Finite sets and frege structures

1999 ◽  
Vol 64 (4) ◽  
pp. 1552-1556 ◽  
Author(s):  
John L. Bell
Keyword(s):  
Set Theory ◽  
Finite Sets ◽  
The Family ◽  
Image Position ◽  
Intuitionistic Set Theory

Call a family of subsets of a set E inductive if and is closed under unions with disjoint singletons, that is, ifA Frege structure is a pair (E, ν) with ν a map to E whose domain dom(ν) is an inductive family of subsets of E such thatIn [2] it is shown in a constructive setting that each Frege structure determines a subset which is the domain of a model of Peano's axioms. In this note we establish, within the same constructive setting, three facts. First, we show that the least inductive family of subsets of a set E is precisely the family of decidable Kuratowski finite subsets of E. Secondly, we establish that the procedure presented in [2] can be reversed, that is, any set containing the domain of a model of Peano's axioms determines a map which turns the set into a minimal Frege structure: here by a minimal Frege structure is meant one in which dom(ν) is the least inductive family of subsets of E. And finally, we show that the procedures leading from minimal Frege structures to models of Peano's axioms and vice-versa are mutually inverse. It follows that the postulation of a (minimal) Frege structure is constructively equivalent to the postulation of a model of Peano's axioms.All arguments will be formulated within constructive (intuitionistic) set theory.

Frege's theorem in a constructive setting

1999 ◽  
Vol 64 (2) ◽  
pp. 486-488 ◽  
Author(s):  
John L. Bell
Keyword(s):  
Set Theory ◽  
Natural Numbers ◽  
The Family ◽  
Power Set ◽  
Law Of Excluded Middle ◽  
The Law ◽  
Image Position ◽  
Excluded Middle ◽  
Intuitionistic Set Theory

By Frege's Theorem is meant the result, implicit in Frege's Grundlagen, that, for any set E, if there exists a map υ from the power set of E to E satisfying the conditionthen E has a subset which is the domain of a model of Peano's axioms for the natural numbers. (This result is proved explicitly, using classical reasoning, in Section 3 of [1].) My purpose in this note is to strengthen this result in two directions: first, the premise will be weakened so as to require only that the map υ be defined on the family of (Kuratowski) finite subsets of the set E, and secondly, the argument will be constructive, i.e., will involve no use of the law of excluded middle. To be precise, we will prove, in constructive (or intuitionistic) set theory, the followingTheorem. Let υ be a map with domain a family of subsets of a set E to E satisfying the following conditions:(i) ø ϵdom(υ)(ii)∀U ϵdom(υ)∀x ϵ E − UU ∪ x ϵdom(υ)(iii)∀UV ϵdom(5) υ(U) = υ(V) ⇔ U ≈ V.Then we can define a subset N of E which is the domain of a model of Peano's axioms.

Replacement and collection in intuitionistic set theory

1985 ◽  
Vol 50 (2) ◽  
pp. 344-348 ◽  
Author(s):  
Nicolas D. Goodman
Keyword(s):  
Set Theory ◽  
Predicate Calculus ◽  
Transfinite Induction ◽  
Separation Axiom ◽  
Classical Form ◽  
Existence Property ◽  
Membership Relation ◽  
Image Position ◽  
Excluded Middle ◽  
Intuitionistic Set Theory

Intuitionistic Zermelo-Fraenkel set theory, which we call ZFI, was introduced by Friedman and Myhill in [3] in 1970. The idea was to have a theory with the same axioms as ordinary classical ZF, but with Heyting's predicate calculus HPC as the underlying logic. Since some classically equivalent statements are intuitionistically inequivalent, however, it was not always obvious which form of a classical axiom to take. For example, the usual formulation of the axiom of foundation had to be replaced with a principle of transfinite induction on the membership relation in order to avoid having excluded middle provable and thus making the logic classical. In [3] the replacement axiom is taken in its most common classical form:In the presence of the separation axiom,this is equivalent to the axiomIt is this schema Rep that we shall call the replacement axiom.Friedman and Myhill were able to show in [3] that ZFI has a number of desirable “constructive” properties, including the existence property for both numbers and sets. They were not able to determine, however, whether ZFI is proof-theoretically as strong as ZF. This is still open.Three years later, in [2], Friedman introduced a theory ZF− which is like ZFI except that the replacement axiom is changed to the classically equivalent collection axiom:He showed that ZF− is proof-theoretically of the same strength as ZF, and he asked whether ZF− is equivalent to ZFI.

Two questions from Dana Scott: Intuitionistic topologies and continuous functions

2009 ◽  
Vol 74 (2) ◽  
pp. 689-692
Author(s):  
Charles McCarty
Keyword(s):  
Set Theory ◽  
Continuous Functions ◽  
Unit Interval ◽  
Natural Numbers ◽  
Double Negation ◽  
Closed Set ◽  
Closed Sets ◽  
Image Position ◽  
Intuitionistic Set Theory ◽  
Formal Theories

Since intuitionistic sets are not generally stable – their membership relations are not always closed under double negation – the open sets of a topology cannot be recovered from the closed sets of that topology via complementation, at least without further ado. Dana Scott asked, first, whether it is possible intuitionistically for two distinct topologies, given as collections of open sets on the same carrier, to share their closed sets. Second, he asked whether there can be intuitionistic functions that are closed continuous in that the inverse of every closed set is closed without being continuous in the usual, open sense. Here, we prove that, as far as intuitionistic set theory is concerned, there can be infinitely-many distinct topologies on the same carrier sharing a single collection of closed sets. The proof employs Heyting-valued sets, and demonstrates that the intuitionistic set theory IZF [4, 624], as well as the theory IZF plus classical elementary arithmetic, are both consistent with the statement that infinitely many topologies on the set of natural numbers share the same closed sets. Without changing models, we show that these formal theories are also consistent with the statement that there are infinitely many endofunctions on the natural numbers that are closed continuous but not open continuous with respect to a single topology.For each prime k ∈ ω, let Ak be this ω-sequence of sets open in the standard topology on the closed unit interval: for each n ∈ ω,

