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Does category theory provide a framework for mathematical structuralism?
 PHILOSOPHIA MATHEMATICA
, 2003
"... Category theory and topos theory have been seen as providing a structuralist framework for mathematics autonomous vis à vis set theory. It is argued here that these theories require a background logic of relations and substantive assumptions addressing mathematical existence of categories themselves ..."
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Category theory and topos theory have been seen as providing a structuralist framework for mathematics autonomous vis à vis set theory. It is argued here that these theories require a background logic of relations and substantive assumptions addressing mathematical existence of categories themselves. We propose a synthesis of Bell’s “manytopoi” view and modalstructuralism. Surprisingly, a combination of mereology and plural quantification suffices to describe hypothetical large domains, recovering the Grothendieck method of universes. Both topos theory and set theory can be carried out relative to such domains; puzzles about “large categories ” and “proper classes ” are handled in a
Complete Quotient Boolean Algebras
"... For I a proper, countably complete ideal on P(X) for some set X, can the quotient Boolean algebra P(X)/I be complete? This question was raised by Sikorski [Si] in 1949. By a simple projection argument as for measurable cardinals, it can be assumed that X is an uncountable cardinal κ, and that I is a ..."
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For I a proper, countably complete ideal on P(X) for some set X, can the quotient Boolean algebra P(X)/I be complete? This question was raised by Sikorski [Si] in 1949. By a simple projection argument as for measurable cardinals, it can be assumed that X is an uncountable cardinal κ, and that I is a κcomplete ideal on P(κ) containing all singletons. In this paper we provide consequences from and consistency results about completeness. Throughout, κ will denote an uncountable cardinal, and by an ideal over κ we shall mean a proper, κcomplete ideal on P(κ) containing all singletons. If κ is a measurable cardinal and I a prime ideal over κ, then of course P(κ)/I is complete, being the twoelement Boolean algebra. The following theorem shows that completeness in itself has strong consistency strength: Theorem A. If κ ≥ ω3 and there is an ideal I over κ such that P(κ)/I is complete, then there is an inner model with a measurable cardinal. The restriction κ ≥ ω3 is necessary for our proof. There is a wellknown situation in which completeness obtains. An ideal over κ is λsaturated iff whenever {Xα  α < λ} ⊆ P(κ) − I, there are α < β < λ such that Xα ∩ Xβ