A normalization theorem for set theory

1988 ◽  
Vol 53 (3) ◽  
pp. 673-695 ◽  
Author(s):  
Sidney C. Bailin
Keyword(s):  
Set Theory ◽  
Natural Deduction ◽  
Inference Rule ◽  
Major Premise ◽  
Elimination Rule ◽  
Introduction Rule ◽  
Normalization Theorem ◽  
Image Position ◽  
Well Foundedness

In this paper we present a normalization theorem for a natural deduction formulation of Zermelo set theory. Our result gets around M. Crabbe's counterexample to normalizability (Hallnäs [3]) by adding an inference rule of the formand requiring that this rule be used wherever it is applicable. Alternatively, we can regard the result as pertaining to a modified notion of normalization, in which an inferenceis never considered reducible if A is T Є T, even if R is an elimination rule and the major premise of R is the conclusion of an introduction rule. A third alternative is to regard (1) as a derived rule: using the general well-foundedness rulewe can derive (1). If we regard (2) as neutral with respect to the normality of derivations (i.e., (2) counts as neither an introduction nor an elimination rule), then the resulting proofs are normal.

Some consistency proofs and a characterization of inconsistency proofs in illative combinatory logic

1987 ◽  
Vol 52 (1) ◽  
pp. 89-110 ◽  
Author(s):  
M. W. Bunder
Keyword(s):  
Normal Form ◽  
Propositional Calculus ◽  
Weak Form ◽  
Combinatory Logic ◽  
Major Premise ◽  
Elimination Rule ◽  
Introduction Rule ◽  
Consistency Proofs ◽  
Arbitrary Term

It is well known that combinatory logic with unrestricted introduction and elimination rules for implication is inconsistent in the strong sense that an arbitrary term Y is provable. The simplest proof of this, now usually called Curry's paradox, involves for an arbitrary term Y, a term X defined by X = Y(CPy).The fact that X = PXY = X ⊃ Y is an essential part of the proof.The paradox can be avoided by placing restrictions on the implication introduction rule or on the axioms from which it can be proved.In this paper we determine the forms that must be taken by inconsistency proofs of systems of propositional calculus based on combinatory logic, with arbitrary restrictions on both the introduction and elimination rules for the connectives. Generally such a proof will involve terms without normal form and cut formulas, i.e. formulas formed by an introduction rule that are immediately removed by an elimination with at most some equality steps intervening. (Such a sequence of steps we call a “cut”.)The above applies not only to the strong form of inconsistency defined above, but also to the weak form of inconsistency defined by: all propositions are provable, if this can be represented in the system.Any inconsistency proof of this kind of system can be reduced to one where the major premise of the elimination rule involved in the cut and its deduction must also appear in the deduction of the minor premise involved in the cut.We can, using this characterization of inconsistency proofs, put appropriate restrictions on certain introduction rules so that the systems, including a full classical propositional one, become provably consistent.

scholarly journals The normalization theorem for extended natural deduction

2017 ◽  
Vol 101 (115) ◽  
pp. 75-98
Author(s):  
Mirjana Borisavljevic
Keyword(s):  
Natural Deduction ◽  
Standard System ◽  
Cut Elimination ◽  
Normalization Theorem

The normalization theorem for the system of extended natural deduction will be proved as a consequence of the cut-elimination theorem, by using the connections between the system of extended natural deduction and a standard system of sequents.

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.

Normalization theorems for full first order classical natural deduction

1991 ◽  
Vol 56 (1) ◽  
pp. 129-149 ◽  
Author(s):  
Gunnar Stålmarck
Keyword(s):  
Normal Form ◽  
Natural Deduction ◽  
Full System ◽  
Strong Normalization ◽  
Logical Constants ◽  
Restricted Version ◽  
First Order ◽  
Form Theorem ◽  
Normalization Theorem ◽  
Normal Form Theorem

