[FOM] Definition of "large cardinal axiom"
JoeShipman@aol.com
JoeShipman at aol.com
Thu Apr 15 10:58:49 EDT 2004
My previous post used some abbreviations which may not be familiar to everyone on the list, so I will elaborate here.
"Inacc" is the statement "there exists an inaccessible cardinal" (kappa is inaccessible if it is not the sum of <kappa smaller cardinals and 2^lambda < kappa if lambda < kappa). Con(Inacc) is the statement "Inacc is consistent with ZFC". Inacc implies ZFC is consistent, so is independent of ZFC by Godel, and Con(Inacc) is independent of ZFC+Inacc.
"MC" is the statement "there exists a measurable cardinal" (kappa is measurable if there exists a kappa-additive 2-valued measure on its powerset; in other words, a collection of "null subsets" of kappa closed under <kappa union and including exactly one of {X, kappa/X} for each subset X of kappa). A measurable cardinal kappa is inaccessible (in fact is the "kappath" inaccessible).
"RVM" is the statement "there exists a real-valued measurable cardinal", which is equivalent to the statement "there exists a countably additive extension of Lebesgue measure to ALL subsets of [0,1]". Solovay showed that RVM is consistent with ZFC iff MC is. But RVM says much less about large sets (it is consistent with Inacc being false) and much more about small sets (it implies that the Continuum Hypothesis fails very badly).
PD, or Projective Determinacy, is the statement "every projective set X of real numbers, when considered as a game in the standard way (identify the reals with subsets of N, 2 players build a real by alternately deciding membership in it of 1,2,3,..., first player wins iff the real number built is in X), admits a winning strategy for one of the two players". A projective set of reals is one definable in the structure {P(N), N, +, *, /epsilon } where /epsilon is the membership relation.
For a more precise explication, see Woodin's excellent presentation on the Continuum Hypothesis, available at this URL:
http://math.berkeley.edu/~woodin/talks/lc2000.1.pdf
Although a statement about sets of real numbers, PD implies the consistency of MC (and of much stronger statements than that).
"0# exists" is a highly technical statement about real numbers, which is implied by MC and implies Con(Inacc). I defer to more experienced logicians like Professor Solovay rather than trying to state it precisely here, and solicit their views on why this statement ought to be regarded as a "Large Cardinal Axiom".
My broader question is whether statements such as Con(Inacc), "0# exists", RVM, and PD may properly be called "Large Cardinal Axioms", or whether that term should be reserved for statements which entail the actual existence of Inaccessibles and not merely their consistency.
-- Joe Shipman
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