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AMALGAMATION, ABSOLUTENESS, AND CATEGORICITY

2011
*
Proceedings of the 11th Asian Logic Conference
*

It reflects discussions with

doi:10.1142/9789814360548_0002
fatcat:doszxnnu5reqnh6gz2amz3v6tq
*set*theorists during my stay at Mittag-Leffler and discussion with*the*Infinity project members*in*Barcelona. ...*In*particular, we illustrate*the*role*of*higher dimensional amalgamations and sketch*the*role*of*a weak extension*of*ZFC*in**the*proof. ... So this is also*arithmetic*. ii) By i),*the**set**of*p such that 'p is*in*S at (A)' is*arithmetic*(a fortiori Σ 1 1 )*in*A, so by Lemma 5.4.ii, each such p is hyperarithmetic*in*A. ...##
###
Partial evaluation of the discrete solution of elliptic boundary value problems

2012
*
Computing and Visualization in Science
*

*The*latter approach is

*in*particular suited for problems with oscillatory coefficients, where one is not interested

*in*all details

*of*

*the*solution. ... This reduces

*the*storage amount and still enables a partial evaluation

*of*

*the*solution (restricted to

*the*skeletons

*of*

*the*remaining subdomains). ...

*The*low rank blocks are selected from a

*hierarchy*

*of*partitions organised

*in*a so-called cluster tree that provide

*hierarchies*

*of*partitionings. ...

##
###
A Robust Unscented Transformation for Uncertain Moments

2019
*
Journal of the Franklin Institute
*

*In*this scenario,

*the*moment matching equations are reformulated as a system

*of*polynomial equations and inequalities, and it is proposed to use

*the*Chebychev center

*of*

*the*solution

*set*as a robust UT. ... This paper proposes a robust version

*of*

*the*unscented transform (UT) for one-dimensional random variables. It is assumed that

*the*moments are not exactly known, but are known to lie

*in*intervals. ... A naive choice

*of*sigma points would be

*the*use

*of*

*the*

*arithmetic*mean

*of*

*the*lower and upper bounds for

*the*moments

*in*(19). ...

##
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Completely determined Borel sets and measurability
[article]

2021
*
arXiv
*
pre-print

We consider

arXiv:2001.01881v2
fatcat:nyyuzn6qqbeqnep5jbxwdcabyq
*the*reverse math strength*of**the*statement 𝖢-𝖣𝖬:"Every completely determined Borel*set*is measurable." ... Whenever M⊆ 2^ω is*the*second-order part*of*an ω-model*of*𝖢-𝖣𝖬, then for every Z ∈ M, there is a R ∈ M such that R is Δ^1_1-random relative to Z. ... Thus there is an ordinal b ∈ O such that H R0 b computes either a jump*hierarchy*on or a descending sequence*in*a * . But recognizing a jump*hierarchy*or a descending sequence is*arithmetic*. ...##
###
Complete L_ω_1,ω-Sentences with Maximal Models in Multiple Cardinalities
[article]

2018
*
arXiv
*
pre-print

*In*[BKS15] examples

*of*incomplete sentences are given with maximal models

*in*more than one cardinality.

*The*question was raised whether one can find similar examples

*of*complete sentences. ... From (homogeneous) characterizability

*of*κ we construct sentences with maximal models

*in*κ and

*in*one

*of*κ^+, κ^ω, 2^κ and more. ... All

*the*examples

*in*this paper have maximal models

*in*some cardinalities and using settheory we can identify

*the*maximality cardinals

*in*

*the*ℵ-

*hierarchy*. ...

##
###
Well Quasiorders and Hierarchy Theory
[article]

2018
*
arXiv
*
pre-print

While

arXiv:1809.02941v1
fatcat:7ztqokcr7ret3fiusnzszegyxa
*the*classical*hierarchies**of**sets*usually degenerate to structures very close to ordinals,*the*extension*of*them to functions requires more complicated WQOs, and*the*same applies to reducibilities ... We discuss some applications*of*WQOs to several fields were*hierarchies*and reducibilities are*the*principal classification tools, notably to Descriptive*Set*Theory, Computability theory and Automata Theory ... I am grateful to Leibniz-Zentrum für Informatik for accepting and taking care*of*Dagstuhl Seminars 08271, 11411, 15392, 16031 which were important for promoting*the*topic*of*this paper. ...##
###
Bi-interpretation in weak set theories
[article]

2020
*
arXiv
*
pre-print

Similarly, using a construction

arXiv:2001.05262v2
fatcat:nbt6d73iwrffvofconvds2si24
*of*Mathias, we prove that every model*of*ZF is bi-interpretable with a model*of*Zermelo*set*theory*in*which*the*replacement axiom fails. ...*In*contrast to*the*robust mutual interpretability phenomenon*in**set*theory, Ali Enayat proved that bi-interpretation is absent: distinct theories extending ZF are never bi-interpretable and models*of*ZF ... Was*the*use*of**the*V α*hierarchy*significant? Can one prove*the*analogues*of**the*theorems for ZFC − , that is, for*set*theory without*the*power*set*axiom? ...##
###
HyperLTL Satisfiability is Σ_1^1-complete, HyperCTL* Satisfiability is Σ_1^2-complete
[article]

