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Paradox, ZF, and the Axiom of Foundation [chapter]

Adam Rieger
2011 The Western Ontario Series in Philosophy of Science  
Impredicative definitions are perfectly allowable in ZF. 8 As I recounted above, most of the axioms of ZF resulted from Zermelo's attempt to defend his proof that every set could be well-ordered.  ...  for an axiomatization of set theory, as is done in von Neumann [1925] . they are paradoxical, whereas the point is not that sets must be below some particular size but that they must not be as big as  ... 
doi:10.1007/978-94-007-0214-1_9 fatcat:5ch7gvxl55dqxctjomljcusffq

The Relative Consistency of the Axiom of Choice Mechanized Using Isabelle⁄zf

Lawrence C. Paulson
2003 LMS Journal of Computation and Mathematics  
It also serves as an example of what to expect when deep mathematics is formalized.  ...  AbstractThe proof of the relative consistency of the axiom of choice has been mechanized using Isabelle⁄ZF, building on a previous mechanization of the reflection theorem.  ...  Isabelle work is supported by the U.K.'s Engineering and Physical Sciences Research Council, grant GR/M75440. Markus Wenzel greatly improved Isabelle's locale construct to support these proofs.  ... 
doi:10.1112/s1461157000000449 fatcat:rfmy4ykqffd7dfk75if7a76hfu

A topological set theory implied by ZF and GPK [article]

Andreas Fackler
2012 arXiv   pre-print
On the other hand, it retains most of the expressiveness of these theories and has the same consistency strength as ZF.  ...  On the one hand, this system is a common weakening of Zermelo-Fraenkel set theory ZF, the positive set theory GPK and the theory of hyperuniverses.  ...  Zermelo-Fraenkel set theory (ZF) is the Limitation of Size Principle: Only small classes are sets.  ... 
arXiv:1206.1927v1 fatcat:len7s3rjhra4lnotc3ohufqwaa

STS: a structural theory of sets

A Baltag
1999 Logic Journal of the IGPL  
A set is a transfinite process of unfolding of an arbitrary (possibly large) binary structure, with identity of sets given by the observational equivalence between such processes.  ...  We explore a non-classical, universal set theory, based on a purely "structural" conception of sets.  ...  We assume V satisfies all the axioms of ZF C. A set is said to be small if it is of the same size as some set in V .  ... 
doi:10.1093/jigpal/7.4.481 fatcat:n5hrzwhmnrdate3hfwk75n4atq

Erdős-Rado without choice

Thomas Forster
2007 Journal of Symbolic Logic (JSL)  
A version of the Erdős-Rado theorem on partitions of the unordered n-tuples from uncountable sets is proved, without using the axiom of choice.  ...  The case with exponent 1 is just the Sierpinski-Hartogs' result that .  ...  The idea is to remove ordered pairs from < to obtain a wellfounded tree with The Nice Path Property: for all a, if b and c lie on the same branch as a and beyond a, then {a, b} and {a, c} are the same  ... 
doi:10.2178/jsl/1191333846 fatcat:lmrok7j3qfdyhmf3kmu5mfkuw4

Mathematical Objects arising from Equivalence Relations and their Implementation in Quine's NF

T. Forster
2014 Philosophia Mathematica  
However that set has an ordinal in turn, which is not a member of the set constructed, so no set of all ordinals is obtained thereby. This "recurrence problem" is discussed.  ...  This essay can be seen as a non-technical companion to [6] and is, like it, a precursor to-and will be subsumed into-the longer book-length treatment of these matters that I have promised myself I will  ...  Sets (in the new model) that are the same size as wellfounded sets (in the new model) are said to be low. In Church's original construction he provides a universal set.  ... 
doi:10.1093/philmat/nku005 fatcat:jldlnkdjsvdfdoz3tiwmzp66zi

Mathematical Objects arising from Equivalence Relations and their Implementation in Quine's NF

T. Forster
2014 Philosophia Mathematica  
However that set has an ordinal in turn, which is not a member of the set constructed, so no set of all ordinals is obtained thereby. This "recurrence problem" is discussed.  ...  This essay can be seen as a non-technical companion to [6] and is, like it, a precursor to-and will be subsumed into-the longer book-length treatment of these matters that I have promised myself I will  ...  Sets (in the new model) that are the same size as wellfounded sets (in the new model) are said to be low. In Church's original construction he provides a universal set.  ... 
doi:10.1093/phimat/nku005 fatcat:nnkbkci64fhkflqvskl4ufryji

