The consistency of one fixed omega

1995 ◽  
Vol 60 (1) ◽  
pp. 172-177 ◽  
Author(s):  
J. M. Henle
Keyword(s):  
Inaccessible Cardinal ◽  
Partition Relation ◽  
Image Position ◽  
The Way

AbstractThe paper “Partitions of Products” [DiPH] investigated the polarized partition relationThe relation is consistent relative to an inaccessible cardinal if every αi is finite, but inconsistent if two are infinite. We show here that it consistent (relative to an inaccessible) for one to be infinite.Along the way, we prove an interesting proposition from ZFC concerning partitions of the finite subsets of ω.

Partitions of products

1993 ◽  
Vol 58 (3) ◽  
pp. 860-871 ◽  
Author(s):  
Carlos A. Di Prisco ◽  
James M. Henle
Keyword(s):  
Finite Number ◽  
Axiom Of Choice ◽  
Inaccessible Cardinal ◽  
Partition Relation ◽  
Partition Relations ◽  
Image Position ◽  
The Axiom Of Choice ◽  
Usual Notation

We will consider some partition properties of the following type: given a function F: ωω →2, is there a sequence H0, H1, … of subsets of ω such that F is constant on ΠiεωHi? The answer is obviously positive if we allow all the Hi's to have exactly one element, but the problem is nontrivial if we require the Hi's to have at least two elements. The axiom of choice contradicts the statement “for all F: ωω→ 2 there is a sequence H0, H1, H2,… of subsets of ω such that {i|(Hi) ≥ 2} is infinite and F is constant on ΠHi”, but the infinite exponent partition relation ω(ω)ω implies it; so, this statement is relatively consistent with an inaccessible cardinal. (See [1] where these partition properties were considered.)We will also consider partitions into any finite number of pieces, and we will prove some facts about partitions into ω-many pieces.Given a partition F: ωω → k, we say that H0, H1…, a sequence of subsets of ω, is homogeneous for F if F is constant on ΠHi. We say the sequence H0, H1,… is nonoverlapping if, for all i ∈ ω, ∪Hi > ∩Hi+1.The sequence 〈Hi: i ∈ ω〉 is of type 〈α0, α1,…〉 if, for every i ∈ ω, ∣Hi∣ = αi.We will adopt the usual notation for polarized partition relations due to Erdös, Hajnal, and Rado.means that for every partition F: κ1 × κ2 × … × κn→δ there is a sequence H0, H1,…, Hn such that Hi ⊂ κi and ∣Hi∣ = αi for every i, 1 ≤ i ≤ n, and F is constant on H1 × H2 × … × Hn.

scholarly journals THE STRONG TREE PROPERTY AT SUCCESSORS OF SINGULAR CARDINALS

2014 ◽  
Vol 79 (01) ◽  
pp. 193-207 ◽  
Author(s):  
LAURA FONTANELLA
Keyword(s):  
Singular Limit ◽  
Inaccessible Cardinal ◽  
Tree Property ◽  
Supercompact Cardinals ◽  
Singular Cardinals ◽  
Image Position ◽  
Strongly Compact Cardinals

Abstract An inaccessible cardinal is strongly compact if, and only if, it satisfies the strong tree property. We prove that if there is a model of ZFC with infinitely many supercompact cardinals, then there is a model of ZFC where ${\aleph _{\omega + 1}}$ has the strong tree property. Moreover, we prove that every successor of a singular limit of strongly compact cardinals has the strong tree property.

scholarly journals LOGICS FOR PROPOSITIONAL DETERMINACY AND INDEPENDENCE

2018 ◽  
Vol 11 (3) ◽  
pp. 470-506 ◽  
Author(s):  
VALENTIN GORANKO ◽  
ANTTI KUUSISTO
Keyword(s):  
Natural Language ◽  
Kripke Semantics ◽  
Dependence Logic ◽  
Team Semantics ◽  
Reasoning Tasks ◽  
Image Position ◽  
The Way

AbstractThis paper investigates formal logics for reasoning about determinacy and independence. Propositional Dependence Logic${\cal D}$and Propositional Independence Logic${\cal I}$are recently developed logical systems, based on team semantics, that provide a framework for such reasoning tasks. We introduce two new logics${{\cal L}_D}$and${{\cal L}_{\,I\,}}$, based on Kripke semantics, and propose them as alternatives for${\cal D}$and${\cal I}$, respectively. We analyse the relative expressive powers of these four logics and discuss the way these systems relate to natural language. We argue that${{\cal L}_D}$and${{\cal L}_{\,I\,}}$naturally resolve a range of interpretational problems that arise in${\cal D}$and${\cal I}$. We also obtain sound and complete axiomatizations for${{\cal L}_D}$and${{\cal L}_{\,I\,}}$.

