Core models with more Woodin cardinals

2002 ◽  
Vol 67 (3) ◽  
pp. 1197-1226 ◽  
Author(s):  
J. R. Steel
Keyword(s):  
Fine Structure ◽  
Model Theory ◽  
Core Model ◽  
Structure Theory ◽  
General Nature ◽  
Woodin Cardinals ◽  
The One ◽  
Iteration Trees

In this paper, we shall prove two theorems involving the construction of core models with infinitely many Woodin cardinals. We assume familiarity with [12], which develops core model theory the one Woodin level, and with [10] and [6], which extend the fine structure theory of [5] to mice having many Woodin cardinals. The most important new problem of a general nature which we must face here concerns the iterability of Kc with respect to uncountable iteration trees.Our first result is the following theorem, a slightly stronger version of which was proved independently and earlier by Woodin. The theorem settles positively a conjecture of Feng, Magidor, and Woodin [2].Theorem. Let Ω be measurable. Then the following are equivalent:(a) for all posets,(b) for every poset,(c) for every poset ℙ ∈ VΩ, Vℙ ⊨ there is no uncountable sequence of distinct reals in L(ℝ)(d) there is an Ω-iterable premouse of height Ω which satisfies “there are infinitely many Woodin cardinals”.It is an immediate corollary that if every set of reals in L(ℝ) is weakly homogeneous, then ADL(ℝ) holds. We shall also indicate some extensions of the theorem to pointclasses beyond L(ℝ), and mice with more than ω Woodin cardinals.

The fine structure of real mice

1998 ◽  
Vol 63 (3) ◽  
pp. 937-994 ◽  
Author(s):  
Daniel W. Cunningham
Keyword(s):  
Fine Structure ◽  
Core Model ◽  
Structure Theory ◽  
The Real

AbstractBefore one can construct scales of minimal complexity in the Real Core Model, K(ℝ), one needs to develop the fine-structure theory of K (ℝ). In this paper, the fine structure theory of mice, first introduced by Dodd and Jensen, is generalized to that of real mice. A relative criterion for mouse iterability is presented together with two theorems concerning the definability of this criterion. The proof of the first theorem requires only fine structure; whereas, the second theorem applies to real mice satisfying AD and follows from a general definability result obtained by abstracting work of John Steel on L(ℝ). In conclusion, we discuss several consequences of the work presented in this paper relevant to two issues: the complexity of scales in K(ℝ)and the strength of the theory ZF + AD + ┐DCℝ.

Fine structure for tame inner models

1996 ◽  
Vol 61 (2) ◽  
pp. 621-639 ◽  
Author(s):  
E. Schimmerling ◽  
J. R. Steel
Keyword(s):  
Fine Structure ◽  
Large Cardinal ◽  
Structure Theory ◽  
Uniqueness Problem ◽  
Covering Property ◽  
Strong Cardinal ◽  
Woodin Cardinals ◽  
Inner Models ◽  
Normal Measure ◽  
Consistency Results

In this paper, we solve the strong uniqueness problem posed in [St2]. That is, we extend the full fine structure theory of [MiSt] to backgrounded models all of whose levels are tame (defined in [St2] and below). As a consequence, more powerful large cardinal properties reflect to fine structural inner models. For example, we get the following extension to [MiSt, Theorem 11.3] and [St2, Theorem 0.3].Suppose that there is a strong cardinal that is a limit of Woodin cardinals. Then there is a good extender sequence such that(1) every level of is a sound, tame mouse, and(2) ⊨ “There is a strong cardinal that is a limit of Woodin cardinals”.Recall that satisfies GCH if all its levels are sound. Another consequence of our work is the following covering property, an extension to [St1, Theorem 1.4] and [St3, Theorem 1.10].Suppose that fi is a normal measure on Ω and that all premice are tame. Then Kc, the background certified core model, exists and is a premouse of height Ω. Moreover, for μ-almost every α < Ω.Ideas similar to those introduced here allow us to extend the fine structure theory of [Sch] to the level of tame mice. The details of this extension shall appear elsewhere. From the extension of [Sch] and Theorem 0.2, new relative consistency results follow. For example, we have the following application.If there is a cardinal κ such that κ is κ+-strongly compact, then there is a premouse that is not tame.

