Inner models for set theory—Part I

1951 ◽  
Vol 16 (3) ◽  
pp. 161-190 ◽  
Author(s):  
J. C. Shepherdson
Keyword(s):  
Set Theory ◽  
Continuum Hypothesis ◽  
Large Family ◽  
Axiom System ◽  
Axiom Of Choice ◽  
Axiomatic System ◽  
Additional Hypothesis ◽  
Inner Models ◽  
Inner Model ◽  
The Axiom Of Choice

One of the standard ways of proving the consistency of additional hypotheses with the basic axioms of an axiom system is by the construction of what may be described as ‘inner models.’ By starting with a domain of individuals assumed to satisfy the basic axioms an inner model is constructed whose domain of individuals is a certain subset of the original individual domain. If such an inner model can be constructed which satisfies not only the basic axioms but also the particular additional hypothesis under consideration, then this affords a proof that if the basic axiom system is consistent then so is the system obtained by adding to this system the new hypothesis. This method has been applied to axiom systems for set theory by many authors, including v. Neumann (4), Mostowski (5), and more recently Gödel (1), who has shown by this method that if the basic axioms of a certain axiomatic system of set theory are consistent then so is the system obtained by adding to these axioms a strong form of the axiom of choice and the generalised continuum hypothesis. Having been shown in this striking way the power of this method it is natural to inquire whether it has any limitations or whether by the construction of a sufficiently ingenious inner model one might hope to decide other outstanding consistency questions, such as the consistency of the negations of the axiom of choice and continuum hypothesis. In this and two following papers we prove some general theorems concerning inner models for a certain axiomatic system of set theory which lead to the result that as far as a fairly large family of inner models are concerned this method of proving consistency has been exhausted, that no essentially new consistency results can be obtained by the use of this kind of model.

On the inadequacy of inner models

1972 ◽  
Vol 37 (3) ◽  
pp. 569-571
Author(s):  
Andreas Blass
Keyword(s):  
Standard Model ◽  
Continuum Hypothesis ◽  
Axiom Of Choice ◽  
Precise Statement ◽  
Relative Consistency ◽  
Inner Models ◽  
Generalized Continuum ◽  
The Axiom Of Choice

The method of inner models, used by Gödel to prove the (relative) consistency of the axiom of choice and the generalized continuum hypothesis [2], cannot be used to prove the (relative) consistency of any statement which contradicts the axiom of constructibility (V = L). A more precise statement of this well-known fact is:(*)For any formula θ(x) of the language of ZF, there is an axiom α of the theory ZF + V ≠ L such that the relativization α(θ) is not a theorem of ZF.On p. 108 of [1], Cohen gives a proof of (*) in ZF assuming the existence of a standard model of ZF, and he indicates that this assumption can be avoided. However, (*) is not a theorem of ZF (unless ZF is inconsistent), because (*) trivially implies the consistency of ZF. What assumptions are needed to prove (*)? We know that the existence of a standard model implies (*) which, in turn, implies the consistency of ZF. Is either implication reversible?From our main result, it will follow that, if the converse of the first implication is provable in ZF, then ZF has no standard model, and if the converse of the second implication is provable in ZF, then so is the inconsistency of ZF. Thus, it is quite improbable that either converse is provable in ZF.

scholarly journals Power Kripke–Platek set theory and the axiom of choice

2020 ◽  
Vol 30 (1) ◽  
pp. 447-457
Author(s):  
Michael Rathjen
Keyword(s):  
Set Theory ◽  
Axiom Of Choice ◽  
Selection Mechanism ◽  
Ordinal Analysis ◽  
Inner Model ◽  
Power Set ◽  
The Axiom Of Choice ◽  
The Relationship ◽  
Set Operation

Abstract While power Kripke–Platek set theory, ${\textbf{KP}}({\mathcal{P}})$, shares many properties with ordinary Kripke–Platek set theory, ${\textbf{KP}}$, in several ways it behaves quite differently from ${\textbf{KP}}$. This is perhaps most strikingly demonstrated by a result, due to Mathias, to the effect that adding the axiom of constructibility to ${\textbf{KP}}({\mathcal{P}})$ gives rise to a much stronger theory, whereas in the case of ${\textbf{KP}}$, the constructible hierarchy provides an inner model, so that ${\textbf{KP}}$ and ${\textbf{KP}}+V=L$ have the same strength. This paper will be concerned with the relationship between ${\textbf{KP}}({\mathcal{P}})$ and ${\textbf{KP}}({\mathcal{P}})$ plus the axiom of choice or even the global axiom of choice, $\textbf{AC}_{\tiny {global}}$. Since $L$ is the standard vehicle to furnish a model in which this axiom holds, the usual argument for demonstrating that the addition of ${\textbf{AC}}$ or $\textbf{AC}_{\tiny {global}}$ to ${\textbf{KP}}({\mathcal{P}})$ does not increase proof-theoretic strength does not apply in any obvious way. Among other tools, the paper uses techniques from ordinal analysis to show that ${\textbf{KP}}({\mathcal{P}})+\textbf{AC}_{\tiny {global}}$ has the same strength as ${\textbf{KP}}({\mathcal{P}})$, thereby answering a question of Mathias. Moreover, it is shown that ${\textbf{KP}}({\mathcal{P}})+\textbf{AC}_{\tiny {global}}$ is conservative over ${\textbf{KP}}({\mathcal{P}})$ for $\varPi ^1_4$ statements of analysis. The method of ordinal analysis for theories with power set was developed in an earlier paper. The technique allows one to compute witnessing information from infinitary proofs, providing bounds for the transfinite iterations of the power set operation that are provable in a theory. As the theory ${\textbf{KP}}({\mathcal{P}})+\textbf{AC}_{\tiny {global}}$ provides a very useful tool for defining models and realizability models of other theories that are hard to construct without access to a uniform selection mechanism, it is desirable to determine its exact proof-theoretic strength. This knowledge can for instance be used to determine the strength of Feferman’s operational set theory with power set operation as well as constructive Zermelo–Fraenkel set theory with the axiom of choice.

