Log in

Relevant bibliographies by topics / Large cardinals (Mathematics) / Journal articles

To see the other types of publications on this topic, follow the link: Large cardinals (Mathematics).

Journal articles on the topic 'Large cardinals (Mathematics)'

Author: Grafiati

Published: 4 June 2021

Last updated: 6 February 2022

Create a spot-on reference in APA, MLA, Chicago, Harvard, and other styles

Select a source type:

Consult the top 50 journal articles for your research on the topic 'Large cardinals (Mathematics).'

Next to every source in the list of references, there is an 'Add to bibliography' button. Press on it, and we will generate automatically the bibliographic reference to the chosen work in the citation style you need: APA, MLA, Harvard, Chicago, Vancouver, etc.

You can also download the full text of the academic publication as pdf and read online its abstract whenever available in the metadata.

Browse journal articles on a wide variety of disciplines and organise your bibliography correctly.

1

Apter, Arthur W., James Cummings, and Joel David Hamkins. "Large cardinals with few measures." Proceedings of the American Mathematical Society 135, no. 07 (July 1, 2007): 2291–301. http://dx.doi.org/10.1090/s0002-9939-07-08786-2.

Full text

APA, Harvard, Vancouver, ISO, and other styles

2

Good, Chris. "Large cardinals and small Dowker spaces." Proceedings of the American Mathematical Society 123, no. 1 (January 1, 1995): 263. http://dx.doi.org/10.1090/s0002-9939-1995-1216813-0.

Full text

APA, Harvard, Vancouver, ISO, and other styles

3

ESKEW, MONROE. "GENERIC LARGE CARDINALS AS AXIOMS." Review of Symbolic Logic 13, no. 2 (May 14, 2019): 375–87. http://dx.doi.org/10.1017/s1755020319000200.

Full text

APA, Harvard, Vancouver, ISO, and other styles

4

FARAH, ILIJAS, and BOBAN VELICKOVIC. "VON NEUMANN'S PROBLEM AND LARGE CARDINALS." Bulletin of the London Mathematical Society 38, no. 06 (December 2006): 907–12. http://dx.doi.org/10.1112/s0024609306018704.

Full text

APA, Harvard, Vancouver, ISO, and other styles

5

Boney, Will. "Model theoretic characterizations of large cardinals." Israel Journal of Mathematics 236, no. 1 (February 12, 2020): 133–81. http://dx.doi.org/10.1007/s11856-020-1971-6.

Full text

APA, Harvard, Vancouver, ISO, and other styles

6

Jensen, Ronald. "Inner Models and Large Cardinals." Bulletin of Symbolic Logic 1, no. 4 (December 1995): 393–407. http://dx.doi.org/10.2307/421129.

Full text

Abstract:

In this paper, we sketch the development of two important themes of modern set theory, both of which can be regarded as growing out of work of Kurt Gödel. We begin with a review of some basic concepts and conventions of set theory. §0. The ordinal numbers were Georg Cantor's deepest contribution to mathematics. After the natural numbers 0, 1, …, n, … comes the first infinite ordinal number ω, followed by ω + 1, ω + 2, …, ω + ω, … and so forth. ω is the first limit ordinal as it is neither 0 nor a successor ordinal. We follow the von Neumann convention, according to which each ordinal number α is identified with the set {ν ∣ ν α} of its predecessors. The ∈ relation on ordinals thus coincides with <. We have 0 = ∅ and α + 1 = α ∪ {α}. According to the usual set-theoretic conventions, ω is identified with the first infinite cardinal ℵ0, similarly for the first uncountable ordinal number ω1 and the first uncountable cardinal number ℵ1, etc. We thus arrive at the following picture: The von Neumann hierarchy divides the class V of all sets into a hierarchy of sets Vα indexed by the ordinal numbers. The recursive definition reads: (where } is the power set of x); Vλ = ∪v<λVv for limit ordinals λ. We can represent this hierarchy by the following picture.

APA, Harvard, Vancouver, ISO, and other styles

7

Foreman, Matthew, and Andras Hajnal. "A partition relation for successors of Large Cardinals." Mathematische Annalen 325, no. 3 (March 1, 2003): 583–623. http://dx.doi.org/10.1007/s00208-002-0323-7.

