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Complexity of reals in inner models of set theory

Boban Velickovic, W.Hugh Woodin
1998 Annals of Pure and Applied Logic  
We consider the possible complexity of the set of reals belonging to an inner model M of set theory. We show that if this set is analytic then either l-2, M is countable or else all reals are in M.  ...  A similar construction shows that there can be an inner model M which computes correctly Ni, contains a perfect set of reals as a subset and yet not all reals are in M.  ...  In fact we prove that if A4 is an inner model of set theory and the set RM of reals in A4 is analytic then either all reals are in M or else N, M is countable.  ... 
doi:10.1016/s0168-0072(98)00010-4 fatcat:af6azjivsfc7xhsxrc7hkpb5xe

The complexity of the reals in inner models of set theory [article]

Boban Velickovic, W. Hugh Woodin
1995 arXiv   pre-print
In fact we prove that if M is an inner model of set theory and the set of reals in M is analytic then either all reals are in M or else _1^M is countable.  ...  Friedman asked if the set of constructible reals can be analytic or even Borel in a nontrivial way. A related problem was posed by K.  ...  In fact we prove that if M is an inner model of set theory and the set R M of reals in M is analytic then either all reals are in M or else ℵ M 1 is countable.  ... 
arXiv:math/9501203v1 fatcat:e44xbwvpzvcgzlmfmytkcjckvm

Page 8243 of Mathematical Reviews Vol. , Issue 2002K [page]

2002 Mathematical Reviews  
In the case of a real or complex spin factor, when its dimension is infinite, there always exist derivations which are not inner.  ...  The paper under review deals with questions of when all derivations on real or complex JB*-triples are inner.  ... 

Page 3695 of Mathematical Reviews Vol. , Issue 99f [page]

1999 Mathematical Reviews  
Hugh (1-CA; Berkeley, CA) Complexity of reals in inner models of set theory. (English summary) Ann. Pure Appl. Logic 92 (1998), no. 3, 283-295.  ...  Summary: “We consider the possible complexity of the set of reals belonging to an inner model M of set theory. We show that if this set is analytic then either 8}!  ... 

An undecidable extension of Morley's theorem on the number of countable models [article]

Christopher J. Eagle and Clovis Hamel and Sandra Müller and Franklin D. Tall
2021 arXiv   pre-print
More generally, we calculate the number of equivalence classes of σ-projective equivalence relations in several models of set theory.  ...  Our methods include random and Cohen forcing, Woodin cardinals and Inner Model Theory.  ...  also assume some knowledge of Inner Model Theory.  ... 
arXiv:2107.07636v1 fatcat:5sj4gs5dcrgcndqfpotxjvcyhu

Page 5917 of Mathematical Reviews Vol. , Issue 99i [page]

1999 Mathematical Reviews  
The descriptive complexities of the reals in inner models of set theory were first determined to establish delimitative results for classical descriptive set theory.  ...  Gédel showed that the set of reals in L is 2}, and Silver, that the set of reals in an inner model of measurability is D}.  ... 

Reality conditions in non-perturbative quantum cosmology

Guillermo A Mena Marugán
1994 Classical and quantum gravity  
We carry out the nonperturbative canonical quantization of several types of cosmological models that have already been studied in the geometrodynamic formulation using the complex path-integral approach  ...  We establish a relation between the choices of complex contours in the path integral and the sets of reality conditions for which the metric representation is well defined, proving that the ambiguity in  ...  This work was supported by funds provided by the Spanish Ministry of Education and Science Grant No. EX92-06996911.  ... 
doi:10.1088/0264-9381/11/3/012 fatcat:4j3xphhql5eqrhl4a6ewgw3bpy

Page 4752 of Mathematical Reviews Vol. , Issue 98H [page]

1998 Mathematical Reviews  
This is enough to prove a comparison theorem for inner models with one Woodin cardinal. In the paper being reviewed the authors intro- duce two new iteration games played on inner models.  ...  Assertions such as “there is an elementary embedding of the universe into an inner model of ZFC”, discussed in the theory of large cardinals, cannot even be expressed in the language of ZFC (because they  ... 

