The monadic theory of ω2

1983 ◽  
Vol 48 (2) ◽  
pp. 387-398 ◽  
Author(s):  
Yuri Gurevich ◽  
Menachem Magidor ◽  
Saharon Shelah
Keyword(s):  
Order Theory ◽  
Second Order ◽  
The Other ◽  
Weakly Compact ◽  
Compact Cardinal ◽  
Monadic Theory ◽  
Weakly Compact Cardinal

AbstractAssume ZFC + “There is a weakly compact cardinal” is consistent. Then:(i) For every S ⊆ ω, ZFC + “S and the monadic theory of ω2 are recursive each in the other” is consistent; and(ii) ZFC + “The full second-order theory of ω2 is interpretable in the monadic theory of ω2” is consistent.

scholarly journals The least weakly compact cardinal can be unfoldable, weakly measurable and nearly $${\theta}$$ θ -supercompact

2015 ◽  
Vol 54 (5-6) ◽  
pp. 491-510 ◽  
Author(s):  
Brent Cody ◽  
Moti Gitik ◽  
Joel David Hamkins ◽  
Jason A. Schanker
Keyword(s):  
Weakly Compact ◽  
Compact Cardinal ◽  
Weakly Compact Cardinal

scholarly journals Simultaneously vanishing higher derived limits

2021 ◽  
Vol 9 ◽  
Author(s):  
Jeffrey Bergfalk ◽  
Chris Lambie-Hanson
Keyword(s):  
Euclidean Space ◽  
Metric Spaces ◽  
Inverse System ◽  
Finite Support ◽  
Partial Functions ◽  
Weakly Compact ◽  
Compact Cardinal ◽  
Strong Homology ◽  
Finite Support Iteration ◽  
Weakly Compact Cardinal

Abstract In 1988, Sibe Mardešić and Andrei Prasolov isolated an inverse system $\textbf {A}$ with the property that the additivity of strong homology on any class of spaces which includes the closed subsets of Euclidean space would entail that $\lim ^n\textbf {A}$ (the nth derived limit of $\textbf {A}$ ) vanishes for every $n>0$ . Since that time, the question of whether it is consistent with the $\mathsf {ZFC}$ axioms that $\lim ^n \textbf {A}=0$ for every $n>0$ has remained open. It remains possible as well that this condition in fact implies that strong homology is additive on the category of metric spaces. We show that assuming the existence of a weakly compact cardinal, it is indeed consistent with the $\mathsf {ZFC}$ axioms that $\lim ^n \textbf {A}=0$ for all $n>0$ . We show this via a finite-support iteration of Hechler forcings which is of weakly compact length. More precisely, we show that in any forcing extension by this iteration, a condition equivalent to $\lim ^n\textbf {A}=0$ will hold for each $n>0$ . This condition is of interest in its own right; namely, it is the triviality of every coherent n-dimensional family of certain specified sorts of partial functions $\mathbb {N}^2\to \mathbb {Z}$ which are indexed in turn by n-tuples of functions $f:\mathbb {N}\to \mathbb {N}$ . The triviality and coherence in question here generalise the classical and well-studied case of $n=1$ .

scholarly journals PROOF THEORY OF WEAK COMPACTNESS

2013 ◽  
Vol 13 (01) ◽  
pp. 1350003 ◽  
Author(s):  
TOSHIYASU ARAI
Keyword(s):  
Set Theory ◽  
Proof Theory ◽  
Weak Compactness ◽  
Weakly Compact ◽  
Compact Cardinal ◽  
Weakly Compact Cardinal

We show that the existence of a weakly compact cardinal over the Zermelo–Fraenkel's set theory ZF is proof-theoretically reducible to iterations of Mostowski collapsings and Mahlo operations.

scholarly journals A Second-Order Theory of the Galilean Satellites of Jupiter

1978 ◽  
Vol 41 ◽  
pp. 209-236
Author(s):  
S. Ferraz-Mello
Keyword(s):  
Order Theory ◽  
Second Order ◽  
The Other ◽  
Reference Plane ◽  
Internal Mass ◽  
Galilean Satellites ◽  
Satellites Of Jupiter ◽  
Trigonometric Solution ◽  
The Mean ◽  
Non Linear Mechanics

AbstractThe theory of the motion of the Galilean satellites of Jupiter is developed up to the second-order terms. The disturbing forces are those due to mutual attractions, to the non-symmetrical internal mass distribution of Jupiter and to the attraction from the Sun. The mean equator of Jupiter is taken as the reference plane and its motion is considered. The integration of the equations is performed. The geometric equations are solved for the case in which the amplitude of libration is zero. The perturbation method is shortly commented on the grounds of some recent advances in non-linear mechanics.In a previous paper (Ferraz-Mello, 1974) one perturbation theory has been constructed with special regard to the problem of the motion of the Galilean satellites of Jupiter. In this problem, the motions are nearly circular and coplanar; on the other hand the quasi-resonances lead to strong perturbations. The main characteristic of the theory is that it allows the main frequencies to be kept fixed from the earlier stages, and so, to have a purely trigonometric solution.

Do children understand that people selectively conceal or express emotion?

