User:Schneelocke/Tmp/Axioms

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[edit] Axioms

  • AC = Axiom of choice
  • AD = Axiom of determinacy
  • ADR = Axiom of real determinacy
  • MA = Martin's axiom
  • MC = Existence of a measurable cardinal
  • PD = Projective determinacy
  • SC = Existence of a supercompact cardinal
  • SU = Existence of a Suslin line
  • ZF = Zermelo-Fraenkel
  • ZFC = ZF + AC
ZF + AD \vdash \neg AC
ZF + GCH \vdash \mbox{weakly inaccessible} = \mbox{inaccessible}
ZF + V=L \vdash AC
ZF + V=L \vdash GCH
ZF + V=L \vdash \diamondsuit
ZF + \diamondsuit \vdash CH
ZFC + CH \vdash MA
ZFC + MA + \neg CH \vdash \neg SU
ZFC + MC \vdash V\neq L
ZFC + SC \vdash PD