Completeness of global intuitionistic set theory

1997 ◽  
Vol 62 (2) ◽  
pp. 506-528 ◽  
Author(s):  
Satoko Titani
Keyword(s):  
Set Theory ◽  
Intuitionistic Logic ◽  
Classical Logic ◽  
Mathematical Object ◽  
The Other ◽  
Logical System ◽  
Logical Operators ◽  
Logical Symbol ◽  
Image Position ◽  
Intuitionistic Set Theory

Gentzen's sequential system LJ of intuitionistic logic has two symbols of implication. One is the logical symbol → and the other is the metalogical symbol ⇒ in sequentsConsidering the logical system LJ as a mathematical object, we understand that the logical symbols ∧, ∨, →, ¬, ∀, ∃ are operators on formulas, and ⇒ is a relation. That is, φ ⇒ Ψ is a metalogical sentence which is true or false, on the understanding that our metalogic is a classical logic. In other words, we discuss the logical system LJ in the classical set theory ZFC, in which φ ⇒ Ψ is a sentence.The aim of this paper is to formulate an intuitionistic set theory together with its metatheory. In Takeuti and Titani [6], we formulated an intuitionistic set theory together with its metatheory based on intuitionistic logic. In this paper we postulate that the metatheory is based on classical logic.Let Ω be a cHa. Ω can be a truth value set of a model of LJ. Then the logical symbols ∧, ∨, →, ¬, ∀x, ∃x are interpreted as operators on Ω, and the sentence φ ⇒ Ψ is interpreted as 1 (true) or 0 (false). This means that the metalogical symbol ⇒ also can be expressed as a logical operators such that φ ⇒ Ψ is interpreted as 1 or 0.

scholarly journals Singular perturbations of the unicritical polynomials with two parameters

2016 ◽  
Vol 37 (6) ◽  
pp. 1997-2016 ◽  
Author(s):  
YINGQING XIAO ◽  
FEI YANG
Keyword(s):  
Julia Set ◽  
Dynamical Behavior ◽  
Rational Maps ◽  
Topological Properties ◽  
Fatou Set ◽  
The Family ◽  
Image Position ◽  
Two Parameters ◽  
Unicritical Polynomials

In this paper, we study the dynamics of the family of rational maps with two parameters $$\begin{eqnarray}f_{a,b}(z)=z^{n}+\frac{a^{2}}{z^{n}-b}+\frac{a^{2}}{b},\end{eqnarray}$$ where $n\geq 2$ and $a,b\in \mathbb{C}^{\ast }$. We give a characterization of the topological properties of the Julia set and the Fatou set of $f_{a,b}$ according to the dynamical behavior of the orbits of the free critical points.

scholarly journals Communication Complexity And Linearly Ordered Sets

2015 ◽  
Vol 29 (1) ◽  
pp. 93-117
Author(s):  
Mieczysław Kula ◽  
Małgorzata Serwecińska
Keyword(s):  
Upper Bound ◽  
Open Problem ◽  
Numerical Experiments ◽  
Communication Complexity ◽  
Ordered Sets ◽  
Finite Sets ◽  
The Family ◽  
Linear Lattices

AbstractThe paper is devoted to the communication complexity of lattice operations in linearly ordered finite sets. All well known techniques ([4, Chapter 1]) to determine the communication complexity of the infimum function in linear lattices disappoint, because a gap between the lower and upper bound is equal to O(log2n), where n is the cardinality of the lattice. Therefore our aim will be to investigate the communication complexity of the function more carefully. We consider a family of so called interval protocols and we construct the interval protocols for the infimum. We prove that the constructed protocols are optimal in the family of interval protocols. It is still open problem to compute the communication complexity of constructed protocols but the numerical experiments show that their complexity is less than the complexity of known protocols for the infimum function.

An extension of Ackermann's set theory

1972 ◽  
Vol 37 (4) ◽  
pp. 703-704
Author(s):  
Donald Perlis
Keyword(s):  
Set Theory ◽  
Free Variables ◽  
Image Position

Ackermann's set theory [1], called here A, involves a schemawhere φ is an ∈-formula with free variables among y1, …, yn and w does not appear in φ. Variables are thought of as ranging over classes and V is intended as the class of all sets.S is a kind of comprehension principle, perhaps most simply motivated by the following idea: The familiar paradoxes seem to arise when the class CP of all P-sets is claimed to be a set, while there exists some P-object x not in CP such that x would have to be a set if CP were. Clearly this cannot happen if all P-objects are sets.Now, Levy [2] and Reinhardt [3] showed that A* (A with regularity) is in some sense equivalent to ZF. But the strong replacement axiom of Gödel-Bernays set theory intuitively ought to be a theorem of A* although in fact it is not (Levy's work shows this). Strong replacement can be formulated asThis lack of A* can be remedied by replacing S above bywhere ψ and φ are ∈-formulas and x is not in ψ and w is not in φ. ψv is ψ with quantifiers relativized to V, and y and z stand for y1, …, yn and z1, …, zm.

Sign in / Sign up

Export Citation Format

Share Document