and axiomatic rather than proving a single isolated theorem; it systematizes how these principles are formulated and compared in axiomatic set theory.
Formula
Examples (schematic): ∃κ inaccessible(κ); ∃κ Mahlo(κ); stronger “large cardinal” existence claims as first-order axioms extending ZFC.
Proof sketch
Not a single proof of one statement: a chapter-length development in axiomatic set theory, relating definitions, implications, and consistency-strength intuitions for large-cardinal hypotheses. Standard secondary references include Jech, Set Theory, and Kanamori, The Higher Infinite.
Source
Lévy, A. Axioms of strong infinity in axiomatic set theory, in Axiomatic Set Theory (Proc. Sympos. Pure Math., Vol. XIII, Part I), Amer. Math. Soc., Providence, R.I., 1971, pp. 1–38.
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- Theorem: Axioms of strong infinity in axiomatic set theory (PSPUM XIII.1)
- Proof: Axioms of strong infinity in axiomatic set theory (PSPUM XIII.1)