Full text

APA, Harvard, Vancouver, ISO, and other styles

8

Fremlin, D. H. "Large correlated families of positive random variables." Mathematical Proceedings of the Cambridge Philosophical Society 103, no. 1 (January 1988): 147–62. http://dx.doi.org/10.1017/s0305004100064707.

Full text

Abstract:

S. Argyros and N. Kalamidas([l], repeated in [2], Theorem 6·15) proved the following. If κ is a cardinal of uncountable cofinality, and 〈Eξ〉ξ<κ is a family of measurable sets in a probability space (X, μ) such that infξ<κμEξ = δ, and if n ≥ 1, , then there is a set Γ ⊆ κ such that #(Γ) = κ and μ(∩ξ∈IEξ) ≥ γ whenever I ⊆ ξ has n members. In Proposition 7 below I refine this result by (i) taking any γ < δn (which is best possible) and (ii) extending the result to infinite cardinals of countable cofinality, thereby removing what turns out to be an irrelevant restriction. The proof makes it natural to perform a further extension to general uniformly bounded families of non-negative measurable functions in place of characteristic functions.

APA, Harvard, Vancouver, ISO, and other styles

9

Eskew, Monroe, and Yair Hayut. "Global Chang’s Conjecture and singular cardinals." European Journal of Mathematics 7, no. 2 (March 24, 2021): 435–63. http://dx.doi.org/10.1007/s40879-021-00459-8.

Full text

Abstract:

AbstractWe investigate the possibilities of global versions of Chang’s Conjecture that involve singular cardinals. We show some $$\mathrm{ZFC} $$ ZFC limitations on such principles and prove relative to large cardinals that Chang’s Conjecture can consistently hold between all pairs of limit cardinals below $$\aleph _{\omega ^\omega }$$ ℵ ω ω .

APA, Harvard, Vancouver, ISO, and other styles

10

Feferman, Solomon. "Operational set theory and small large cardinals." Information and Computation 207, no. 10 (October 2009): 971–79. http://dx.doi.org/10.1016/j.ic.2008.04.007.

Full text

APA, Harvard, Vancouver, ISO, and other styles

11

Apter, Arthur W. "Some results on consecutive large cardinals II: Applications of radin forcing." Israel Journal of Mathematics 52, no. 4 (December 1985): 273–92. http://dx.doi.org/10.1007/bf02774081.

Full text

APA, Harvard, Vancouver, ISO, and other styles

12

Gitik, Moti, and Jiri Witzany. "Consistency strength of the axiom of full reflection at large cardinals." Israel Journal of Mathematics 93, no. 1 (December 1996): 113–24. http://dx.doi.org/10.1007/bf02761096.

Full text

APA, Harvard, Vancouver, ISO, and other styles

13

WEBER, ZACH. "TRANSFINITE NUMBERS IN PARACONSISTENT SET THEORY." Review of Symbolic Logic 3, no. 1 (January 14, 2010): 71–92. http://dx.doi.org/10.1017/s1755020309990281.

Full text

Abstract:

This paper begins an axiomatic development of naive set theory—the consequences of a full comprehension principle—in a paraconsistent logic. Results divide into two sorts. There is classical recapture, where the main theorems of ordinal and Peano arithmetic are proved, showing that naive set theory can provide a foundation for standard mathematics. Then there are major extensions, including proofs of the famous paradoxes and the axiom of choice (in the form of the well-ordering principle). At the end I indicate how later developments of cardinal numbers will lead to Cantor’s theorem, the existence of large cardinals, and a counterexample to the continuum hypothesis.

APA, Harvard, Vancouver, ISO, and other styles

14

CUMMINGS, JAMES, MATTHEW FOREMAN, and MENACHEM MAGIDOR. "SQUARES, SCALES AND STATIONARY REFLECTION." Journal of Mathematical Logic 01, no. 01 (May 2001): 35–98. http://dx.doi.org/10.1142/s021906130100003x.

