FOM: models of FA

[Robert Black]

Just an addendum to what Alen Hazen and Richard Heck have said: In my
article 'Nothing matters too much' (_Analysis_ July 2000) I point out that
if one takes seriously Frege's notion that numbers are objects and his
hierarchy of concepts of different levels, then we will need enough numbers
to count not only collections of objects, but also collections of concepts,
including higher-level concepts. And whether HP extended in this way is
satisfiable *at all* is a question undecided in ZFC. (If you have k objects
you have 2^k first-level concepts, 2^2^k second-level concepts etc. So
going up all finite levels k will have to be such that there are at most k
cardinals less than beth_omega(k). If GCH holds every infinite k is such,
but if GCH fails badly enough there may be no such k.)

Robert

Robert Black
Dept of Philosophy
University of Nottingham
Nottingham NG7 2RD

tel. 0115-951 5845