A Mountain-Climbing Problem

1966 ◽  
Vol 18 ◽  
pp. 873-882 ◽  
Author(s):  
James V. Whittaker
Keyword(s):  
Mountain Range ◽  
Mountain Climbing ◽  
Continuous Mappings ◽  
Image Position ◽  
The Way

Suppose that two men stand at the same elevation on opposite sides of a mountain range and begin to climb in such a way that their elevations remain equal at all times. Will they ever meet along the way? It is this question, restated in mathematical terms, that we shall consider. We replace the mountain range by the graph of a continuous, real-valued function f(x) defined for x ∈ [0, 1], where f(0) = f(1) = 0, and we ask whether there exist continuous mappings ϕ(t), ψ(t) from [0, 1] into [0, 1] such that12

scholarly journals FINITARY REDUCIBILITY ON EQUIVALENCE RELATIONS

2016 ◽  
Vol 81 (4) ◽  
pp. 1225-1254 ◽  
Author(s):  
RUSSELL MILLER ◽  
KENG MENG NG
Keyword(s):  
Equivalence Relation ◽  
Equivalence Relations ◽  
Arithmetical Hierarchy ◽  
Natural Equivalence ◽  
Image Position ◽  
The Way

AbstractWe introduce the notion of finitary computable reducibility on equivalence relations on the domainω. This is a weakening of the usual notion of computable reducibility, and we show it to be distinct in several ways. In particular, whereas no equivalence relation can be${\rm{\Pi }}_{n + 2}^0$-complete under computable reducibility, we show that, for everyn, there does exist a natural equivalence relation which is${\rm{\Pi }}_{n + 2}^0$-complete under finitary reducibility. We also show that our hierarchy of finitary reducibilities does not collapse, and illustrate how it sharpens certain known results. Along the way, we present several new results which use computable reducibility to establish the complexity of various naturally defined equivalence relations in the arithmetical hierarchy.

scholarly journals HYPERCLASS FORCING IN MORSE-KELLEY CLASS THEORY

2017 ◽  
Vol 82 (2) ◽  
pp. 549-575 ◽  
Author(s):  
CAROLIN ANTOS ◽  
SY-DAVID FRIEDMAN
Keyword(s):  
Second Order ◽  
Inaccessible Cardinal ◽  
Second Order Arithmetic ◽  
Class Theory ◽  
Order Arithmetic ◽  
Image Position ◽  
Class Forcing

AbstractIn this article we introduce and study hyperclass-forcing (where the conditions of the forcing notion are themselves classes) in the context of an extension of Morse-Kelley class theory, called MK**. We define this forcing by using a symmetry between MK** models and models of ZFC− plus there exists a strongly inaccessible cardinal (called SetMK**). We develop a coding between β-models ${\cal M}$ of MK** and transitive models M+ of SetMK** which will allow us to go from ${\cal M}$ to M+ and vice versa. So instead of forcing with a hyperclass in MK** we can force over the corresponding SetMK** model with a class of conditions. For class-forcing to work in the context of ZFC− we show that the SetMK** model M+ can be forced to look like LK*[X], where κ* is the height of M+, κ strongly inaccessible in M+ and $X \subseteq \kappa$. Over such a model we can apply definable class forcing and we arrive at an extension of M+ from which we can go back to the corresponding β-model of MK**, which will in turn be an extension of the original ${\cal M}$. Our main result combines hyperclass forcing with coding methods of [3] and [4] to show that every β-model of MK** can be extended to a minimal such model of MK** with the same ordinals. A simpler version of the proof also provides a new and analogous minimality result for models of second-order arithmetic.

Some remarkable integrals derived from a simple algebraic identity

2013 ◽  
Vol 97 (539) ◽  
pp. 205-209
Author(s):  
Graham J. O. Jameson ◽  
Timothy P. Jameson
Keyword(s):  
Definite Integral ◽  
Algebraic Identity ◽  
Special Cases ◽  
Long Time ◽  
Comprehensive Survey ◽  
Image Position ◽  
The Way

The identity in question really is simple: it says, for u ≠ −1,We describe two types of definite integral that look quite formidable, but dissolve into a much simpler form by an application of (1) in a way that seems almost magical.Both types, or at least special cases of them, have been mathematical folklore for a long time. For example, case (10) below appears in [1, p. 262], published in 1922 (we are grateful to Donald Kershaw for showing us this example). However, they do not seem to figure in most books on calculus except possibly tucked away as an exercise The comprehensive survey [2] mentions the second type on p. 253, but only as a lemma on the way to an identity the authors call the ‘master formula’ We come back to this formula later, but only after describing a number of other more immediate applications.

scholarly journals OPTIMAL RESULTS ON RECOGNIZABILITY FOR INFINITE TIME REGISTER MACHINES

