Set Theory Jottings 28. Forcing Roadmap

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Cohen introduced forcing in 1963 to prove the independence of the continuum hypothesis from ZFC. The next few years witnessed simplifications, other applications, and other perspectives. Despite the simplifications, forcing in set theory comes encrusted with technicalities.

In set theory, the basic forcing concepts occur entangled with the cumulative hierarchy, the level-by-level build-up of the universe of sets from the empty set. This makes for a messy situation. The most “barnacle-free” setting for forcing is Peano arithmetic. Only three levels demand attention:

1. Numbers.
2. Sets of numbers.
3. Classes of sets of numbers. (I use the word ‘class’ just for smoothness; these are bona fide sets in ZF, not proper classes.)

Both in ZF and in the cases treated here, forcing deals with definability. Gödel had shown that if all sets are constructible (V=L), then AC and GCH hold. So Cohen started off by trying to show the relative consistency of V≠L. As we’ve seen, constructibility is all about definability.

We’ll look at four kinds of definability for Peano arithmetic (or really three):

• Arithmetic sets of numbers. A set is arithmetic if it can be defined by a formula of ℒ(PA).
• Implicitly definable sets of numbers. A set is implicitly definable if it can be defined by a formula in the language of PA augmented by a predicate symbol (say S) standing for the set. I’ll write ℒ(PA+S) for the augmented language.
• Arithmetic classes of sets of numbers. A class is arithmetic if it can be defined by a formula of ℒ(PA+S). Implicit definability is the special case where a singleton {A} is an arithmetic class.
• Hyperarithmetic sets of numbers (aka Δ¹₁ subsets). A set is hyperarithmetic if it can be defined by a certain kind of formula in the second-order language of PA, denoted by ℒ²(PA).

Using forcing, Feferman showed the existence of a hyperarithmetic set that is not implicitly definable, and Addison showed that the class of arithmetic sets is not an arithmetic class.

Our roadmap:

1. We define the four kinds of definability.
2. In the context of PA, we offer intuition for the notions generic set and forcing.
3. We proceed with the formal development, and prove the Feferman and Addison theorems.
4. We generalize the notions of forcing and generic sets.
5. We turn to ZF set theory.

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