FOM: Friedman's result on necessary use of large cardinals

Mon Mar 16 01:32:56 EST 1998

Torkel Franzen has trouble understanding Harvey's results.  They're not that
complicated to state.  About the only practical obstacle to understanding what
is going on is having never seen a "Ramsey"-style combinatorial theorem.  These
theorems all have the form that any sufficiently large object or collection of
objects satisfying some structural condition has a subobject or subcollection
satisfying a stronger condition, and they are the bread-and-butter of
combinatorics.  Harvey's statements fit this pattern perfectly and there is
nothing at all artificial about them, they sound just like many real theorems
of combinatorics.  They happen to have the exact logical strength of subtle
cardinals which are not as well-known as the smaller inaccessibles or the
larger measurables (though you can find references to them on the Web if you use
Infoseek rather than AltaVista, concerning  a paper by Kanamori in the Journal
of Pure and Applied Logic), but surely the fact that inaccessible cardinals
(the smallest and easiest-to-define cardinals too large for ZFC to show
consistent) are necessary is already extremely striking? -- Joe Shipman

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