In this paper we prove the strong normalization theorem for full first order classical N.D. (natural deduction)—full in the sense that all logical constants are taken as primitive. We also give a syntactic proof of the normal form theorem and (weak) normalization for the same system.The theorem has been stated several times, and some proofs appear in the literature. The first proof occurs in Statman [1], where full first order classical N.D. (with the elimination rules for ∨ and ∃ restricted to atomic conclusions) is embedded in a system for second order (propositional) intuitionistic N.D., for which a strong normalization theorem is proved using strongly impredicative methods.A proof of the normal form theorem and (weak) normalization theorem occurs in Seldin [1] as an extension of a proof of the same theorem for an N.D.-system for the intermediate logic called MH.The proof of the strong normalization theorem presented in this paper is obtained by proving that a certain kind of validity applies to all derivations in the system considered.The notion “validity” is adopted from Prawitz [2], where it is used to prove the strong normalization theorem for a restricted version of first order classical N.D., and is extended to cover the full system. Notions similar to “validity” have been used earlier by Tait (convertability), Girard (réducibilité) and Martin-Löf (computability).In Prawitz [2] the N.D. system is restricted in the sense that ∨ and ∃ are not treated as primitive logical constants, and hence the deductions can only be seen to be “natural” with respect to the other logical constants. To spell it out, the strong normalization theorem for the restricted version of first order classical N.D. together with the well-known results on the definability of the rules for ∨ and ∃ in the restricted system does not imply the normalization theorem for the full system.

scholarly journals THE CONSISTENCY STRENGTH OF LONG PROJECTIVE DETERMINACY

2019 ◽  
Vol 85 (1) ◽  
pp. 338-366 ◽  
Author(s):  
JUAN P. AGUILERA ◽  
SANDRA MÜLLER
Keyword(s):  
Set Theory ◽  
Consistency Strength ◽  
Axiom Of Determinacy ◽  
Woodin Cardinals ◽  
Inner Model ◽  
Image Position ◽  
Woodin Cardinal

AbstractWe determine the consistency strength of determinacy for projective games of length ω2. Our main theorem is that $\Pi _{n + 1}^1 $-determinacy for games of length ω2 implies the existence of a model of set theory with ω + n Woodin cardinals. In a first step, we show that this hypothesis implies that there is a countable set of reals A such that Mn (A), the canonical inner model for n Woodin cardinals constructed over A, satisfies $$A = R$$ and the Axiom of Determinacy. Then we argue how to obtain a model with ω + n Woodin cardinal from this.We also show how the proof can be adapted to investigate the consistency strength of determinacy for games of length ω2 with payoff in $^R R\Pi _1^1 $ or with σ-projective payoff.

scholarly journals SQUARES, ASCENT PATHS, AND CHAIN CONDITIONS

2018 ◽  
Vol 83 (04) ◽  
pp. 1512-1538 ◽  
Author(s):  
CHRIS LAMBIE-HANSON ◽  
PHILIPP LÜCKE
Keyword(s):  
Partial Order ◽  
Set Theory ◽  
Consistency Strength ◽  
Chain Conditions ◽  
Regular Cardinals ◽  
Aronszajn Tree ◽  
Square Principles ◽  
Image Position ◽  
Aronszajn Trees ◽  
Consistency Strengths

AbstractWith the help of various square principles, we obtain results concerning the consistency strength of several statements about trees containing ascent paths, special trees, and strong chain conditions. Building on a result that shows that Todorčević’s principle $\square \left( {\kappa ,\lambda } \right)$ implies an indexed version of $\square \left( {\kappa ,\lambda } \right)$, we show that for all infinite, regular cardinals $\lambda < \kappa$, the principle $\square \left( \kappa \right)$ implies the existence of a κ-Aronszajn tree containing a λ-ascent path. We then provide a complete picture of the consistency strengths of statements relating the interactions of trees with ascent paths and special trees. As a part of this analysis, we construct a model of set theory in which ${\aleph _2}$-Aronszajn trees exist and all such trees contain ${\aleph _0}$-ascent paths. Finally, we use our techniques to show that the assumption that the κ-Knaster property is countably productive and the assumption that every κ-Knaster partial order is κ-stationarily layered both imply the failure of $\square \left( \kappa \right)$.