Model Theory Methods for Topological Groups

Tomás Ibarlucía
2018 Bulletin of Symbolic Logic  
Chapter IX, "Some Model Theory of Classically Valued Fields," applies some ideas from classification theory to a specific AEC: the class of classically valued fields.  ...  This discrete analogue requires an infinitary description to ensure the range of the (analogue of the) metric has range in the real numbers.  ...  The first key result is that if there is no inner model with a Woodin cardinal and all strong cardinals of the core model K are countable in V , then there is a stationary set preserving forcing extension  ... 
doi:10.1017/bsl.2018.32 fatcat:wgj65guzljh6xazhthsokheyv4

Agent-Based Modeling Simulation of Social Adaptation and Long-Term Change in Inner Asia [chapter]

Claudio Cioffi-Revilla, Sean Luke, Dawn C. Parker, J. Daniel Rogers, William W. Fitzhugh, William Honeychurch, Bruno Fröhlich, Paula De Priest, Chunag Amartuvshin
2007 Advancing Social Simulation: The First World Congress  
We present a new international project to develop temporally and spatially calibrated agent-based models of the rise and fall of polities in Inner Asia (Central Eurasia) in the past 5,000 years.  ...  Gaps in theory, data, and computational models for explaining long-term sociopolitical change-both growth and decaymotivate this project.  ...  , and methodological contributions for understanding social complexity and long-term change and adaptation in real and artificial societies.  ... 
doi:10.1007/978-4-431-73167-2_18 dblp:conf/wcss/Cioffi-RevillaL06 fatcat:uzpfqzg3mnalflwdawy5wlm52i

The Construction of Hilbert Spaces over the Non-Newtonian Field

Uğur Kadak, Hakan Efe
2014 International Journal of Analysis  
Also we give the definitions of real and complex inner product spaces and study Hilbert spaces which are special type of normed space and complete inner product spaces in the sense of*-calculus.  ...  In this paper we introduce vector spaces over real and complex non-Newtonian field with respect to the*-calculus which is a branch of non-Newtonian calculus.  ...  Conflict of Interests The authors declare that there is no conflict of interests regarding the publication of this paper.  ... 
doi:10.1155/2014/746059 fatcat:xhelzmq5yfgxxcyuw7ljxevgkq

Minisuperspaces: Symmetries and Quantization [article]

Abhay Ashtekar, Ranjeet S. Tate, Claes Uggla
1993 arXiv   pre-print
In several of the class A Bianchi models, minisuperspaces admit symmetries. It is pointed out that they can be used effectively to complete the Dirac quantization program.  ...  The resulting quantum theory provides a useful platform to investigate a number of conceptual and technical problems of quantum gravity.  ...  This J is a real-linear operator on V with J 2 = −1. It enables us to "multiply" real solutions φ ∈ V by complex numbers: (a + ib) • φ := aφ + bJ • φ, which is again in V .  ... 
arXiv:gr-qc/9302026v1 fatcat:7zynmut6jzczbo656b4iyjzuym

Page 2859 of Mathematical Reviews Vol. , Issue 87f [page]

1987 Mathematical Reviews  
The author establishes a theorem that in effect shows that there are no infinite exponent relations intermediary in strength; in particular, if there is no inner model 87£:03145 O3E Set theory 87£:03147  ...  of the concept of rank for set theory with non-well-founded sets.  ... 

ModulE: Module Embedding for Knowledge Graphs [article]

Jingxuan Chai, Guangming Shi
2022 arXiv   pre-print
However, existing methods mainly focus on modeling relation patterns, while simply embed entities to vector spaces, such as real field, complex field and quaternion space.  ...  To model the embedding space from a more rigorous and theoretical perspective, we propose a novel general group theory-based embedding framework for rotation-based models, in which both entities and relations  ...  Complex field C is a finite extension of real field R. Given α = a + bi ∈ C, the norm of α is N C/R (a + bi) = a 2 + b 2 . Inner Product Map Definition 4.  ... 
arXiv:2203.04702v1 fatcat:blsgvshu55fnpnplfbgymy6bza

BPS Domain Walls in Models with Flat Directions

M. Naganuma, M. Nitta
2001 Progress of theoretical physics  
The moduli space of the supersymmetric vacua in such models have non-compact flat directions, and the complex BPS walls interpolating between two disjoint flat directions can exist.  ...  We examine this possibility in two models with global O(2) symmetry, and construct the solution of such BPS walls.  ...  The work of M. Nitta is supported in part by JSPS Research Fellowships.  ... 
doi:10.1143/ptp.105.501 fatcat:l4zuzd2de5eehgr6l5juijybwe
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