Shoenfield's Absoluteness Theorem, proved by Joseph Shoenfield in 1961, states that every Sigma-1-2 and Pi-1-2 statement of the analytical hierarchy has the same truth value in any transitive model of set theory and in that model's own constructible universe, provided both contain the same ordinals. The theorem shows that a wide class of statements, including most classical questions of analysis and descriptive set theory, cannot be shown independent of the standard ZFC axioms by the method of forcing, since a forcing extension and its ground model must agree on every statement in this class.
Facts
StatementEvery Sigma-1-2 and Pi-1-2 sentence of the analytical hierarchy, interpreted as a statement about the natural numbers, has the same truth value in a model V of ZF set theory and in that model's constructible universe L. 1 Connections
In Branch
Source Absoluteness (logic) (Wikipedia)
Sources
1. Shoenfield's absoluteness theorem, Wikipedia
Lead section
Shoenfield's absoluteness theorem shows that Π₂¹ and Σ₂¹ sentences in the analytical hierarchy are absolute between a model V of ZF and the constructible universe L of the model, when interpreted as statements about the natural numbers in each model.
References section, Shoenfield 1961 citation
The problem of predicativity, Essays on the foundations of mathematics, Y. Bar-Hillel et al., eds., pp. 132-142.
View the SourceAbsoluteness (logic) (Wikipedia)
In Branch: Logic and Foundations, Lead sentenceQuote, In Branch: Logic and Foundations, Lead sentence
In mathematical logic, a formula is said to be absolute to some class of structures (also called models), if it has the same truth
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