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The strength of extensionality I — weak weak set theories with infinity

Kentaro Sato
2009 Annals of Pure and Applied Logic  
We first introduce a weak weak set theory Basic (which has the axioms of infinity and of collapsing) as a base over which to clarify the strength of these axioms.  ...  proof-theoretic strength of the axioms of set theory which make the theory look really like a "theory of sets", namely, the axiom of extensionality Ext, separation axioms and the axiom of regularity Reg  ...  pages than prescribed in the submission rule of the competition.  ... 
doi:10.1016/j.apal.2008.09.010 fatcat:ml2xcyi5hnfwjndekffrkhrsou

On the strength of a weak variant of the Axiom of Counting [article]

Zachiri McKenzie
2016 arXiv   pre-print
This paper shows that NFU^-AC+AxCount_≥ proves the consistency of the Simple Theory of Types with Infinity (TSTI).  ...  In this paper NFU^-AC is used to denote Ronald Jensen's modification of Quine's 'New Foundations' Set Theory (NF) fortified with a type-level pairing function but without the Axiom of Choice.  ...  In the presence of (2) the symbol S becomes redundant and the Weak Extensionality axiom reduces to the usual extensionality axiom for set theory.  ... 
arXiv:1601.04168v2 fatcat:36pyljzobrdmloh7h5zdbdp7fu

Page 4637 of Mathematical Reviews Vol. , Issue 2002G [page]

2002 Mathematical Reviews  
Applying a method of Forster and Kaye, he gives the first complete proof of the equiconsistency of MAC with the simple theory of types (together with the axiom of infinity).  ...  A central role is assigned to what the author calls Mac Lane set theory, MAC for short, whose axioms are extensionality, empty set, pairing, union, infinity, power set, bounded (Ao-) separa- tion, foundation  ... 

Alternative Set Theories [chapter]

M. Randall Holmes, Thomas Forster, Thierry Libert
2012 Handbook of the History of Logic  
An interest in the range of alternative set theories does not presuppose an interest in replacing the dominant set theory with one of the alternatives; acquainting ourselves with foundations of mathematics  ...  The study of alternative set theories can dispel a facile identification of "set theory" with "Zermelo-Fraenkel set theory"; they are not the same thing.  ...  It is interesting to observe that Mac Lane set theory is precisely equivalent in consistency strength and expressive power to TST with the Axiom of Infinity.  ... 
doi:10.1016/b978-0-444-51621-3.50008-6 fatcat:tyea73zijjcoxc23qgowqgkq5u

A new model construction by making a detour via intuitionistic theories II: Interpretability lower bound of Feferman's explicit mathematics T0

Kentaro SATO
2015 Annals of Pure and Applied Logic  
subsystems and supersystems of Kripke-Platek set theory KP, of Martin-Löf type theories, and of constructive Zermelo-Fraenkel set theory CZF.  ...  First, it has the applicative nature, which allows us to treat operations directly (unlike in set theory where we have to encode A[f (x)] by ∃y("y = f (x)"∧A[y]) with quantifiers).  ...  Acknowledgements The author would like to express his gratitude to Rico Zumbrunnen, the co-author of the preceding paper [37] .  ... 
doi:10.1016/j.apal.2015.04.002 fatcat:whmtsigbf5bb7lc3dcgpigbe3q

Type theory and homotopy [article]

Steve Awodey
2010 arXiv   pre-print
type theory of Martin-L\"of into homotopy theory, resulting in new examples of higher-dimensional categories.  ...  The purpose of this survey article is to introduce the reader to a connection between Logic, Geometry, and Algebra which has recently come to light in the form of an interpretation of the constructive  ...  As a consequence, the extensional version of the theory is essentially a dependent type theory with a standard, extensional equality relation.  ... 
arXiv:1010.1810v1 fatcat:tafuayc4pbeaxcq54lyvxrwlky

Type Theory and Homotopy [chapter]

Steve Awodey
2012 Epistemology versus Ontology  
As a consequence, the extensional version of the theory is essentially a dependent type theory with a standard, extensional equality relation.  ...  [Str91] for a discussion of the difference between the intensional and extensional forms of the theory).  ... 
doi:10.1007/978-94-007-4435-6_9 dblp:series/leus/Awodey12 fatcat:dzg7yeegdrch7bvakqwqjuaf34

On interpretations of bounded arithmetic and bounded set theory [article]

Richard Pettigrew
2008 arXiv   pre-print
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.  ...  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  ...  Because of the weakness of this theory of sets, I cannot straightforwardly adapt Kaye and Wong's interpretation of the arithmetic in the set theory.  ... 
arXiv:0807.4850v3 fatcat:uthv4jwd2bayxowvg2fkzwuihy