Full text

Abstract:

Since the work of Gödel and Cohen, which showed that Hilbert's First Problem (the Continuum Hypothesis) was independent of the usual assumptions of mathematics (axiomatized by Zermelo–Fraenkel Set Theory with the Axiom of Choice, ZFC), there have been a myriad of independence results in many areas of mathematics. These results have led to the systematic study of several combinatorial principles that have proven effective at settling many of the important independent statements. Among the most prominent of these are the principles diamond(♢) and square(□) discovered by Jensen. Simultaneously, attempts have been made to find suitable natural strengthenings of ZFC, primarily by Large Cardinal or Reflection Axioms. These two directions have tension between them in that Jensen's principles, which tend to suggest a rather rigid mathematical universe, are at odds with reflection properties. A third development was the discovery by Shelah of "PCF Theory", a generalization of cardinal arithmetic that is largely determined inside ZFC. In this paper we consider interactions between these three theories in the context of singular cardinals, focusing on the various implications between square and scales (a fundamental notion in PCF theory), and on consistency results between relatively strong forms of square and stationary set reflection.

APA, Harvard, Vancouver, ISO, and other styles

15

PARÉ, R., and J. ROSICKÝ. "Colimits of accessible categories." Mathematical Proceedings of the Cambridge Philosophical Society 155, no. 1 (January 28, 2013): 47–50. http://dx.doi.org/10.1017/s0305004113000030.

Full text

Abstract:

AbstractWe show that any directed colimit of accessible categories and accessible full embeddings is accessible and, assuming the existence of arbitrarily large strongly compact cardinals, any directed colimit of accessible categories and accessible embeddings is accessible.

APA, Harvard, Vancouver, ISO, and other styles

16

Shelah, Saharon, and Hugh Woodin. "Large cardinals imply that every reasonably definable set of reals is lebesgue measurable." Israel Journal of Mathematics 70, no. 3 (October 1990): 381–94. http://dx.doi.org/10.1007/bf02801471.

Full text

APA, Harvard, Vancouver, ISO, and other styles

17

Levy, Azriel. "Alfred Tarski's work in set theory." Journal of Symbolic Logic 53, no. 1 (March 1988): 2–6. http://dx.doi.org/10.1017/s0022481200028887.

Full text

Abstract:

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.

APA, Harvard, Vancouver, ISO, and other styles

18

Balanda, Kevin P. "Families of partial representing sets." Journal of the Australian Mathematical Society. Series A. Pure Mathematics and Statistics 38, no. 2 (April 1985): 198–206. http://dx.doi.org/10.1017/s1446788700023053.

Full text

Abstract:

AbstractAssume GCH. Let κ, μ, Σ be cardinals, with κ infinite. Let be a family consisting of λ pairwise almost disjoint subsets of Σ each of size κ, whose union is Σ. In this note it is shown that for each μ with 1 ≤ μ ≤min(λ, Σ), there is a “large” almost disjoint family of μ-sized subsets of Σ, each member of having non-empty intersection with at least μ members of the family .

APA, Harvard, Vancouver, ISO, and other styles

19

Bagaria, Joan. "Saharon Shelah and Hugh Woodin. Large cardinals imply that every reasonably definable set of reals is Lebesgue measurable. Israel journal of mathematics, vol. 70 (1990), pp. 381–394." Bulletin of Symbolic Logic 8, no. 4 (December 2002): 543–45. http://dx.doi.org/10.2178/bsl/1182353934.

Full text

APA, Harvard, Vancouver, ISO, and other styles

20

Corazza, Paul. "The Axiom of Infinity and Transformations j: V → V." Bulletin of Symbolic Logic 16, no. 1 (March 2010): 37–84. http://dx.doi.org/10.2178/bsl/1264433797.

Full text

Abstract:

AbstractWe suggest a new approach for addressing the problem of establishing an axiomatic foundation for large cardinals. An axiom asserting the existence of a large cardinal can naturally be viewed as a strong Axiom of Infinity. However, it has not been clear on the basis of our knowledge of ω itself, or of generally agreed upon intuitions about the true nature of the mathematical universe, what the right strengthening of the Axiom of Infinity is—which large cardinals ought to be derivable? It was shown in the 1960s by Lawvere that the existence of an infinite set is equivalent to the existence of a certain kind of structure-preserving transformation from V to itself, not isomorphic to the identity. We use Lawvere's transformation, rather than ω, as a starting point for a reasonably natural sequence of strengthenings and refinements, leading to a proposed strong Axiom of Infinity. A first refinement was discussed in later work by Trnková–Blass, showing that if the preservation properties of Lawvere's tranformation are strengthened to the point of requiring it to be an exact functor, such a transformation is provably equivalent to the existence of a measurable cardinal. We propose to push the preservation properties as far as possible, short of inconsistency. The resulting transformation V → V is strong enough to account for virtually all large cardinals, but is at the same time a natural generalization of an assertion about transformations V → V known to be equivalent to the Axiom of Infinity.

