scholarly journals Constructing illoyal algebra-valued models of set theory

2021 ◽  
Vol 82 (3) ◽  
Author(s):  
Benedikt Löwe ◽  
Robert Paßmann ◽  
Sourav Tarafder
Keyword(s):  
Set Theory ◽  
Propositional Logic ◽  
Truth Values ◽  
Models Of Set Theory

AbstractAn algebra-valued model of set theory is called loyal to its algebra if the model and its algebra have the same propositional logic; it is called faithful if all elements of the algebra are truth values of a sentence of the language of set theory in the model. We observe that non-trivial automorphisms of the algebra result in models that are not faithful and apply this to construct three classes of illoyal models: tail stretches, transposition twists, and maximal twists.

scholarly journals A Functional Representation Model Facilitating Design Space Expansion

2013 ◽  
Vol 2013 ◽  
pp. 1-15
Author(s):  
Wei Xu ◽  
Ke Zhao ◽  
Yatao Li ◽  
Peitao Cheng
Keyword(s):  
Set Theory ◽  
Propositional Logic ◽  
Design Space ◽  
Functional Representation ◽  
Functional Reasoning ◽  
Variation Range ◽  
Relational Calculus ◽  
Event Model ◽  
Space Expansion ◽  
The Stability

This paper addresses the functional representation based on the event model. In the event model, the ontology is defined based on the theory of propositional logic to describe the connotation of the event, and the variant is defined based on the theories of domain relational calculus and set theory to express the variation range of the event, which is alterable part of the event under the constraints of the ontology. Function is an important concept in conceptual design and has its connotation and extension. The functional representation is proposed based on the event model. The ontology of event is used to describe the connotation of function and to reflect the stability of function. The variant of the event is used to represent the extension and to incarnate the variety of function. The extension of function is the change range of function under the constraints of the connotation. The proposed functional representation divides the function into the immutable part and the alterable part, facilitating the expansion of design space. A functional reasoning model is also put forward based on the event model to support the function reasoning on the computers. Finally, a simple case validates the feasibility of the model.

scholarly journals Models of Set Theory in which Nonconstructible Reals First Appear at a Given Projective Level

Mathematics ◽  
2020 ◽  
Vol 8 (6) ◽  
pp. 910 ◽  
Author(s):  
Vladimir Kanovei ◽  
Vassily Lyubetsky
Keyword(s):  
Lower Limit ◽  
Set Theory ◽  
Generic Extension ◽  
Projective Class ◽  
Almost Disjoint ◽  
Projective Hierarchy ◽  
Models Of Set Theory

Models of set theory are defined, in which nonconstructible reals first appear on a given level of the projective hierarchy. Our main results are as follows. Suppose that n ≥ 2 . Then: 1. If it holds in the constructible universe L that a ⊆ ω and a ∉ Σ n 1 ∪ Π n 1 , then there is a generic extension of L in which a ∈ Δ n + 1 1 but still a ∉ Σ n 1 ∪ Π n 1 , and moreover, any set x ⊆ ω , x ∈ Σ n 1 , is constructible and Σ n 1 in L . 2. There exists a generic extension L in which it is true that there is a nonconstructible Δ n + 1 1 set a ⊆ ω , but all Σ n 1 sets x ⊆ ω are constructible and even Σ n 1 in L , and in addition, V = L [ a ] in the extension. 3. There exists an generic extension of L in which there is a nonconstructible Σ n + 1 1 set a ⊆ ω , but all Δ n + 1 1 sets x ⊆ ω are constructible and Δ n + 1 1 in L . Thus, nonconstructible reals (here subsets of ω ) can first appear at a given lightface projective class strictly higher than Σ 2 1 , in an appropriate generic extension of L . The lower limit Σ 2 1 is motivated by the Shoenfield absoluteness theorem, which implies that all Σ 2 1 sets a ⊆ ω are constructible. Our methods are based on almost-disjoint forcing. We add a sufficient number of generic reals to L , which are very similar at a given projective level n but discernible at the next level n + 1 .

Elementary extensions of models of set theory

2000 ◽  
Vol 39 (7) ◽  
pp. 509-514 ◽  
Author(s):  
James H. Schmerl
Keyword(s):  
Set Theory ◽  
Models Of Set Theory

Subdifferentials in Boolean-valued models of set theory

1984 ◽  
Vol 24 (5) ◽  
pp. 735-746 ◽  
Author(s):  
A. G. Kusraev ◽  
S. S. Kutateladze
Keyword(s):  
Set Theory ◽  
Models Of Set Theory

scholarly journals Chains of end elementary extensions of models of set theory

1998 ◽  
Vol 63 (3) ◽  
pp. 1116-1136 ◽  
Author(s):  
Andrés Villaveces
Keyword(s):  
Set Theory ◽  
Large Cardinals ◽  
Elementary Extension ◽  
Models Of Set Theory

AbstractLarge cardinals arising from the existence of arbitrarily long end elementary extension chains over models of set theory are studied here. In particular, we show that the large cardinals obtained in this fashion (‘unfoldable cardinals’) lie in the boundary of the propositions consistent with ‘V = L’ and the existence of 0#. We also provide an ‘embedding characterisation’ of the unfoldable cardinals and study their preservation and destruction by various forcing constructions.

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