[FOM] Eliminating AC

[Ali Enayat]

This is a remark on the recent posting of Robert Lubarasky (March 29),
who has written:

>One can use class forcing to get AC to be true. The integers do not change, >so the truth of arithmetical statements remain unchanged.

Alas, the above promising idea does not succeed in showing the
conservativity of ZFC over ZF for arithmetical statements, since there
are models of ZF that cannot be class-generically extended to models
of ZFC. Such models were constructed by Douglass Bert Morris in his
1970-dissertation (University of Wisconsin). For more detail, see
problem 14 on p.80 of Jech's book "The Axiom of Choice" (reprinted by
Dover).

Morris' thesis is entitled "Adding Total Indiscernibles to Models of
Set Theory"; but the results of his thesis never appeared in a journal
publication and have not reached the audience they deserve.

Best regards,

Ali Enayat