Scales in K(ℝ) at the end of a weak gap

2008 ◽  
Vol 73 (2) ◽  
pp. 369-390 ◽  
Author(s):  
J. R. Steel
Keyword(s):  
Induction Hypothesis ◽  
The Other ◽  
Core Model ◽  
The Core ◽  
Woodin Cardinals ◽  
Other Hand

In this note we shall proveTheorem 0.1. Letbe a countably ω-iterable-mouse which satisfies AD, and [α, β] a weak gap of. Supposeis captured by mice with iteration strategies in ∣α. Let n be least such that ; then we have that believes that has the Scale Property.This complements the work of [5] on the construction of scales of minimal complexity on sets of reals in K(ℝ). Theorem 0.1 was proved there under the stronger hypothesis that all sets definable over are determined, although without the capturing hypothesis. (See [5, Theorem 4.14].) Unfortunately, this is more determinacy than would be available as an induction hypothesis in a core model induction. The capturing hypothesis, on the other hand, is available in such a situation. Since core model inductions are one of the principal applications of the construction of optimal scales, it is important to prove 0.1 as stated.Our proof will incorporate a number of ideas due to Woodin which figure prominently in the weak gap case of the core model induction. It relies also on the connection between scales and iteration strategies with the Dodd-Jensen property first discovered in [3]. Let be the pointclass at the beginning of the weak gap referred to in 0.1. In section 1, we use Woodin's ideas to construct a Γ-full a mouse having ω Woodin cardinals cofinal in its ordinals, together with an iteration strategy Σ which condenses well in the sense of [4, Def. 1.13]. In section 2, we construct the desired scale from and Σ.

The Fine Structure Theory

2017 ◽  
pp. 225-302
Author(s):  
Keith J. Devlin
Keyword(s):  
Fine Structure ◽  
Structure Theory

Custodial Sentences

2019 ◽  
Author(s):  
John Sprack ◽  
Michael Engelhardt–Sprack
Keyword(s):  
General Nature ◽  
Age Group ◽  
Legal Aid ◽  
The One

This Chapter and the one following describe the sentencing options open to the courts. The general nature of a particular sentence, the age group for which it is intended, and any restrictions on its imposition are explained. The special limitations on the powers of the magistrates’ courts, and on the powers of an adult magistrates’ court when dealing with a juvenile have already been explained (see 10.65 to 10.73, 11.29 to 11.31, and 12.02 to 12.05). This Chapter concentrates upon those sentences which can be described as ‘custodial’. The Legal Aid, Sentencing and Punishment of Offenders Act 2012 (LASPO Act 2012) made a number of changes in this field, and those changes are described in the relevant paragraphs.

scholarly journals The stark effect of the fine-structure of hydrogen

1928 ◽  
Vol 119 (782) ◽  
pp. 313-334 ◽  
Keyword(s):  
Fine Structure ◽  
Hydrogen Atom ◽  
Electric Field ◽  
Quantum Dynamics ◽  
Stark Effect ◽  
Energy Levels ◽  
Structure Theory ◽  
Weak Electric Field ◽  
Classical Dynamics ◽  
Balmer Series

One of the earliest successes of classical quantum dynamics in a field where ordinary methods had proved inadequate was the solution, by Schwarzschild and Epstein, of the problem of the hydrogen atom in an electric field. It was shown by them that under the influence of the electric field each of the energy levels in which the unperturbed atom can exist on Bohr’s original theory breaks up into a number of equidistant levels whose separation is proportional to the strength of the field. Consequently, each of the Balmer lines splits into a number of components with separations which are integral multiples of the smallest separation. The substitution of the dynamics of special relativity for classical dynamics in the problem of the unperturbed hydrogen atom led Sommerfeld to his well-known theory of the fine-structure of the levels; thus, in the absence of external fields, the state n = 1 ( n = 2 in the old notation) is found to consist of two levels very close together, and n = 2 of three, so that the line H α of the Balmer series, which arises from a transition between these states, has six fine-structure components, of which three, however, are found to have zero intensity. The theory of the Stark effect given by Schwarzschild and Epstein is adequate provided that the electric separation is so much larger than the fine-structure separation of the unperturbed levels that the latter may be regarded as single; but in weak fields, when this is no longer so, a supplementary investigation becomes necessary. This was carried out by Kramers, who showed, on the basis of Sommerfeld’s original fine-structure theory, that the first effect of a weak electric field is to split each fine-structure level into several, the separation being in all cases proportional to the square of the field so long as this is small. When the field is so large that the fine-structure is negligible in comparison with the electric separation, the latter becomes proportional to the first power of the field, in agreement with Schwarzschild and Epstein. The behaviour of a line arising from a transition between two quantum states will be similar; each of the fine-structure components will first be split into several, with a separation proportional to the square of the field; as the field increases the separations increase, and the components begin to perturb each other in a way which leads ultimately to the ordinary Stark effect.

