[FOM] Ranks of failure of well-ordering in ZF

Colin McLarty colin.mclarty at case.edu
Mon Jun 18 11:03:58 EDT 2012

What is known about what can be the smallest rank of a
non-well-orderable set in ZF?

I mean, of course non-well-orderabilty can begin with the set of reals
at rank omega plus1, the lowest rank where ZF does not obviously prove
well-orderability,  But are there also models of ZF where all sets of
rank omega plus 1 are well-orderable, but not all sets of rank omega
plus 2?  Are there any restrictions about the rank where well-ordering
first fails in ZF, besides that it must be above omega?

thanks, Colin

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