[FOM] New axiom dichotomy

[joeshipman@aol.com]

Axioms that have been seriously proposed as additions to the ZFC system 
seem to fall into 2 classes:

1) Provably relatively consistent with ZFC, having no new arithmetical 
consequences
2) Implying Con(ZF)

Has an axiom augmenting ZFC ever been seriously proposed which falls 
into one of these other classes?

3) Not known whether it implies Con(ZF)
4) Provably relatively consistent with ZFC, but having other new 
arithmetical consequences
5) Provably relatively consistent with ZFC, but not known whether it 
has any new arithmetical consequences

Note that I am only talking about enhancements to ZFC. not alternative 
systems of set theory like NF in which some axioms of ZFC are 
technically false.


-- JS