Introduction to Forcing

Set Theory · Axiom Academy

LESSON Introduction to Forcing Cohen's revolutionary technique for constructing new models of set theory 1. Cohen's Revolutionary Idea (1963) Before Cohen, Gödel had shown that if ZF is consistent, then so is ZFC + CH (the continuum hypothesis). But could CH be false in some model? Cohen's insight: we can force new sets into existence by carefully extending a model. Think of it as "building on top of" an existing model of set theory. 2. Forcing Conditions and Generic Filters A forcing condition is a finite piece of information about the object we want to add. For example, if we're adding a new subset G of ω, a condition might specify that certain numbers are in G or not in G. We organize conditions into a partially ordered set (poset) P, where p ≤ q means p gives more information than q. Upward closed (if p ∈ G and p ≤ q, then q ∈ G) Directed (any two conditions in G have a common extension in G) Generic: G meets every dense subset of P that exists in M Given a generic filter G (which cannot exist inside M by a counting argument), we construct the forcing extension M[G]. M[G] contains all the sets from M, plus the new object determined by G, plus all sets definable from these. Remarkably, ZFC still holds in M[G]! Every ordinal in M is an ordinal in M[G] 4. Applications: Proving Independence The power of forcing lies in its flexibility. By choosing different posets P, we can force different properties: Cohen forcing: Add many new reals to violate CH

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