[FOM] Inconsistency of Inaccessibility

[MartDowd at aol.com]

In a message dated 10/24/2011 5:01:50 P.M. Pacific Daylight Time,  
meskew at math.uci.edu writes:

Are you  claiming that giving a convincing philosophical argument for
the addition  of a mathematical axiom makes it likely that the axiom  is
consistent?
It adds to the evidence.  Statements which are independent of ZFC can  only 
be accepted by agreement that they are true.  At this point, such  
agreement is being argued for, but is by no means inevitable.  A candidate  for 
acceptance should already enjoy substantial likelihood that it is  consistent.
 
Existence of inaccessible cardinals seems consistent.  They have  appeared 
in logic, in settings such as Grothendieck universes, monster  sets, etc.  
Most mathematicians would probably agree that consistency  holds.  Claiming 
that truth holds is a bolder leap, although various  mathematicians have been 
inclined to make it.  Actually, it seems that  likelihood of consistency is 
a prerequisite to possibility of truth.
 
The existence of measurable cardinals provides another example.  Many  set 
theorists are inclined to accept the truth of this (I think the opposite  
view should be well-considered, though).  So far, attempts to prove  
inconsistency have failed; but to me at least, there are few if any arguments  for 
consistency, other than that inconsistency has not been  proved.  Truth, in 
fact, for inaccessible cardinals, on the  other hand, can be given various 
arguments.
 
I might add that some of the arguments in my paper are mathematical, for  
example that certain axioms imply that Ord is Mahlo.  This could be seen as  
"a posteriori" evidence, as so highly favored by the advocates of measurable 
 cardinal existence.
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