Large cardinals

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عدد المتطلبات السابقة الإلزامية (حتى الأساس): 24

ستتعلّم

• 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.

تعريف

• 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.

مبرهنة

• 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.

مفاهيم خاطئة شائعة

• 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.

اكتشافات اختيارية

• 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

قراءات إضافية

• Akihiro Kanamori, The Higher Infinite (1994)
• Thomas Jech, Set Theory (Third Millennium Edition) (2003)