[FOM] Alternative foundations?

jkennedy at mappi.helsinki.fi jkennedy at mappi.helsinki.fi
Mon Feb 24 03:06:02 EST 2014



"If Juliette Kennedy is right that all category-theoretic and set-theoretic
approaches are bi-interpretable, this this refutes claims (2) and (3)."

If you need confirmation check the writings of Colin McCarty, he is  
certainly an expert on this.

The foundational aspects of set theory aside---e.g. I believe it is
possible that the set theoretic formalism
is not optimal for certain kinds of proof-checking---if your concern
is with the mathematics of large cardinals, infinitary
combinatorics, inner models, and so on, that formalism is
going to be very useful.


Quoting Mitchell Spector <spector at alum.mit.edu>:

> Carl Hewitt wrote:
>> Dana Scott [1967] pointed out that foundations need **both** types  
>> and **sets**:
>> “there is only one satisfactory way of avoiding the paradoxes:  
>> namely, the use of some form of the
>> theory of types... the best way to regard Zermelo's theory is as a  
>> simplification and extension of
>> Russell's ...simple theory of types. Now Russell made his types  
>> explicit in his notation and Zermelo
>> left them implicit. It is a mistake to leave something so important  
>> invisible...”
>> “As long as an idealistic manner of speaking about abstract objects  
>> is popular in mathematics,
>> people will speak about collections of objects, and then  
>> collections of collections of ... of
>> collections. In other words **set theory is inevitable**.”
> Yes, as Scott said, a theory of types appears to be needed to escape  
> the paradoxes (while still having an interestingly large  
> mathematical universe).
> This shows a compelling foundational feature of ZF: a theory of  
> types, in the form of well-ordered set-theoretic ranks, arises  
> naturally and elegantly, emerging surprisingly from simply stated  
> and straightforward properties of the membership relation.
> Mitchell
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