61 research outputs found
Systems of combinatory logic related to Quine's ‘New Foundations’
AbstractSystems TRC and TRCU of illative combinatory logic are introduced and shown to be equivalent in consistency strength and expressive power to Quine's set theory ‘New Foundations’ (NF) and the fragment NFU + Infinity of NF described by Jensen, respectively. Jensen demonstrated the consistency of NFU + Infinity relative to ZFC; the question of the consistency of NF remains open. TRC and TRCU are presented here as classical first-order theories, although they can be presented as equational theories; they are not constructive
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
About the coexistence of “classical sets” with “non-classical” ones: A survey
This is a survey of some possible extensions of ZF to a larger universe, closer to the “naive set theory” (the universes discussed here concern, roughly speaking : stratified sets, partial sets, positive sets, paradoxical sets and double sets)
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