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partial order with chain condition does not collapse cardinals (Theorem)

If $ P$ is a partial order which satisfies the $ \kappa$ chain condition and $ G$ is a generic subset of $ P$ then for any $ \kappa<\lambda\in\mathfrak{M}$, $ \lambda$ is also a cardinal in $ \mathfrak{M}[G]$, and if $ \operatorname{cf}(\alpha)=\lambda$ in $ \mathfrak{M}$ then also $ \operatorname{cf}(\alpha)=\lambda$ in $ \mathfrak{M}[G]$.

This theorem is the simplest way to control a notion of forcing, since it means that a notion of forcing does not have an effect above a certain point. Given that any $ P$ satisfies the $ \vert P\vert^+$ chain condition, this means that most forcings leaves all of $ \mathfrak{M}$ above a certain point alone. (Although it is possible to get around this limit by forcing with a proper class.)



"partial order with chain condition does not collapse cardinals" is owned by mathcam. [ full author list (2) | owner history (1) ]
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See Also: partial order, chain condition


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proof of partial order with chain condition does not collapse cardinals (Proof) by Henry
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Cross-references: proper class, point, forcing, cardinal, subset, generic, chain condition, satisfies, partial order
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This is version 3 of partial order with chain condition does not collapse cardinals, born on 2002-07-30, modified 2004-03-27.
Object id is 3242, canonical name is PartialOrderWithChainConditionDoesNotCollapseCardinals.
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Classification:
AMS MSC03E35 (Mathematical logic and foundations :: Set theory :: Consistency and independence results)

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