APA, Harvard, Vancouver, ISO, and other styles

21

Welch, P. D. "A. Kanamori The higher infinite: large cardinals in set theory from their beginnings (Perspectives in Mathematical Logic, Springer-Verlag, Berlin-Heidelberg-New York-London-Paris-Tokyo-Hong Kong, 1994), xxiv + 536 pp., 3 540 57071 3, £77.50." Proceedings of the Edinburgh Mathematical Society 41, no. 1 (February 1998): 208–9. http://dx.doi.org/10.1017/s0013091500019532.

Full text

APA, Harvard, Vancouver, ISO, and other styles

22

Macpherson, Dugald, Alan H. Mekler, and Saharon Shelah. "The number of infinite substructures." Mathematical Proceedings of the Cambridge Philosophical Society 109, no. 1 (January 1991): 193–209. http://dx.doi.org/10.1017/s0305004100069668.

Full text

Abstract:

AbstractGiven a relational structure M and a cardinal λ < |M|, let øλ denote the number of isomorphism types of substructures of M of size λ. It is shown that if μ < λ are cardinals, and |M| is sufficiently larger than λ, then øμ ≤ øλ. A description is also given for structures with few substructures of given infinite cardinality.

APA, Harvard, Vancouver, ISO, and other styles

23

Kanamori, Akihiro. "Regressive partitions and Borel diagonalization." Journal of Symbolic Logic 54, no. 2 (June 1989): 540–52. http://dx.doi.org/10.2307/2274868.

Full text

Abstract:

Several rather concrete propositions about Borel measurable functions of several variables on the Hilbert cube (countable sequences of reals in the unit interval) were formulated by Harvey Friedman [F1] and correlated with strong set-theoretic hypotheses. Most notably, he established that a “Borel diagonalization” proposition P is equivalent to: for any a ⊆ co and n ⊆ ω there is an ω-model of ZFC + ∃κ(κ is n-Mahlo) containing a. In later work (see the expository Stanley [St] and Friedman [F2]), Friedman was to carry his investigations further into propositions about spaces of groups and the like, and finite propositions. He discovered and analyzed mathematical propositions which turned out to have remarkably strong consistency strength in terms of large cardinal hypotheses in set theory.In this paper, we refine and extend Friedman's work on the Borel diagonalization proposition P. First, we provide more combinatorics about regressive partitions and n-Mahlo cardinals and extend the approach to the context of the Erdös cardinals In passing, a combinatorial proof of a well-known result of Silver about these cardinals is given. Incorporating this work and sharpening Friedman's proof, we then show that there is a level-by-level analysis of P which provides for each n ⊆ ω a proposition almost equivalent to: for any a ⊆ co there is an ω-model of ZFC + ∃κ(κ is n-Mahlo) containing a. Finally, we use the combinatorics to bracket a natural generalization Sω of P between two large cardinal hypotheses.

APA, Harvard, Vancouver, ISO, and other styles

24

McCallum, Rupert. "Intrinsic Justifications for Large-Cardinal Axioms." Philosophia Mathematica 29, no. 2 (April 5, 2021): 195–213. http://dx.doi.org/10.1093/philmat/nkaa038.

Full text

Abstract:

ABSTRACT We shall defend three philosophical theses about the extent of intrinsic justification based on various technical results. We shall present a set of theorems which indicate intriguing structural similarities between a family of “weak” reflection principles roughly at the level of those considered by Tait and Koellner and a family of “strong” reflection principles roughly at the level of those of Welch and Roberts, which we claim to lend support to the view that the stronger reflection principles are intrinsically justified as well as the weaker ones. We consider connections with earlier work of Marshall.

