398 research outputs found
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
CZF does not have the Existence Property
Constructive theories usually have interesting metamathematical properties
where explicit witnesses can be extracted from proofs of existential sentences.
For relational theories, probably the most natural of these is the existence
property, EP, sometimes referred to as the set existence property. This states
that whenever (\exists x)\phi(x) is provable, there is a formula \chi(x) such
that (\exists ! x)\phi(x) \wedge \chi(x) is provable. It has been known since
the 80's that EP holds for some intuitionistic set theories and yet fails for
IZF. Despite this, it has remained open until now whether EP holds for the most
well known constructive set theory, CZF. In this paper we show that EP fails
for CZF
On interpretations of bounded arithmetic and bounded set theory
In a recent paper, Kaye and Wong proved the following result, which they
considered to belong to the folklore of mathematical logic.
THEOREM: The first-order theories of Peano arithmetic and ZF with the axiom
of infinity negated are bi-interpretable: that is, they are mutually
interpretable with interpretations that are inverse to each other.
In this note, I describe a theory of sets that stands in the same relation to
the bounded arithmetic IDelta0 + exp. Because of the weakness of this theory of
sets, I cannot straightforwardly adapt Kaye and Wong's interpretation of
arithmetic in set theory. Instead, I am forced to produce a different
interpretation.Comment: 12 pages; section on omega-models removed due to error; references
added and typos correcte
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
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