[FOM] Replacement

[A. Mani]

On Thursday 23 August 2007 10:44, Bill Taylor wrote:
>
> Yes; but why would one want to formalize vagueness, in math, of all places?

Mathematics is all of formal mathematics and most of mathematics if not all 
comes from the real world, so it must be done. It depends on what you see as 
foundations and that I believe includes any relevant subset of any set of 
axioms seen as 'possible foundations'.


> I can see how one might regard the "completed cumulative hierarchy"
> as incoherent; but I cannot see how one might derive a contradiction
> from the supposition thereof.
>
> You say it is easy.   Can you elaborate please?

The way it happens in rough set theory is by abstraction.
Let <X, R> ( 'Approximation space') be a pair of a set in ZFC and R an 
equivalence relation on it.
If A is a subset of X, then it is definite iff it is a union of classes of R. 
The lower and upper approximation of a set B is defined as the union of 
classes that are included in A and those that intersect A respectively.
Now if F : X \mapsto X is a map (in the usual ZFC sense) with image A and A is 
not a definite set, then relative to the rough universe this map fails to 
have an image. According to that universe a sentence like "Is A a set ?" will 
not have a 2 valued answer.

An example of this scheme of things is in the way mathematicians perceive 
mathematical knowledge. If a set is a union of classes, then (relative to the 
perspective induced by the indiscernibility relation R),  it can be taken to 
represent definite knowledge.  

The weakening of replacement happens by way of abstraction of the context. 
Many of the associated rough logics have ZF models and semantics. That does 
not mean that there is any incoherence anywhere. It is a question of imposing 
ZF or not. If we get a contradiction to ZFC by describing say semi-set theory 
in ZFC, then there is much to be considered.        


Best

A. Mani



-- 
A. Mani
Member, Cal. Math. Soc