Proper forcing and remarkable cardinals II

2001 ◽  
Vol 66 (3) ◽  
pp. 1481-1492 ◽  
Author(s):  
Ralf-Dieter Schindler
Keyword(s):  
Large Cardinal ◽  
Current Paper ◽  
Proper Forcing ◽  
Remarkable Cardinals

AbstractThe current paper proves the results announced in [5].We isolate a new large cardinal concept, “remarkability.” Consistencywise, remarkable cardinals are between ineffable and ω-Erdös cardinals. They are characterized by the existence of “0#-like” embeddings; however, they relativize down to L. It turns out that the existence of a remarkable cardinal is equiconsistent with L(ℝ) absoluteness for proper forcings. In particular, said absoluteness does not imply determinacy.

Proper Forcing and Remarkable Cardinals

2000 ◽  
Vol 6 (2) ◽  
pp. 176-184 ◽  
Author(s):  
Ralf-Dieter Schindler
Keyword(s):  
Measurable Cardinal ◽  
Large Cardinal ◽  
Exact Formulation ◽  
Axiom Of Determinacy ◽  
Slow Pace ◽  
Woodin Cardinals ◽  
Inner Model ◽  
Proper Forcing ◽  
Remarkable Cardinals ◽  
The Universe

The present paper investigates the power of proper forcings to change the shape of the universe, in a certain well-defined respect. It turns out that the ranking among large cardinals can be used as a measure for that power. However, in order to establish the final result I had to isolate a new large cardinal concept, which I dubbed “remarkability.” Let us approach the exact formulation of the problem—and of its solution—at a slow pace.Breathtaking developments in the mid 1980s found one of its culminations in the theorem, due to Martin, Steel, and Woodin, that the existence of infinitely many Woodin cardinals with a measurable cardinal above them all implies that AD, the axiom of determinacy, holds in the least inner model containing all the reals, L(ℝ) (cf. [6[, p. 91). One of the nice things about AD is that the theory ZF + AD + V = L(ℝ) appears as a choiceless “completion” of ZF in that any interesting question (in particular, about sets of reals) seems to find an at least attractive answer in that theory (cf., for example, [5] Chap. 6). (Compare with ZF + V = L!) Beyond that, AD is very canonical as may be illustrated as follows.Let us say that L(ℝ) is absolute for set-sized forcings if for all posets P ∈ V, for all formulae ϕ, and for all ∈ ℝ do we have thatwhere is a name for the set of reals in the extension.

scholarly journals Ramsey-like cardinals II

2011 ◽  
Vol 76 (2) ◽  
pp. 541-560 ◽  
Author(s):  
Victoria Gitman ◽  
P. D. Welch
Keyword(s):  
Upper Bound ◽  
Large Cardinals ◽  
Large Cardinal ◽  
Core Model ◽  
Consistency Strength ◽  
The Core ◽  
Countable Ordinal ◽  
Elementary Embeddings ◽  
Remarkable Cardinals

AbstractThis paper continues the study of the Ramsey-like large cardinals introduced in [5] and [14]. Ramsey-like cardinals are defined by generalizing the characterization of Ramsey cardinals via the existence of elementary embeddings. Ultrafilters derived from such embeddings are fully iterable and so it is natural to ask about large cardinal notions asserting the existence of ultrafilters allowing only α-many iterations for some countable ordinal α. Here we study such α-iterable cardinals. We show that the α-iterable cardinals form a strict hierarchy for α ≤ ω1, that they are downward absolute to L for , and that the consistency strength of Schindler's remarkable cardinals is strictly between 1-iterable and 2-iterable cardinals.We show that the strongly Ramsey and super Ramsey cardinals from [5] are downward absolute to the core model K. Finally, we use a forcing argument from a strongly Ramsey cardinal to separate the notions of Ramsey and virtually Ramsey cardinals. These were introduced in [14] as an upper bound on the consistency strength of the Intermediate Chang's Conjecture.

Strongly unfoldable cardinals made indestructible

2008 ◽  
Vol 73 (4) ◽  
pp. 1215-1248 ◽  
Author(s):  
Thomas A. Johnstone
Keyword(s):  
Large Cardinals ◽  
Large Cardinal ◽  
Weakly Compact ◽  
Proper Forcing ◽  
Class Forcing

AbstractI provide indestructibility results for large cardinals consistent with V = L, such as weakly compact, indescribable and strongly unfoldable cardinals. The Main Theorem shows that any strongly unfoldable cardinal κ can be made indestructible by <κ-closed, κ-proper forcing. This class of posets includes for instance all <κ-closed posets that are either κ−-c.c. or <κ-strategically closed as well as finite iterations of such posets. Since strongly unfoldable cardinals strengthen both indescribable and weakly compact cardinals, the Main Theorem therefore makes these two large cardinal notions similarly indestructible. Finally. I apply the Main Theorem to obtain a class forcing extension preserving all strongly unfoldable cardinals in which every strongly unfoldable cardinal κ is indestructible by <κ-closed, κ-proper forcing.

