[FOM] ch consistency and existence

Mark Lance lancem at georgetown.edu
Sat Jan 19 22:47:50 EST 2008

Responding to the attributed claim of Alex Blum that consistency implies 
existence, Bill Taylor writes:

I came back with a response that, surely this
couldn't be the case, because ZFC (for example) could be extended in various
ways, *using the same language with the same intended interpretation*,
(an important point, I think, which I didn't stress at the time),
all of which might well be consistent separately, but not together.

But this, I think, cannot be both responsive and right. The key question 
is what "with the same intended interpretation" means here.  If we can 
start by stipulating a specific intended interpretation, that is some 
structure or other can be uniquely intended, then it is certainly true 
that being able to say something consistent about that is not the same 
as being able to say something true about it.  Surely no one would have 
meant that there is exactly one structure such that any consistent thing 
that can be attributed to it is true. 

I take it that the claim of Blum had to be that for any consistent 
theory there is a real structure.  Now in the case of ZFC, I have no 
idea what someone means by THE intended interpretation.  In fact, I 
don't think it is possible to say (or think) anything that would pick 
out a single interpretation of ZFC, so I don't think anyone has any such 
idea.  But if you could specify an intended interpretation, then it 
certainly wouldn't be both an interpretation of ZFC + CH and ZFC + ~CH.  
One thing we know is that there is no such structure as that.

So clearly the claim that Blum is making must be -- forget a priori Blum 
interpretation, this is the view of what we might call the realist 
pluralist -- that given any consistent theory, there are structures 
corresponding to the models of that theory.  So since ZFC +CH and ZFC + 
~CH don't share any models, the realist pluralist will take each to pick 
out different models, and so will take the language to mean something 
different in the two theories.  epsilon would pick out a different 
relation when functioning in the one theory than it does in the other.  
One would be about sets and the other "sets".  But in both cases epsilon 
picks out a relation that exists within a structure. 

I'm not endorsing this view or denying it.  But it is hard for me to see 
what is incoherent about it.  Indeed, if you are a realist about things 
like standard models of ZFC, it is a bit hard to see why you would think 
that some consistent theories pick out structures and others don't.  
Again, if you thought we had some idea of what THE intended model was, 
you might think that this was also all there is.  But you would need an 
independent argument for both parts of this view.

Mark Lance

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