APA, Harvard, Vancouver, ISO, and other styles

25

Boney, Will, and Spencer Unger. "Large cardinal axioms from tameness in AECs." Proceedings of the American Mathematical Society 145, no. 10 (April 7, 2017): 4517–32. http://dx.doi.org/10.1090/proc/13555.

Full text

APA, Harvard, Vancouver, ISO, and other styles

26

Casacuberta, Carles, Dirk Scevenels, and Jeffrey H. Smith. "Implications of large-cardinal principles in homotopical localization." Advances in Mathematics 197, no. 1 (October 2005): 120–39. http://dx.doi.org/10.1016/j.aim.2004.10.001.

Full text

APA, Harvard, Vancouver, ISO, and other styles

27

Drake, F. R. "THE HIGHER INFINITE. LARGE CARDINALS IN SET THEORY FROM THEIR BEGINNINGS (Perspectives in Mathematical Logic)." Bulletin of the London Mathematical Society 29, no. 1 (January 1997): 111–13. http://dx.doi.org/10.1112/s0024609396221678.

Full text

APA, Harvard, Vancouver, ISO, and other styles

28

Todorcevic, Stevo. "Combinatorial Dichotomies in Set Theory." Bulletin of Symbolic Logic 17, no. 1 (March 2011): 1–72. http://dx.doi.org/10.2178/bsl/1294186662.

Full text

Abstract:

AbstractWe give an overview of a research line concentrated on finding to which extent compactness fails at the level of first uncountable cardinal and to which extent it could be recovered on some other perhaps not so large cardinal. While this is of great interest to set theorists, one of the main motivations behind this line of research is in its applicability to other areas of mathematics. We give some details about this and we expose some possible directions for further research.

APA, Harvard, Vancouver, ISO, and other styles

29

Mignone, R. "The relative consistency of a \large cardinal" property for ω 1." Rocky Mountain Journal of Mathematics 20, no. 1 (March 1990): 209–13. http://dx.doi.org/10.1216/rmjm/1181073173.

Full text

APA, Harvard, Vancouver, ISO, and other styles

30

Apter, Arthur W., and Grigor Sargsyan. "Can a large cardinal be forced from a condition implying its negation?" Proceedings of the American Mathematical Society 133, no. 10 (May 4, 2005): 3103–8. http://dx.doi.org/10.1090/s0002-9939-05-07840-8.

Full text

APA, Harvard, Vancouver, ISO, and other styles

31

Sivakumar, N. "A note on the Gaussian cardinal-interpolation operator." Proceedings of the Edinburgh Mathematical Society 40, no. 1 (February 1997): 137–49. http://dx.doi.org/10.1017/s0013091500023506.

Full text

Abstract:

Suppose λ is a positive number, and let , x∈Rd, denote the d-dimensional Gaussian. Basic theory of cardinal interpolation asserts the existence of a unique function , x∈Rd, satisfying the interpolatory conditions , k∈Zd, and decaying exponentially for large argument. In particular, the Gaussian cardinal-interpolation operator, given by , x∈Rd, , is a well-defīned linear map from ℓ2(Zd) into L2(Rd). It is shown here that its associated operator-norm is , implying, in particular, that is contractive. Some sidelights are also presented.

APA, Harvard, Vancouver, ISO, and other styles

32

Litkowski, Ellen C., Robert J. Duncan, Jessica A. R. Logan, and David J. Purpura. "Alignment Between Children’s Numeracy Performance, the Kindergarten Common Core State Standards for Mathematics, and State-Level Early Learning Standards." AERA Open 6, no. 4 (July 2020): 233285842096854. http://dx.doi.org/10.1177/2332858420968546.

Full text

Abstract:

The current study examined preschoolers’ (N = 801) age-related performance on one measure of verbal counting and two measures of cardinality (“how many” and “give n”) aligned with the kindergarten Common Core State Standards for Mathematics (CCSSM) and included in the majority of states’ early learning guidelines for mathematics. Children were grouped into five age categories (3, 3.5, 4, 4.5, 5, 5.5), and within-age-group average rates of correct responses for each item within these three measures were calculated. Results demonstrated that the majority of children were already successfully meeting the CCSSM standards for both cardinal number knowledge tasks (86.5% and 53.3%, respectively) prior to kindergarten entry but that only 18.9% of the children were meeting the standard for verbal counting. Findings indicate potential misalignment between children’s existing capabilities and the CCSSM standards for cardinality and underscore the need to conduct large, nationally representative studies measuring children’s abilities on items that more closely assess the specific mathematics skills included in the CCSSM and early learning guidelines.

