A copy of this work was available on the public web and has been preserved in the Wayback Machine. The capture dates from 2017; you can also visit the original URL.
The file type is `application/pdf`

.

## Filters

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

2011
*
The Western Ontario Series in Philosophy of Science
*

Impredicative definitions are perfectly allowable in

doi:10.1007/978-94-007-0214-1_9
fatcat:5ch7gvxl55dqxctjomljcusffq
*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*...##
###
The Relative Consistency of the Axiom of Choice Mechanized Using Isabelle⁄zf

2003
*
LMS Journal of Computation and Mathematics
*

It also serves

doi:10.1112/s1461157000000449
fatcat:rfmy4ykqffd7dfk75if7a76hfu
*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. ...##
###
A topological set theory implied by ZF and GPK
[article]

2012
*
arXiv
*
pre-print

On

arXiv:1206.1927v1
fatcat:len7s3rjhra4lnotc3ohufqwaa
*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*. ...##
###
STS: a structural theory of sets

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 . ...

##
###
Erdős-Rado without choice

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*...

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

2014
*
Philosophia Mathematica
*

However that

doi:10.1093/philmat/nku005
fatcat:jldlnkdjsvdfdoz3tiwmzp66zi
*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*. ...##
###
Mathematical Objects arising from Equivalence Relations and their Implementation in Quine's NF

2014
*
Philosophia Mathematica
*

However that

doi:10.1093/phimat/nku005
fatcat:nnkbkci64fhkflqvskl4ufryji
*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*. ...##
###
Pseudo-superstructures as nonstandard universes

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. ...

##
###
Determinacy separations for class games
[article]

2016
*
arXiv
*
pre-print

Our argument

arXiv:1607.05515v1
fatcat:sc5mhh52aneerodyfsix3kvcva
*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 ...##
###
Church's Set Theory with a Universal Set
[chapter]

2001
*
Logic, Meaning and Computation
*

Notice that in V σ

doi:10.1007/978-94-010-0526-5_4
fatcat:nahnfwcv75envlexrp4l7pu4w4
*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*. ...##
###
Nonstandard set theories and information management

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 ...

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

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. ...

##
###
Measurable cardinals and choiceless axioms
[article]

2021
*
arXiv
*
pre-print

We prove that if there

arXiv:2106.05916v3
fatcat:f2qq6lw4rjff3jh4vfnb33r5wa
*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 . ...##
###
Determinacy in third order arithmetic

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. ...

##
###
Martin's Maximum and tower forcing

2013
*
Israel Journal of Mathematics
*

First,

doi:10.1007/s11856-013-0004-0
fatcat:rsqi373jorgtllmp7y6nn7diw4
*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 . ...
« Previous

*Showing results 1 — 15 out of 258 results*