[FOM] The applicability of forcing

[joeshipman@aol.com]

"Forcing" is a general term for certain kinds of arguments used in 
establishing the consistency of propositions with base theories; but I 
don't recall a seeing clear delineation of what is, and what is not, a 
forcing argument. In order to clarify this, I ask for examples of the 
following:

1) a well-known use of forcing to establish the independence of a 
theorem of ZF from a weaker system than ZF

2) a well-known use of forcing which does not conform to the pattern 
"Proposition D is relatively consistent with ZF plus a subset of 
{A,B,C}" where A is consistent with ZFC, B is a large cardinal axiom, C 
is the axiom of choice

3) a well-known independence result which was first proved by forcing, 
but which can be proved "without using forcing" in some reasonable sense

4) a well-known result of the form "Proposition X is consistent with 
and independent of ZFC" where neither side of the result uses forcing

-- JS
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