335 research outputs found
On Tarski's fixed point theorem
A concept of abstract inductive definition on a complete lattice is
formulated and studied. As an application, a constructive and predicative
version of Tarski's fixed point theorem is obtained.Comment: Proc. Amer. Math. Soc., to appea
Derived rules for predicative set theory: an application of sheaves
We show how one may establish proof-theoretic results for constructive
Zermelo-Fraenkel set theory, such as the compactness rule for Cantor space and
the Bar Induction rule for Baire space, by constructing sheaf models and using
their preservation properties
On the logical structure of choice and bar induction principles
We develop an approach to choice principles and their contrapositive
bar-induction principles as extensionality schemes connecting an "intensional"
or "effective" view of respectively ill-and well-foundedness properties to an
"extensional" or "ideal" view of these properties. After classifying and
analysing the relations between different intensional definitions of
ill-foundedness and well-foundedness, we introduce, for a domain , a
codomain and a "filter" on finite approximations of functions from
to , a generalised form GDC of the axiom of dependent choice and
dually a generalised bar induction principle GBI such that:
GDC intuitionistically captures the strength of
the general axiom of choice expressed as when is a
filter that derives point-wise from a relation on without
introducing further constraints,
the Boolean Prime Filter Theorem / Ultrafilter Theorem if is
the two-element set (for a constructive definition of prime
filter),
the axiom of dependent choice if ,
Weak K{\"o}nig's Lemma if and (up
to weak classical reasoning)
GBI intuitionistically captures the strength of
G{\"o}del's completeness theorem in the form validity implies
provability for entailment relations if ,
bar induction when ,
the Weak Fan Theorem when and .
Contrastingly, even though GDC and GBI smoothly capture
several variants of choice and bar induction, some instances are inconsistent,
e.g. when is and is .Comment: LICS 2021 - 36th Annual Symposium on Logic in Computer Science, Jun
2021, Rome / Virtual, Ital
Constructive set theory and Brouwerian principles
The paper furnishes realizability models of constructive Zermelo-Fraenkel set theory, CZF, which also validate Brouwerian principles such as the axiom of continuous choice (CC), the fan theorem (FT), and monotone bar induction (BIM), and thereby determines the proof-theoretic strength of CZF augmented by these principles. The upshot is that CZF+CC+FT possesses the same strength as CZF, or more precisely, that CZF+CC+FTis conservative over CZF for 02 statements of arithmetic, whereas the addition of a restricted version of bar induction to CZF (called decidable bar induction, BID) leads to greater proof-theoretic strength in that CZF+BID proves the consistency of CZF
Heyting-valued interpretations for Constructive Set Theory
We define and investigate Heyting-valued interpretations for Constructive Zermelo–Frankel set theory (CZF). These interpretations provide models for CZF that are analogous to Boolean-valued models for ZF and to Heyting-valued models for IZF. Heyting-valued interpretations are defined here using set-generated frames and formal topologies. As applications of Heyting-valued interpretations, we present a relative consistency result and an independence proof
Brouwer's Fan Theorem as an axiom and as a contrast to Kleene's Alternative
The paper is a contribution to intuitionistic reverse mathematics. We
introduce a formal system called Basic Intuitionistic Mathematics BIM, and then
search for statements that are, over BIM, equivalent to Brouwer's Fan Theorem
or to its positive denial, Kleene's Alternative to the Fan Theorem. The Fan
Theorem is true under the intended intuitionistic interpretation and Kleene's
Alternative is true in the model of BIM consisting of the Turing-computable
functions. The task of finding equivalents of Kleene's Alternative is,
intuitionistically, a nontrivial extension of finding equivalents of the Fan
Theorem, although there is a certain symmetry in the arguments that we shall
try to make transparent.
We introduce closed-and-separable subsets of Baire space and of the set of
the real numbers. Such sets may be compact and also positively noncompact. The
Fan Theorem is the statement that Cantor space, or, equivalently, the unit
interval, is compact, and Kleene's Alternative is the statement that Cantor
space, or, equivalently, the unit interval is positively noncompact. The class
of the compact closed-and-separable sets and also the class of the
closed-and-separable sets that are positively noncompact are characterized in
many different ways and a host of equivalents of both the Fan Theorem and
Kleene's Alternative is found
Mathematical Logic: Proof theory, Constructive Mathematics
The workshop “Mathematical Logic: Proof Theory, Constructive Mathematics” was centered around proof-theoretic aspects of current mathematics, constructive mathematics and logical aspects of computational complexit
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