2014 ◽  
Vol 39 (1) ◽  
pp. 1-8 ◽  
Author(s):  
Hajimu Hayashi ◽  
Yuki Shiomi
Keyword(s):  
First Graders ◽  
Negative Emotion ◽  
Fifth Graders ◽  
Order Theory ◽  
Third Graders ◽  
Second Order ◽  
The Other ◽  
The Real ◽  
Express Emotion ◽  
Negative Feelings

This study examined whether children understand that people selectively conceal or express emotion depending upon the context. We prepared two contexts for a verbal display task for 70 first-graders, 80 third-graders, 64 fifth-graders, and 71 adults. In both contexts, protagonists had negative feelings because of the behavior of the other character. In the prosocial context, children were instructed that the protagonist wished to spare the other character’s feelings. In contrast, in the real-emotion context, children were told that the protagonist was fed up with the other character’s behavior. Participants were asked to imagine what the protagonists would say. Adults selected utterances with positive or neutral emotion in the prosocial context but chose utterances with negative emotion in the real-emotion context, whereas first-graders selected utterances with negative emotion in both contexts. In the prosocial context, the proportion of utterances with negative emotion decreased from first-graders to adults, whereas in the real-emotion context the proportion was U-shaped, decreasing from first- to third-graders and increasing from fifth-graders to adults. Further, performance on both contexts was associated with second-order false beliefs as well as second-order intention understanding. These results indicate that children begin to understand that people selectively conceal or express emotion depending upon context after 8 to 9 years. This ability is also related to second-order theory of mind.

Peano arithmetic may not be interpretable in the monadic theory of linear orders

1997 ◽  
Vol 62 (3) ◽  
pp. 848-872 ◽  
Author(s):  
Shmuel Lifsches ◽  
Saharon Shelah
Keyword(s):  
Real Line ◽  
Order Theory ◽  
Peano Arithmetic ◽  
Second Order ◽  
Linear Orders ◽  
The Real ◽  
Monadic Theory

AbstractGurevich and Shelah have shown that Peano Arithmetic cannot be interpreted in the monadic second-order theory of short chains (hence, in the monadic second-order theory of the real line). We will show here that it is consistent that the monadic second-order theory of no chain interprets Peano Arithmetic.

Two weak consequences of 0#

1985 ◽  
Vol 50 (3) ◽  
pp. 597-603
Author(s):  
M. Gitik ◽  
M. Magidor ◽  
H. Woodin
Keyword(s):  
Limit Point ◽  
Inaccessible Cardinal ◽  
Weakly Compact ◽  
Compact Cardinal ◽  
Weakly Compact Cardinal

AbstractIt is proven that the following statement:“there exists a club C ⊆ κ such that every α ∈ C is an inaccessible cardinal in L and, for every δ a limit point of C, C ∩ δ is almost contained in every club of δ of L”is equiconsistent with a weakly compact cardinal if δ = ℵ1, and with a weakly compact cardinal of order 1 if δ = ℵ2.

On the ordering of certain large cardinals

1979 ◽  
Vol 44 (4) ◽  
pp. 563-565
Author(s):  
Carl F. Morgenstern
Keyword(s):  
Measurable Cardinal ◽  
Large Cardinals ◽  
Large Cardinal ◽  
Inaccessible Cardinal ◽  
Elementary Embedding ◽  
Strongly Compact Cardinal ◽  
Weakly Compact ◽  
Compact Cardinal ◽  
The Universe ◽  
Weakly Compact Cardinal

It is well known that the first strongly inaccessible cardinal is strictly less than the first weakly compact cardinal which in turn is strictly less than the first Ramsey cardinal, etc. However, once one passes the first measurable cardinal the inequalities are no longer strict. Magidor [3] has shown that the first strongly compact cardinal may be equal to the first measurable cardinal or equal to the first super-compact cardinal (the first supercompact cardinal is strictly larger than the first measurable cardinal). In this note we will indicate how Magidor's methods can be used to show that it is undecidable whether one cardinal (the first strongly compact) is greater than or less than another large cardinal (the first huge cardinal). We assume that the reader is familiar with the ultrapower construction of Scott, as presented in Drake [1] or Kanamori, Reinhardt and Solovay [2].Definition. A cardinal κ is huge (or 1-huge) if there is an elementary embedding j of the universe V into a transitive class M such that M contains the ordinals, is closed under j(κ) sequences, j(κ) > κ and j ↾ Rκ = id. Let κ denote the first huge cardinal, and let λ = j(κ).One can see from easy reflection arguments that κ and λ are inaccessible in V and, in fact, that κ is measurable in V.

On the consistency of the Definable Tree Property on ℵ1

2000 ◽  
Vol 65 (3) ◽  
pp. 1204-1214 ◽  
Author(s):  
Amir Leshem
Keyword(s):  
Tree Property ◽  
Consistency Strength ◽  
Weakly Compact ◽  
Compact Cardinal ◽  
First Order ◽  
Weakly Compact Cardinal

AbstractIn this paper we prove the equiconsistency of “Every ω1 –tree which is first order definable over (, ε) has a cofinal branch” with the existence of a reflecting cardinal. We also prove that the addition of MA to the definable tree property increases the consistency strength to that of a weakly compact cardinal. Finally we comment on the generalization to higher cardinals.

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