The inconsistency of a certain axiom system for set theory

1972 ◽  
Vol 37 (3) ◽  
pp. 538-542
Author(s):  
James D. Davis
Keyword(s):  
Set Theory ◽  
Axiom System ◽  
Reflection Principle ◽  
Axiom Of Choice ◽  
Standard Notation ◽  
The Axiom Of Choice

We prove in this paper that the -system, an axiom system for set theory suggested for investigation by Takeuti in [2], is inconsistent. We also show that this system without the ω-rule is consistent if Zermelo-Fraenkel set theory with the axiom of choice and an axiom due to Reinhardt and Silver is consistent. The -system is an effort to strengthen Bernays-Gödel set theory by adding a reflection principle.In addition to the standard notation of set theory, we write X″{x) to mean {y∣〈x, y〉 Є X}.

An addendum to “The work of Kurt Gödel”

1978 ◽  
Vol 43 (3) ◽  
pp. 613-613 ◽  
Author(s):  
Stephen C. Kleene
Keyword(s):  
Set Theory ◽  
Continuum Hypothesis ◽  
Axiom System ◽  
Von Neumann ◽  
Main Achievement ◽  
Constructible Sets ◽  
Kurt Gödel ◽  
The One ◽  
The Axiom Of Choice ◽  
The Continuum

Gödel has called to my attention that p. 773 is misleading in regard to the discovery of the finite axiomatization and its place in his proof of the consistency of GCH. For the version in [1940], as he says on p. 1, “The system Σ of axioms for set theory which we adopt [a finite one] … is essentially due to P. Bernays …”. However, it is not at all necessary to use a finite axiom system. Gödel considers the more suggestive proof to be the one in [1939], which uses infinitely many axioms.His main achievement regarding the consistency of GCH, he says, really is that he first introduced the concept of constructible sets into set theory defining it as in [1939], proved that the axioms of set theory (including the axiom of choice) hold for it, and conjectured that the continuum hypothesis also will hold. He told these things to von Neumann during his stay at Princeton in 1935. The discovery of the proof of this conjecture On the basis of his definition is not too difficult. Gödel gave the proof (also for GCH) not until three years later because he had fallen ill in the meantime. This proof was using a submodel of the constructible sets in the lowest case countable, similar to the one commonly given today.

scholarly journals Cardinality of the sets of all bijections, injections and surjections

2020 ◽  
Vol 11 ◽  
pp. 143-151
Author(s):  
Marcin Zieliński
Keyword(s):  
Set Theory ◽  
Continuum Hypothesis ◽  
Axiom Of Choice ◽  
Generalized Continuum ◽  
The Axiom Of Choice

The results of Zarzycki for the cardinality of the sets of all bijections, surjections, and injections are generalized to the case when the domains and codomains are infinite and different. The elementary proofs the cardinality of the sets of bijections and surjections are given within the framework of the Zermelo-Fraenkel set theory with the axiom of choice. The case of the set of all injections is considered in detail and more explicit an expression is obtained when the Generalized Continuum Hypothesis is assumed.

The constructible universe

2018 ◽  
Author(s):  
John P. Burgess
Keyword(s):  
Set Theory ◽  
Continuum Hypothesis ◽  
Axiom Of Choice ◽  
Constructible Sets ◽  
Kurt Gödel ◽  
The Axiom Of Choice ◽  
The Universe ◽  
The Continuum

the ‘universe’ of constructible sets was introduced by Kurt Gödel in order to prove the consistency of the axiom of choice (AC) and the continuum hypothesis (CH) with the basic (ZF) axioms of set theory. The hypothesis that all sets are constructible is the axiom of constructibility (V = L). Gödel showed that if ZF is consistent, then ZF + V = L is consistent, and that AC and CH are provable in ZF + V = L.

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