Well Founded Set Theoretic Truth

In the course of failing to come up with an abstract for the upcoming ITP workshop in Cambridge, I came across something I think rather nice which is only to be found in the overheads of the last talk I gave on X-Logic.

So I thought I would post it here, to give it a better home.

First just a few words of context.

I am unusual in considering the most important role for set theory to be its role as a foundation for abstract semantics.
From this point of view it is unfortunate that mathematical logic (where set theory belongs) has a negative attitude towards semantics, and tolerates a situation  in which set theory has no definite semantics (or at least, no single generally agreed semantics, there are many possibilities).

Anyway, I came up with a neat semantics for set theory as follows.
This is a definition for the truth conditions of sentences in set theory.
It assumes understood the notion of truth in an interpretation.

• a well-founded set is a definite collection of well-founded sets
  
• an interpretation of set theory is a transitive well-founded set
  
• a sentence is false if the collection of interpretations in which it is true is definite
  
• a sentence is true if the collection of interpretations in which it is false is definite
  

This is what I wrote about this at the time.

This semantics:

• is maximally rich (large cardinal axioms are true)
  
• is definite (CH has a truth value, particular facts about cardinal arithmetic have truth values)
  
• makes ZFC neither true nor false
  
• is not limited to first order languages (will work for infinitary set theory)
  
• is self defining (conjecture)
  

This probably all needs explaining in much greater detail, I will follow up with a bit more explanation.

RBJ