2021
*
arXiv
*
pre-print

Finally, we show that

arXiv:2105.04176v1
fatcat:of66ktmjtrhcrja43lgrqyl4wi
*the*membership problem for every level*of**the*HyperLTL quantifier alternation*hierarchy*is Π_1^1-complete. ...*In*this paper we settle*the*exact complexity*of*these problems, showing that both are*in*fact highly undecidable: we prove that HyperLTL satisfiability is Σ_1^1-complete and HyperCTL* satisfiability is ... So, Σ 0 1 is part*of**the*first level*of**the**arithmetic**hierarchy*, Σ 1 1 is part*of**the*first level*of**the*analytical*hierarchy*, while Σ 2 1 is not even analytical. ...##
###
Can logic be combined with probability? Probably

2009
*
Journal of Applied Logic
*

*In*

*the*nineteen sixties seminal work was done by Gaifman and then Scott and Krauss

*in*adapting

*the*concepts, tools and procedures

*of*

*the*model theory

*of*modern logic to provide a corresponding model theory ...

*in*which notions

*of*probabilistic consistency and consequence are defined analogously to

*the*way they are defined (semantically)

*in*

*the*deductive case. ... various classes

*of*probability functions definable

*in*L(N) appear

*in*

*the*

*arithmetical*

*hierarchy*. ...

##
###
Fixpoints, games and the difference hierarchy

2003
*
RAIRO - Theoretical Informatics and Applications
*

This both provides a simple characterization

doi:10.1051/ita:2003011
fatcat:4hwj5rljynabbm7vc7af2y4kim
*of**the*fixpoint*hierarchy*, and refines existing results on*the*power*of**the*game quantifier*in*descriptive*set*theory. ... Drawing on an analogy with temporal fixpoint logic, we relate*the**arithmetic*fixpoint definable*sets*to*the*winning positions*of*certain games, namely games whose winning conditions lie*in**the*difference ...*The*following "well-known" result on*the*transfinite difference*hierarchy*over Kleene pointclasses is*the*motivation for extending Theorem 8: Fact 12. ∂ <*ω1*(Σ 0 n ) = ∆ 0 n+1 and*in*particular, Σ ∂ <*ω1*...##
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"Mathematics is the Logic of the Infinite": Zermelo's Project of Infinitary Logic

2021
*
Studies in Logic, Grammar and Rhetoric
*

I also stress

doi:10.2478/slgr-2021-0042
fatcat:nnowhogl3bhklm234xa7dion7u
*the*fact that*the*second axiomatization*of**set*theory provided by Zermelo*in*1930 involved*the*use*of*extremal axioms*of*a very specific sort.1 ...*In*this paper I discuss Ernst Zermelo's ideas concerning*the*possibility*of*developing a system*of*infinitary logic that,*in*his opinion, should be suitable for mathematical inferences. ...*The*standard model*of*Peano*arithmetic*can be characterized*in*L*ω1*ω ;*the*same concerns*the*class*of*all ﬁnite*sets*. 2. ...##
###
Noncommutative polynomial optimization under symmetry
[article]

2021
*
arXiv
*
pre-print

We also apply our technique to

arXiv:2112.10803v1
fatcat:ksue5itlq5dclfjob4lio6ymv4
*the*Sliwa inequalities*in**the*Bell scenario with three parties with binary measurements*settings*/outcomes. ... Symmetry reduction is key to scale*the*applications*of**the*NPA relaxation, and our formalism encompasses and generalizes*the*approaches found*in**the*literature. ... First*of*all, it reduces*the*problem size so that it can be handled*in*exact*arithmetic*. ...##
###
COMPLETENESS AND CATEGORICITY (IN POWER): FORMALIZATION WITHOUT FOUNDATIONALISM

2014
*
Bulletin of Symbolic Logic
*

Moreover, these methods (e.g.,

doi:10.1017/bsl.2014.3
fatcat:mgy3v22mz5b6fhk6h7s5je47d4
*the*stability*hierarchy*) provide an organization for much mathematics which gives specific content to dreams*of*Bourbaki about*the*architecture*of*mathematics. ...*The*stability*hierarchy*extends this virtue to other complete theories. ... with tools*of**the**arithmetic*algebraic geometry. ...##
###
The determined property of Baire in reverse math
[article]

2018
*
arXiv
*
pre-print

We define

arXiv:1809.03940v2
fatcat:gcdbwftcwvffxcxrj6haz3xypa
*the*notion*of*a determined Borel code*in*reverse math, and consider*the*principle DPB, which states that every determined Borel*set*has*the*property*of*Baire. ... We show that whenever M⊆ 2^ω is*the*second-order part*of*an ω-model*of*DPB, then for every Z ∈ M, there is a G ∈ M such that G is Δ^1_1-generic relative to Z. ... This is because,*in*general, it might take a jump*hierarchy**the*height*of**the*rank*of*T*in*order to produce an evaluation map. ...##
###
Incomparable ω_1-like models of set theory
[article]

2015
*
arXiv
*
pre-print

We show that

arXiv:1501.01022v1
fatcat:ixsqgzst7fb3dj6h3twoeiwlfi
*the*analogues*of**the*Hamkins embedding theorems, proved for*the*countable models*of**set*theory, do not hold when extended to*the*uncountable realm*of*ω_1-like models*of**set*theory. ...*sets*is not universal for*the*ω_1-like models*of**set*theory. ...*The*research*of**the*third author has been supported*in*part by NSF grant DMS-0800762, PSC-CUNY grant 64732-00-42, CUNY Collaborative Incentive Award 80209-06 20 and Simons Foundation grant 209252. ...
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