A copy of this work was available on the public web and has been preserved in the Wayback Machine. The capture dates from 2019; you can also visit the original URL.
The file type is `application/pdf`

.

## Filters

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

2009
*
Annals of Pure and Applied Logic
*

We first introduce a

doi:10.1016/j.apal.2008.09.010
fatcat:ml2xcyi5hnfwjndekffrkhrsou
*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. ...##
###
On the strength of a weak variant of the Axiom of Counting
[article]

2016
*
arXiv
*
pre-print

This paper shows that NFU^-AC+AxCount_≥ proves

arXiv:1601.04168v2
fatcat:36pyljzobrdmloh7h5zdbdp7fu
*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*. ...##
###
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]

2012
*
Handbook of the History of Logic
*

An interest in

doi:10.1016/b978-0-444-51621-3.50008-6
fatcat:tyea73zijjcoxc23qgowqgkq5u
*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*. ...##
###
A new model construction by making a detour via intuitionistic theories II: Interpretability lower bound of Feferman's explicit mathematics T0

2015
*
Annals of Pure and Applied Logic
*

subsystems and supersystems

doi:10.1016/j.apal.2015.04.002
fatcat:whmtsigbf5bb7lc3dcgpigbe3q
*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] . ...##
###
Type theory and homotopy
[article]

2010
*
arXiv
*
pre-print

type

arXiv:1010.1810v1
fatcat:tafuayc4pbeaxcq54lyvxrwlky
*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. ...##
###
Type Theory and Homotopy
[chapter]

2012
*
Epistemology versus Ontology
*

As a consequence,

doi:10.1007/978-94-007-4435-6_9
dblp:series/leus/Awodey12
fatcat:dzg7yeegdrch7bvakqwqjuaf34
*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*). ...##
###
On interpretations of bounded arithmetic and bounded set theory
[article]

2008
*
arXiv
*
pre-print

Because

arXiv:0807.4850v3
fatcat:uthv4jwd2bayxowvg2fkzwuihy
*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*. ...##
###
On Interpretations of Bounded Arithmetic and Bounded Set Theory

2009
*
Notre Dame Journal of Formal Logic
*

Because

doi:10.1215/00294527-2009-003
fatcat:wdocp4ehljekxbxpvd444b5ht4
*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*. ...##
###
Rudimentary and arithmetical constructive set theory

2013
*
Annals of Pure and Applied Logic
*

Here (

doi:10.1016/j.apal.2012.10.004
fatcat:yojpov5aj5cixpoun33qmek72q
*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. ...##
###
EXPLICIT MATHEMATICS AND OPERATIONAL SET THEORY: SOME ONTOLOGICAL COMPARISONS

2014
*
Bulletin of Symbolic Logic
*

We discuss several ontological properties

doi:10.1017/bsl.2014.21
fatcat:bd2wiphc2jchlm7x7tiuxy7o6y
*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*...##
###
The Usual Model Construction for NFU Preserves Information

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*...

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

2021
*
arXiv
*
pre-print

Boolean Prime Filter Theorem / Ultrafilter Theorem if B is

arXiv:2105.08951v2
fatcat:r7f62hxb2jd77aa6rmwlgdm74q
*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 ...##
###
The 1910 *Principia*'s Theory of Functions and Classes and the Theory of Descriptions

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 ...

##
###
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,* ...
« Previous

*Showing results 1 — 15 out of 965 results*