2015 ◽  
Vol 80 (4) ◽  
pp. 1116-1130 ◽  
Author(s):  
MERLIN CARL
Keyword(s):  
Detailed Analysis ◽  
Infinite Time ◽  
Image Position ◽  
The Way

AbstractExploring further the properties of ITRM-recognizable reals started in [1], we provide a detailed analysis of recognizable reals and their distribution in Gödels constructible universe L. In particular, we show that new unrecognizable reals are generated at every index $\gamma \, \ge \,\omega _\omega ^{CK}$. We give a machine-independent characterization of recognizability by proving that a real r is recognizable if and only if it is ${\Sigma _1}$-definable over ${L_{\omega _\omega ^{CK,\,r}}}$ and that $r\, \in \,{L_{\omega _\omega ^{CK,\,r}}}$ for every recognizable real r and show that either every or no r with $r\, \in \,{L_{\omega _\omega ^{CK,\,r}}}$ generated over an index stage ${L_\gamma }$ is recognizable. Finally, the techniques developed along the way allow us to prove that the halting number for ITRMs is recognizable and that the set of ITRM-computable reals is not ITRM-decidable.

Death and Immortality: A Study of the Heraclitus Epigram of Callimachus

Ramus ◽  
1982 ◽  
Vol 11 (1) ◽  
pp. 48-56 ◽  
Author(s):  
J. G. MacQueen
Keyword(s):  
Good Deal ◽  
The Sun ◽  
Classical Literature ◽  
English Version ◽  
Long Time ◽  
Image Position ◽  
The Way

Somebody mentioned your fate, Heraclitus, and he brought me to a tear; and I remembered how often we both made the sun sink in conversation. But you, my guest-friend from Halicarnassus, have, I suppose, been ashes for a very long time. But your nightingales are alive, on which Hades, plunderer of all things, will not lay his hand.This epigram of Callimachus is one of the best known poems in Classical literature, but it suffers more than most from the misfortune of having to live permanently in the shadow of its own translation. It may no longer be the case that every schoolboy knows ‘They told me, Heraclitus, they told me you were dead’, but it is certainly true that Cory's English version is much more widely known, and much more widely quoted, than Callimachus's Greek original. One result of this has been that a good deal of attention has often been given to comparing the two poems, but little time has been spent on examining the Callimachus as a poem in itself in an effort to see what its virtues are. One may occasionally find a few remarks on the restraint or simplicity of the Greek, as opposed to the English, or a note suggesting that Heraclitus of Halicarnassus, the poet to whom the verses are addressed, wrote a volume of verse the title of which was actuallyAēdones(‘Nightingales’) — hence the ‘nightingales’ of the second last line. Occasionally a commentator will go a little further. K. J. Mckay for instance remarks: ‘The high respect in which this epigram is held is fully justified. The way in which the thoughts spill over their barriers in the first four lines, the magic ofkatedusamen(suggestive of a communion of uncommon power), the skilful location of key thoughts (teon moron, katedusamen, aēdones), the pathos of an unknown grave and an abiding grief cannot but move us. Above all, the suggestion of unfathomable sorrow.’

Alfred Tarski's work in set theory

1988 ◽  
Vol 53 (1) ◽  
pp. 2-6 ◽  
Author(s):  
Azriel Levy
Keyword(s):  
Set Theory ◽  
Axiom System ◽  
Large Cardinals ◽  
Inaccessible Cardinal ◽  
Promised Land ◽  
And Mathematics ◽  
The University ◽  
Axiomatic Set Theory ◽  
The Way

Alfred Tarski started contributing to set theory at a time when the Zermelo-Fraenkel axiom system was not yet fully formulated and as simple a concept as that of the inaccessible cardinal was not yet fully defined. At the end of Tarski's career the basic concepts of the three major areas and tools of modern axiomatic set theory, namely constructibility, large cardinals and forcing, were already clearly defined and were in the midst of a rapid successful development. The role of Tarski in this development was somewhat similar to the role of Moses showing his people the way to the Promised Land and leading them along the way, while the actual entry of the Promised Land was done mostly by the next generation. The theory of large cardinals was started mostly by Tarski, and developed mostly by his school. The mathematical logicians of Tarski's school contributed much to the development of forcing, after its discovery by Paul Cohen, and to a lesser extent also to the development of the theory of constructibility, discovered by Kurt Gödel. As in other areas of logic and mathematics Tarski's contribution to set theory cannot be measured by his own results only; Tarski was a source of energy and inspiration to his pupils and collaborators, of which I was fortunate to be one, always confronting them with new problems and pushing them to gain new ground.Tarski's interest in set theory was probably aroused by the general emphasis on set theory in Poland after the First World War, and by the influence of Wactaw Sierpinski, who was one of Tarski's teachers at the University of Warsaw. The very first paper published by Tarski, [21], was a paper in set theory.

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