Forcing (mathematics)
Rigorous definitions

The concepts of P {\displaystyle \mathbb {P} } -names, interpretations, and x ˇ {\displaystyle {\check {x}}} may be defined by transfinite recursion. With ∅ {\displaystyle \varnothing } the empty set, α + 1 {\displaystyle \alpha +1} the successor ordinal to ordinal α {\displaystyle \alpha }, P {\displaystyle {\mathcal {P}}} the power-set operator, and λ {\displaystyle \lambda } a limit ordinal, define the following hierarchy:

Then the class of P {\displaystyle \mathbb {P} } -names is defined as

The interpretation map and the map x ↦ x ˇ {\displaystyle x\mapsto {\check {x}}} can similarly be defined with a hierarchical construction.
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