Existence of classes and value specification of variables

1950 ◽  
Vol 15 (2) ◽  
pp. 103-112 ◽  
Author(s):  
Hao Wang
Keyword(s):  
Mathematical Logic ◽  
Set Theory ◽  
Definition Of ◽  
Image Position

In mathematics, when we want to introduce classes which fulfill certain conditions, we usually prove beforehand that classes fulfilling such conditions do exist, and that such classes are uniquely determined by the conditions. The statements which state such unicity and existence of classes are in mathematical logic consequences of the principles of extensionality and class existence. In order to illustrate how these principles enable us to introduce classes into systems of mathematical logic, let us consider the manner in which Gödel introduces classes in his book on set theory.For instance, before introducing the definition of the non-ordered pair of two classesGödel puts down as its justification the following two axioms:By A4, for every two classesyandzthere exists at least one non-ordered pairwof them; and by A3,wis uniquely determined in A4.

scholarly journals AN EXTENSION OF A THEOREM OF ZERMELO

2019 ◽  
Vol 25 (2) ◽  
pp. 208-212 ◽  
Author(s):  
JOUKO VÄÄNÄNEN
Keyword(s):  
Set Theory ◽  
First Order ◽  
Membership Relation ◽  
Image Position

AbstractWe show that if $(M,{ \in _1},{ \in _2})$ satisfies the first-order Zermelo–Fraenkel axioms of set theory when the membership relation is ${ \in _1}$ and also when the membership relation is ${ \in _2}$, and in both cases the formulas are allowed to contain both ${ \in _1}$ and ${ \in _2}$, then $\left( {M, \in _1 } \right) \cong \left( {M, \in _2 } \right)$, and the isomorphism is definable in $(M,{ \in _1},{ \in _2})$. This extends Zermelo’s 1930 theorem in [6].

scholarly journals IDEAL PROJECTIONS AND FORCING PROJECTIONS

2014 ◽  
Vol 79 (4) ◽  
pp. 1247-1285 ◽  
Author(s):  
SEAN COX ◽  
MARTIN ZEMAN
Keyword(s):  
Set Theory ◽  
Negative Answer ◽  
Large Cardinals ◽  
Normal Ideal ◽  
List Type ◽  
Type Number ◽  
Inner Model ◽  
Image Position ◽  
Open Question ◽  
Woodin Cardinal

AbstractIt is well known that saturation of ideals is closely related to the “antichain-catching” phenomenon from Foreman–Magidor–Shelah [10]. We consider several antichain-catching properties that are weaker than saturation, and prove:(1)If${\cal I}$is a normal ideal on$\omega _2 $which satisfiesstationary antichain catching, then there is an inner model with a Woodin cardinal;(2)For any$n \in \omega $, it is consistent relative to large cardinals that there is a normal ideal${\cal I}$on$\omega _n $which satisfiesprojective antichain catching, yet${\cal I}$is not saturated (or even strong). This provides a negative answer to Open Question number 13 from Foreman’s chapter in the Handbook of Set Theory ([7]).

On a set theory of bernays

1967 ◽  
Vol 32 (3) ◽  
pp. 319-321 ◽  
Author(s):  
Leslie H. Tharp
Keyword(s):  
Set Theory ◽  
Free Variable ◽  
Reflection Principle ◽  
Formal Definition ◽  
Von Neumann ◽  
Starting Point ◽  
Definition Of ◽  
Image Position

We are concerned here with the set theory given in [1], which we call BL (Bernays-Levy). This theory can be given an elegant syntactical presentation which allows most of the usual axioms to be deduced from the reflection principle. However, it is more convenient here to take the usual Von Neumann-Bernays set theory [3] as a starting point, and to regard BL as arising from the addition of the schema where S is the formal definition of satisfaction (with respect to models which are sets) and ┌φ┐ is the Gödel number of φ which has a single free variable X.

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