Proper forcing and L(ℝ)

2001 ◽  
Vol 66 (2) ◽  
pp. 801-810 ◽  
Author(s):  
Itay Neeman ◽  
Jindřich Zapletal
Keyword(s):  
Large Cardinals ◽  
Large Cardinal ◽  
Consistency Strength ◽  
Woodin Cardinals ◽  
Proper Forcing

AbstractWe present two ways in which the model L(ℝ) is canonical assuming the existence of large cardinals. We show that the theory of this model, with ordinal parameters, cannot be changed by small forcing: we show further that a set of ordinals in V cannot be added to L(ℝ) by small forcing. The large cardinal needed corresponds to the consistency strength of ADL(ℝ): roughly ω Woodin cardinals.

MRP, tree properties and square principles

2011 ◽  
Vol 76 (4) ◽  
pp. 1441-1452 ◽  
Author(s):  
Remi Strullu
Keyword(s):  
Direct Proof ◽  
Large Cardinal ◽  
Supercompact Cardinal ◽  
Strongly Compact Cardinal ◽  
Compact Cardinal ◽  
Square Principles ◽  
Proper Forcing ◽  
The Relationship

AbstractWe show that MRP + MA implies that ITP(λ,ω2) holds for all cardinal λ ≥ ω2. This generalizes a result by Weiβ who showed that PFA implies that ITP(λ, ω2) holds for all cardinal λ ≥ ω2. Consequently any of the known methods to prove MRP + MA consistent relative to some large cardinal hypothesis requires the existence of a strongly compact cardinal. Moreover if one wants to force MRP + MA with a proper forcing, it requires at least a supercompact cardinal. We also study the relationship between MRP and some weak versions of square. We show that MRP implies the failure of □(λ, ω) for all λ ≥ ω2 and we give a direct proof that MRP + MA implies the failure of □(λ, ω1) for all λ ≥ ω2.

Zero Marking of Past Tense in Child African American English

2014 ◽  
Vol 21 (4) ◽  
pp. 173-181 ◽  
Author(s):  
Ryan Lee ◽  
Janna B. Oetting
Keyword(s):  
African American ◽  
American English ◽  
African American English ◽  
Current Paper ◽  
Past Tense ◽  
Typically Developing ◽  
Grammatical Function

Zero marking of the simple past is often listed as a common feature of child African American English (AAE). In the current paper, we review the literature and present new data to help clinicians better understand zero marking of the simple past in child AAE. Specifically, we provide information to support the following statements: (a) By six years of age, the simple past is infrequently zero marked by typically developing AAE-speaking children; (b) There are important differences between the simple past and participle morphemes that affect AAE-speaking children's marking options; and (c) In addition to a verb's grammatical function, its phonetic properties help determine whether an AAE-speaking child will produce a zero marked form.

Ethical Issues Relevant to the Assessment of Suicide Risk in Nonclinical Research Settings

Crisis ◽  
2012 ◽  
Vol 33 (1) ◽  
pp. 54-59 ◽  
Author(s):  
Carolyn M. Wilson ◽  
Bruce K. Christensen
Keyword(s):  
Undergraduate Students ◽  
Suicide Risk ◽  
Ethical Issues ◽  
Current Paper ◽  
Self Report ◽  
Ethical Guidelines ◽  
Schizotypal Personality ◽  
Increased Risk ◽  
Genetic Features ◽  
Conducting Research

Background: Our laboratory recently confronted this issue while conducting research with undergraduate students at the University of Waterloo (UW). Although our main objective was to examine cognitive and genetic features of individuals with schizotypal personality disorder (SPD), the study protocol also entailed the completion of various self-report measures to identify participants deemed at increased risk for suicide. Aims and Methods: This paper seeks to review and discuss the relevant ethical guidelines and legislation that bear upon a psychologist’s obligation to further assess and intervene when research participants reveal that they are at increased risk for suicide. Results and Conclusions: In the current paper we argue that psychologists are ethically impelled to assess and appropriately intervene in cases of suicide risk, even when such risk is revealed within a research context. We also discuss how any such obligation may potentially be modulated by the research participant’s expectations of the role of a psychologist, within such a context. Although the focus of the current paper is on the ethical obligations of psychologists, specifically those practicing within Canada, the relevance of this paper extends to all regulated health professionals conducting research in nonclinical settings.

The Persnickety Pervasiveness of Rating Enhancement in Personality Assessment

2020 ◽  
pp. 1-13
Author(s):  
Alicia A. Stachowski ◽  
John T. Kulas
Keyword(s):  
Response Bias ◽  
Personality Assessment ◽  
Current Paper ◽  
Psychological Measurement ◽  
Shared Sets ◽  
Self Reports ◽  
Original Target ◽  
Target Profile ◽  
Do So ◽  
Normative Influences

Abstract. The current paper explores whether self and observer reports of personality are properly viewed through a contrasting lens (as opposed to a more consonant framework). Specifically, we challenge the assumption that self-reports are more susceptible to certain forms of response bias than are informant reports. We do so by examining whether selves and observers are similarly or differently drawn to socially desirable and/or normative influences in personality assessment. Targets rated their own personalities and recommended another person to also do so along shared sets of items diversely contaminated with socially desirable content. The recommended informant then invited a third individual to additionally make ratings of the original target. Profile correlations, analysis of variances (ANOVAs), and simple patterns of agreement/disagreement consistently converged on a strong normative effect paralleling item desirability, with all three rater types exhibiting a tendency to reject socially undesirable descriptors while also endorsing desirable indicators. These tendencies were, in fact, more prominent for informants than they were for self-raters. In their entirety, our results provide a note of caution regarding the strategy of using non-self informants as a comforting comparative benchmark within psychological measurement applications.

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