Constructible Universe - Constructible Sets Are Definable From The Ordinals

Constructible Sets Are Definable From The Ordinals

There is a formula of set theory which expresses the idea that X=L_α. It has only free variables for X and α. Using this we can expand the definition of each constructible set. If s∈L_α+1, then s = {y|y∈L_α and Φ(y,z₁,...,z_n) holds in (L_α,∈)} for some formula Φ and some z₁,...,z_n in L_α. This is equivalent to saying that: for all y, y∈s if and only if where Ψ(X,...) is the result of restricting each quantifier in Φ(...) to X. Notice that each z_k∈L_β+1 for some β<α. Combine formulas for the z's with the formula for s and apply existential quantifiers over the z's outside and one gets a formula which defines the constructible set s using only the ordinals α which appear in expressions like X=L_α as parameters.

Example: The set {5,ω} is constructible. It is the unique set, s, which satisfies the formula:
,
where is short for:

Actually, even this complex formula has been simplified from what the instructions given in the first paragraph would yield. But the point remains, there is a formula of set theory which is true only for the desired constructible set s and which contains parameters only for ordinals.