स्नातकोत्तर प्रथम वर्षFoundationsनियोजित
Forcing and independence
Relative consistency and independence: countable transitive models, posets of conditions, generic filters and the forcing relation, leading to Cohen's proof that the continuum hypothesis is independent of ZFC.
math.found.forcing
इस विषय का पाठ अभी लिखा नहीं गया है। नीचे दी गई रूपरेखा पाठ्यक्रम-मानचित्र का हिस्सा है और इसकी समीक्षा नहीं हुई है।
इस पर आधारित
23 अनिवार्य पूर्वापेक्षाएँ (बिल्कुल नीचे तक)
आप सीखेंगे
- Explain what it means for a statement to be independent of ZFC and why independence proofs are relative-consistency proofs.
- Construct a generic extension M[G] by Cohen forcing and verify that it satisfies ZFC + ¬CH, given a countable transitive model M of ZFC.
- State Easton's theorem and give other examples of statements shown independent by forcing, such as Suslin's hypothesis.
परिभाषा
- A statement φ is independent of ZFC if neither φ nor ¬φ is provable from ZFC.
- A forcing notion is a poset (P, ≤) of conditions; a filter G ⊆ P is M-generic if it meets every dense subset of P that belongs to M.
- The forcing relation p ⊩ φ means that φ holds in every generic extension M[G] with p ∈ G.
प्रमेय
- Cohen (1963): if ZFC is consistent, so is ZFC + ¬CH. Together with Gödel's work on L, CH is independent of ZFC.
- Truth lemma: φ holds in M[G] if and only if some p ∈ G forces φ.
- Easton (1970): on regular cardinals, the function κ ↦ 2^κ can consistently be any non-decreasing function F with cf(F(κ)) > κ.
आम भ्रांतियाँ
- Thinking independence makes a question meaningless; it means ZFC does not settle it.
- Believing forcing requires a countable transitive model of ZFC to exist; formal treatments avoid this with Boolean-valued models or by working with finite fragments of ZFC.
आगे ले जाता है
वैकल्पिक खोजें
- Descriptive set theory Many independent statements concern sets of reals, where descriptive set theory gives a finer classification of which sets are well behaved.
- Elementary topos theory Cohen's construction can be presented as passing to a topos of sheaves on a poset of conditions, as in Mac Lane and Moerdijk's treatment of the independence of CH.
- ← Countability and Cantor's diagonal argument
- ← Ordinals and cardinals
- ← Model theory
आगे पढ़ने के लिए
- Kenneth Kunen, Set Theory: An Introduction to Independence Proofs (1980)
- Paul J. Cohen, Set Theory and the Continuum Hypothesis (1966)
- Thomas Jech, Set Theory (Third Millennium Edition) (2003)