研究路径Foundations计划中
Large cardinals
Optional advanced material: inaccessible, measurable and larger cardinals, elementary embeddings of the universe, and the hierarchy of consistency strength that organises much of modern set theory.
math.found.large-cardinals
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前置知识
24 项必需的前置知识（一直追溯到底）
你将学会
- Define inaccessible and measurable cardinals and show that V_κ satisfies ZFC when κ is inaccessible.
- Explain why ZFC cannot prove that an inaccessible cardinal exists (if ZFC is consistent), and what 'consistency strength' means.
- Describe how measurable cardinals correspond to non-trivial elementary embeddings j : V → M.
定义
- κ is (strongly) inaccessible if it is uncountable, regular, and 2^λ < κ for every λ < κ.
- κ is measurable if there is a κ-complete non-principal ultrafilter on κ.
- A theory T has at least the consistency strength of S if the consistency of T implies the consistency of S, provably in a weak base theory.
定理
- If κ is inaccessible then V_κ is a model of ZFC.
- Scott (1961): if a measurable cardinal exists, then V ≠ L.
- Every measurable cardinal is inaccessible and is a limit of inaccessible cardinals.
常见误解
- Thinking large cardinals are merely very big numbers; their significance lies in consistency strength and their consequences, for instance for sets of reals.
- Assuming large-cardinal axioms are known to be consistent; their consistency cannot be proved in ZFC and rests on mathematical evidence and judgement.
可选发现
- Higher categories Grothendieck universes, which correspond to inaccessible cardinals, are a common way of handling size issues in category theory.
- ← ZFC: axiomatic set theory
- ← Ordinals and cardinals
- ← Gödel's incompleteness theorems
- ← Descriptive set theory
延伸阅读
- Akihiro Kanamori, The Higher Infinite (1994)
- Thomas Jech, Set Theory (Third Millennium Edition) (2003)