On Interpretations of Bounded Arithmetic and Bounded Set Theory

Richard Pettigrew
2009 Notre Dame Journal of Formal Logic  
Because of the weakness of this theory of sets, I cannot straightforwardly adapt Kaye and Wong's interpretation of the arithmetic in the set theory.  ...  In this note, I describe a theory of sets that is bi-interpretable with the theory of bounded arithmetic I∆ 0 + exp.  ...  Because of the weakness of this theory of sets, I cannot straightforwardly adapt Kaye and Wong's interpretation of the arithmetic in the set theory.  ... 
doi:10.1215/00294527-2009-003 fatcat:wdocp4ehljekxbxpvd444b5ht4

Rudimentary and arithmetical constructive set theory

Peter Aczel
2013 Annals of Pure and Applied Logic  
Here (i) the axiom system BCST for a basic CST is obtained by leaving out from CZF the axiom of Infinity and the axiom schemes of Strong Collection, Subset Collection and Set Induction, while adding the  ...  The rudimentary functions were originally introduced by Ronald Jensen, see [Jen72] , in the context of classical set theory, in order to develop a good fine structure theory for Goedel's constructible  ...  The paper was completed while a fellow of the Newton Instute, Cambridge, in January, 2012. I am grateful for help at the final stages of completing the paper from Michael Rathjen and Albert Visser.  ... 
doi:10.1016/j.apal.2012.10.004 fatcat:yojpov5aj5cixpoun33qmek72q

EXPLICIT MATHEMATICS AND OPERATIONAL SET THEORY: SOME ONTOLOGICAL COMPARISONS

GERHARD JÄGER, RICO ZUMBRUNNEN
2014 Bulletin of Symbolic Logic  
We discuss several ontological properties of explicit mathematics and operational set theory: global choice, decidable classes, totality and extensionality of operations, function spaces, class and set  ...  formation via formulas that contain the definedness predicate and applications.  ...  (iii) The nonuniform versions of these separations are consistent with OST, but adding them to OST as further axioms increases the proof-theoretic strength from that of Kripke-Platek set theory with infinity  ... 
doi:10.1017/bsl.2014.21 fatcat:bd2wiphc2jchlm7x7tiuxy7o6y

The Usual Model Construction for NFU Preserves Information

M. Randall Holmes
2012 Notre Dame Journal of Formal Logic  
The main result of this paper is that the restriction of the membership relation of the original model of set theory with automorphism to the domain of the Boffa model is first-order definable in the language  ...  The usual construction of models of NFU (New Foundations with urelements, introduced by Jensen) is due to Maurice Boffa.  ...  We briefly explain why ranks of the cumulative hierarchy are definable in the weak set theory we are using (recall that this is Mac Lane set theory (Zermelo set theory with bounded separation) with the  ... 
doi:10.1215/00294527-1722764 fatcat:odpcgneeqfemrd7xh6mrqxk7qm

On the logical structure of choice and bar induction principles [article]

Nuria Brede, Hugo Herbelin
2021 arXiv   pre-print
Boolean Prime Filter Theorem / Ultrafilter Theorem if B is the two-element set 𝔹 (for a constructive definition of prime filter), ∙ the axiom of dependent choice if A = ℕ, ∙ Weak König's Lemma if A =  ...  ℕ and B = 𝔹 (up to weak classical reasoning) GBI_A,B,T intuitionistically captures the strength of ∙ Gödel's completeness theorem in the form validity implies provability for entailment relations if  ...  ACKNOWLEDGMENTS We thank the communities of researchers who contributed to develop the material we built on, and in particular Camille Noûs, from the Cogitamus Lab, who embodies the collective and collaborative  ... 
arXiv:2105.08951v2 fatcat:r7f62hxb2jd77aa6rmwlgdm74q

The 1910 *Principia*'s Theory of Functions and Classes and the Theory of Descriptions

William Demopoulos
2007 European Journal of Analytic Philosophy  
The application of contextual analysis to ontological questions is widely viewed as the central philosophical innovation of Russell's theory of descriptions.  ...  The present paper develops a reconstruction of Principia's theory of functions and classes that is based on Russell's epistemological applications of the method of contextual analysis.  ...  In connection with (i), there must be a "simple infinity" of objects i of lowest type for the construction of the numbers to return their Dedekind infinity when they are represented as objects of (simple  ... 
doaj:6087390beafc452994290b56f9e49957 fatcat:7l53d3bsq5bh3in4m7hjemidki

Page 7680 of Mathematical Reviews Vol. , Issue 2001K [page]

2001 Mathematical Reviews  
It is known that the degrees of the polynomials involved in expressing | as a combination of polyno- mials of the base, must tend to infinity with NV.  ...  The above translation is further expanded to constructive set theories in all finite types, and it is shown that the axiom of extensionality for type 2 is not interpretable by functionals of T, and T,*  ... 
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