SMT-based verification of data-aware processes: a model-theoretic approach

2020 ◽  
Vol 30 (3) ◽  
pp. 271-313
Author(s):  
Diego Calvanese ◽  
Silvio Ghilardi ◽  
Alessandro Gianola ◽  
Marco Montali ◽  
Andrey Rivkin
Keyword(s):  
Model Theory ◽  
Satisfiability Modulo Theories ◽  
Theoretic Approach ◽  
Model Checker ◽  
Present Contribution ◽  
Relational Systems ◽  
Infinite State Systems ◽  
Algorithmic Techniques ◽  
The One ◽  
Infinite State

AbstractIn recent times, satisfiability modulo theories (SMT) techniques gained increasing attention and obtained remarkable success in model-checking infinite-state systems. Still, we believe that whenever more expressivity is needed in order to specify the systems to be verified, more and more support is needed from mathematical logic and model theory. This is the case of the applications considered in this paper: we study verification over a general model of relational, data-aware processes, to assess (parameterized) safety properties irrespectively of the initial database (DB) instance. Toward this goal, we take inspiration from array-based systems and tackle safety algorithmically via backward reachability. To enable the adoption of this technique in our rich setting, we make use of the model-theoretic machinery of model completion, which surprisingly turns out to be an effective tool for verification of relational systems and represents the main original contribution of this paper. In this way, we pursue a twofold purpose. On the one hand, we isolate three notable classes for which backward reachability terminates, in turn witnessing decidability. Two of such classes relate our approach to conditions singled out in the literature, whereas the third one is genuinely novel. On the other hand, we are able to exploit SMT technology in implementations, building on the well-known MCMT (Model Checker Modulo Theories) model checker for array-based systems and extending it to make all our foundational results fully operational. All in all, the present contribution is deeply rooted in the long-standing tradition of the application of model theory in computer science. In particular, this paper applies these ideas in an original mathematical context and shows how these techniques can be used for the first time to empower algorithmic techniques for the verification of infinite-state systems based on arrays, so as to make such techniques applicable to the timely, challenging settings of data-aware processes.

A minimal counterexample to universal baireness

1999 ◽  
Vol 64 (4) ◽  
pp. 1601-1627 ◽  
Author(s):  
Kai Hauser
Keyword(s):  
Fine Structure ◽  
Set Theory ◽  
Canonical Model ◽  
Core Model ◽  
Real Numbers ◽  
The Real ◽  
Minimal Counterexample ◽  
General Method ◽  
The Way

AbstractFor a canonical model of set theory whose projective theory of the real numbers is stable under set forcing extensions, a set of reals of minimal complexity is constructed which fails to be universally Baire. The construction uses a general method for generating non-universally Baire sets via the Levy collapse of a cardinal, as well as core model techniques. Along the way it is shown (extending previous results of Steel) how sufficiently iterable fine structure models recognize themselves as global core models.

Tame failures of the unique branch hypothesis and models of ADℝ + Θ is regular

2016 ◽  
Vol 16 (02) ◽  
pp. 1650007
Author(s):  
Grigor Sargsyan ◽  
Nam Trang
Keyword(s):  
Symbolic Logic ◽  
Generic Extension ◽  
Transitive Model ◽  
Woodin Cardinals ◽  
Iteration Trees

In this paper, we show that the failure of the unique branch hypothesis ([Formula: see text]) for tame iteration trees implies that in some homogenous generic extension of [Formula: see text] there is a transitive model [Formula: see text] containing [Formula: see text] such that [Formula: see text] is regular. The results of this paper significantly extend earlier works from [Non-tame mice from tame failures of the unique branch bypothesis, Canadian J. Math. 66(4) (2014) 903–923; Core models with more Woodin cardinals, J. Symbolic Logic 67(3) (2002) 1197–1226] for tame trees.

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