Pseudo-superstructures as nonstandard universes

Mauro Di Nasso
1998 Journal of Symbolic Logic (JSL)  
A definition of nonstandard universe which gets over the limitation to the finite levels of the cumulative hierarchy is proposed.  ...  Though necessarily nonwellfounded, nonstandard universes are arranged in strata in the likeness of superstructures and allow a rank function taking linearly ordered values.  ...  The author wishes to thank Marco Forti for many useful discussions on the subject and wishes to express his deep gratitude to Karel Hrbacek for his helpful remarks on preliminary drafts of this paper.  ... 
doi:10.2307/2586597 fatcat:yxk4u266j5g5fpssplzax2o6pe

Determinacy separations for class games [article]

Sherwood Hachtman
2016 arXiv   pre-print
Our argument is sufficiently general to give a family of determinacy separation results applying in any setting where the universal class is sufficiently closed; e.g., in third, seventh, or (ω+2)th order  ...  We show, assuming weak large cardinals, that in the context of games played in a proper class of moves, clopen determinacy is strictly weaker than open determinacy.  ...  Note that this F is well-defined: Given β with κ < β < θ, the set W = {T ∈ J β | T is wellfounded} is an element of J θ by Σ 0 -Comprehension, and since W has size κ we may regard the direct sum of these  ... 
arXiv:1607.05515v1 fatcat:sc5mhh52aneerodyfsix3kvcva

Church's Set Theory with a Universal Set [chapter]

Thomas Forster
2001 Logic, Meaning and Computation  
Notice that in V σ every set is (externally) the same size as a wellfounded set, since the members (in V σ ) of x are the members (in V ) of σ −1 (x), and σ −1 (x) is certainly in V .  ...  The point is that in that case everything is the same size as an old set or the complement of an old set.  ... 
doi:10.1007/978-94-010-0526-5_4 fatcat:nahnfwcv75envlexrp4l7pu4w4

Nonstandard set theories and information management

Varol Akman, M�jdat Pakkan
1996 Journal of Intelligent Information Systems  
The merits of set theory as a foundational tool in mathematics stimulate its use in various areas of artificial intelligence, in particular intelligent information systems.  ...  In this paper, a study of various nonstandard treatments of set theory from this perspective is offered.  ...  As usual, we are solely responsible for this final version. The first author's research is supported in part by the Scientific and Technical Research Council of Turkey under grant no. TBAG-992. Note  ... 
doi:10.1007/bf00712384 fatcat:tuxvlxig6bbzlcxqop5styeoi4

Large Cardinals, Inner Models, and Determinacy: An Introductory Overview

P. D. Welch
2015 Notre Dame Journal of Formal Logic  
ZF A is then a formulation of ZF with instances of the predicateȦ allowed in the axioms. A set is transitive, Trans(x), if every element of x is at the same time a subset of x.  ...  It is an exercise to show that if a G A for A ⊆ ω ω is determined, A is countable, or else contains a perfect subset and hence is the size of the continuum.  ... 
doi:10.1215/00294527-2835083 fatcat:su5ti4hikbfgrdzzl3adeh23zq

Measurable cardinals and choiceless axioms [article]

Gabriel Goldberg
2021 arXiv   pre-print
We prove that if there is an elementary embedding from the universe to itself, then there is a proper class of measurable successor cardinals.  ...  on wellfounded filters, whose wellfoundedness can be proved in ZF alone.  ...  Then there is a closed unbounded class of almost extendible cardinals. Lemma 2 . 6 . ( 1 ) 261 Every almost extendible cardinal is almost supercompact. the same as the Ketonen rank of U .  ... 
arXiv:2106.05916v3 fatcat:f2qq6lw4rjff3jh4vfnb33r5wa

Determinacy in third order arithmetic

Sherwood Hachtman
2017 Annals of Pure and Applied Logic  
The main result of that paper is a proof, using the method of forcing, that in the context of two-person perfect information games with moves in R, open determinacy (Σ R 1 -DET) is not implied by clopen  ...  In recent work, Schweber [7] introduces a framework for reverse mathematics in a third order setting and investigates several natural principles of transfinite recursion.  ...  In M set ,every set is hereditarily of size at most 2 ω ; that is, for all x ∈ M set , there is an onto map f : P(ω) M set → tcl(x) in M set , where tcl(x) denotes the transitive closure of x. 4.  ... 
doi:10.1016/j.apal.2017.05.004 fatcat:65l5kfnpnnfkvngmsp367ecnci

Martin's Maximum and tower forcing

Sean Cox, Matteo Viale
2013 Israel Journal of Mathematics  
First, the Reflection Principle (RP) implies that if I is a tower of ideals which concentrates on the class GICω 1 of ω 1 -guessing, internally club sets, then I is not presaturated (a set is ω 1 -guessing  ...  iff its transitive collapse has the ω 1approximation property as defined in Hamkins [10] ).  ...  A tower I is called precipitous iff ult(V, G) is wellfounded for every generic G ⊂ P I .  ... 
doi:10.1007/s11856-013-0004-0 fatcat:rsqi373jorgtllmp7y6nn7diw4
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