Cardinal arithmetic and cardinal characteristics

The arithmetic of infinite cardinals, including the continuum hypothesis and the singular cardinals hypothesis, together with cardinal invariants of the continuum and the cofinal structure of directed sets.

• Bounding number
• Continuum hypothesis
• Tukey reducibility and cofinal types

DefinitionSingular cardinals hypothesis#✓fully proved

For all singular cardinal \(\kappa\), if $2^{cof(\kappa)}\lt \kappa, then \(\kappa^{cof(\kappa)} = \kappa^+\).

def SingularCardinalsHypothesis.{u} : Prop :=
  ∀ κ : Cardinal.{u}, ℵ₀ ≤ κ → κ.ord.cof < κ →
    (2 : Cardinal.{u}) ^ κ.ord.cof < κ → κ ^ κ.ord.cof = Order.succ κ

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