scholarly journals Upper bounds for metapredicative Mahlo in explicit mathematics and admissible set theory

2001 ◽  
Vol 66 (2) ◽  
pp. 935-958 ◽  
Author(s):  
Gerhard Jäger ◽  
Thomas Strahm
Keyword(s):  
Set Theory ◽  
Upper Bounds ◽  
Admissible Set

AbstractIn this article we introduce systems for metapredicative Mahlo in explicit mathematics and admissible set theory. The exact upper proof-theoretic bounds of these systems are established.

scholarly journals Wellordering proofs for metapredicative Mahlo

2002 ◽  
Vol 67 (1) ◽  
pp. 260-278 ◽  
Author(s):  
Thomas Strahm
Keyword(s):  
Set Theory ◽  
Upper Bounds ◽  
Transfinite Induction ◽  
Admissible Set ◽  
Two Systems ◽  
Universe Of Discourse

AbstractIn this article we provide wellordering proofs for metapredicative systems of explicit mathematics and admissible set theory featuring suitable axioms about the Mahloness of the underlying universe of discourse. In particular, it is shown that in the corresponding theories EMA of explicit mathematics and KPm0 of admissible set theory, transfinite induction along initial segments of the ordinal φω00, for φ being a ternary Veblen function, is derivable. This reveals that the upper bounds given for these two systems in the paper Jäger and Strahm [11] are indeed sharp.

scholarly journals AN ORDINAL ANALYSIS OF ADMISSIBLE SET THEORY USING RECURSION ON ORDINAL NOTATIONS

2002 ◽  
Vol 02 (01) ◽  
pp. 91-112 ◽  
Author(s):  
JEREMY AVIGAD
Keyword(s):  
Set Theory ◽  
Proof Theory ◽  
Admissible Set ◽  
Ordinal Analysis ◽  
Rate Of Growth ◽  
Ordinal Notations ◽  
Universe Of Sets ◽  
The Universe

The notion of a function from ℕ to ℕ defined by recursion on ordinal notations is fundamental in proof theory. Here this notion is generalized to functions on the universe of sets, using notations for well orderings longer than the class of ordinals. The generalization is used to bound the rate of growth of any function on the universe of sets that is Σ1-definable in Kripke–Platek admissible set theory with an axiom of infinity. Formalizing the argument provides an ordinal analysis.

scholarly journals Fuzzy Modeling for Uncertainty Nonlinear Systems with Fuzzy Equations

2017 ◽  
Vol 2017 ◽  
pp. 1-10 ◽  
Author(s):  
Raheleh Jafari ◽  
Wen Yu
Keyword(s):  
Nonlinear Systems ◽  
Set Theory ◽  
Nonlinear Modeling ◽  
Upper Bounds ◽  
Equation Model ◽  
Uncertain Nonlinear Systems ◽  
Modeling Process ◽  
Modeling Errors ◽  
The Neural Networks ◽  
Fuzzy Equations

The uncertain nonlinear systems can be modeled with fuzzy equations by incorporating the fuzzy set theory. In this paper, the fuzzy equations are applied as the models for the uncertain nonlinear systems. The nonlinear modeling process is to find the coefficients of the fuzzy equations. We use the neural networks to approximate the coefficients of the fuzzy equations. The approximation theory for crisp models is extended into the fuzzy equation model. The upper bounds of the modeling errors are estimated. Numerical experiments along with comparisons demonstrate the excellent behavior of the proposed method.

Sacks forcing sometimes needs help to produce a minimal upper bound

1989 ◽  
Vol 54 (2) ◽  
pp. 490-498
Author(s):  
Robert S. Lubarsky
Keyword(s):  
Los Angeles ◽  
Upper Bound ◽  
Upper Bounds ◽  
Admissible Set ◽  
Sacks Forcing ◽  
The Right

Does every countable set of hyperdegrees have a minimal upper bound?This question remains unanswered. In this paper, we extend the known results.The standard way to construct minimal upper bounds for degrees is to force with pointed perfect trees. This works for hyperdegrees, in the right context. Sacks [Sa] showed that if an admissible set A satisfies Σ1 DC, then forcing with its uniformly hyperarithmetically pointed perfect trees yields a minimal upper bound for the degrees in A.A next question is whether Σ1DC is necessary. Abramson [A] built an admissible set such that Sacks forcing, or anything like it, would not produce a minimal upper bound. He left open the question, though, whether there is such a bound for his set.We answer this question affirmatively.In §II we summarize the previous relevant results, including Steel forcing. In §III we give a construction different from Abramson's of an admissible set for which Sacks forcing does not produce the desired bound. We present this alternative because it is different (although still based on Steel forcing), simpler than the original, and fully illustrates the technique of finding the bound, which applies equally well to the earlier example. §IV describes the construction of the bound. §V closes with questions.I thank Professor Sy Friedman for bringing this problem to my attention. This paper is dedicated to Professor Alexander Kechris on the occasion of his fortieth birthday, and to the Los Angeles VIGOL on its tenth anniversary.

HC of an admissible set

1979 ◽  
Vol 44 (1) ◽  
pp. 95-102
Author(s):  
Sy D. Friedman
Keyword(s):  
Set Theory ◽  
Admissible Set ◽  
The Real ◽  
Admissible Sets

AbstractIf A is an admissible set, let HC(A) = {x∣x ∈ A and x is hereditarily countable in A}. Then HC(A) is admissible. Corollaries are drawn characterizing the “real parts” of admissible sets and the analytical consequences of admissible set theory.

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