70,074 research outputs found
Realms: A Structure for Consolidating Knowledge about Mathematical Theories
Since there are different ways of axiomatizing and developing a mathematical
theory, knowledge about a such a theory may reside in many places and in many
forms within a library of formalized mathematics. We introduce the notion of a
realm as a structure for consolidating knowledge about a mathematical theory. A
realm contains several axiomatizations of a theory that are separately
developed. Views interconnect these developments and establish that the
axiomatizations are equivalent in the sense of being mutually interpretable. A
realm also contains an external interface that is convenient for users of the
library who want to apply the concepts and facts of the theory without delving
into the details of how the concepts and facts were developed. We illustrate
the utility of realms through a series of examples. We also give an outline of
the mechanisms that are needed to create and maintain realms.Comment: As accepted for CICM 201
Type decomposition in NIP theories
We prove that any type in an NIP theory can be decomposed into a stable part
(a generically stable partial type) and a distal-like quotient.Comment: Several improvements made following the referee repor
On the relative strengths of fragments of collection
Let be the basic set theory that consists of the axioms of
extensionality, emptyset, pair, union, powerset, infinity, transitive
containment, -separation and set foundation. This paper studies the
relative strength of set theories obtained by adding fragments of the
set-theoretic collection scheme to . We focus on two common
parameterisations of collection: -collection, which is the usual
collection scheme restricted to -formulae, and strong
-collection, which is equivalent to -collection plus
-separation. The main result of this paper shows that for all ,
(1) proves the consistency of Zermelo Set Theory plus
-collection,
(2) the theory is
-conservative over the theory .
It is also shown that (2) holds for when the Axiom of Choice is
included in the base theory. The final section indicates how the proofs of (1)
and (2) can be modified to obtain analogues of these results for theories
obtained by adding fragments of collection to a base theory (Kripke-Platek Set
Theory with Infinity and ) that does not include the powerset axiom.Comment: 22 page
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