APA, Harvard, Vancouver, ISO, and other styles

33

Harmse, J. E. "Antipodal coincidence sets and stronger forms of connectedness." Bulletin of the Australian Mathematical Society 31, no. 2 (April 1985): 271–84. http://dx.doi.org/10.1017/s0004972700004743.

Full text

Abstract:

A new notion of α-connectedness (α-path connectedness) in general topological spaces is introduced and it is proved that for a real-valued function defined on a space with this property, the cardinality of the antipodal coincidence set is at least as large as the cardinal number α. In particular, in linear topological spaces, the antipodal coincidence set of a real-valued function has cardinality at least that of the continuum. This could be regarded as a treatment of some Borsuk-Ulam type results in the setting of general topology.

APA, Harvard, Vancouver, ISO, and other styles

34

Matet, Pierre. "Yoshihiro Abe. Weakly normal filters and the closed unbounded filter on Pkλ. Proceedings of the American Mathematical Society, vol. 104 (1998), pp. 1226–1234. - Yoshihiro Abe. Weakly normal filters and large cardinals. Tsukuba journal of mathematics, vol. 16 (1992), pp. 487–494. - Yoshihiro Abe. Weakly normal ideals on Pkλ and the singular cardinal hypothesis. Fundamenta mathematicae, vol. 143 (1993), pp. 97–106. - Yoshihiro Abe. Saturation of fundamental ideals on Pkλ. Journal of the Mathematical Society of Japan, vol. 48 (1996), pp. 511–524. - Yoshihiro Abe. Strongly normal ideals on Pkλ and the Sup-function. opology and its applications, vol. 74 (1996), pp. 97–107. - Yoshihiro Abe. Combinatorics for small ideals on Pkλ. Mathematical logic quarterly, vol. 43 (1997), pp. 541–549. - Yoshihiro Abe and Masahiro Shioya. Regularity of ultrafilters and fixed points of elementary embeddings. Tsukuba journal of mathematics, vol. 22 (1998), pp. 31–37." Bulletin of Symbolic Logic 8, no. 2 (June 2002): 309–11. http://dx.doi.org/10.2178/bsl/1182353882.

Full text

APA, Harvard, Vancouver, ISO, and other styles

35

ADÁMEK, JIŘÍ, and VĚRA TRNKOVÁ. "Initial algebras and terminal coalgebras in many-sorted sets." Mathematical Structures in Computer Science 21, no. 2 (March 25, 2011): 481–509. http://dx.doi.org/10.1017/s0960129510000502.

Full text

Abstract:

We prove that the iterative construction of initial algebras converges for endofunctors F of many-sorted sets whenever F has an initial algebra. In the case of one-sorted sets, the convergence takes n steps where n is either an infinite regular cardinal or is at most 3. Dually, the existence of a many-sorted terminal coalgebra implies that the iterative construction of a terminal coalgebra converges. Moreover, every endofunctor with a fixed-point pair larger than the number of sorts is proved to have a terminal coalgebra. As demonstrated by James Worell, the number of steps here need not be a cardinal even in the case of a single sort: it is ω + ω for the finite power-set functor. The above results do not hold for related categories, such as graphs: we present non-constructive initial algebras and terminal coalgebras.

APA, Harvard, Vancouver, ISO, and other styles

36

Göbel, Rüdiger. "Abelian groups with small cotorsion images." Journal of the Australian Mathematical Society. Series A. Pure Mathematics and Statistics 50, no. 2 (April 1991): 243–47. http://dx.doi.org/10.1017/s1446788700032729.

Full text

Abstract:

AbstractEpimorphic images of compact (algebraically compact) abelian groups are called cotorsion groups after Harrison. In a recent paper, Ph. Schultz raised the question whether “cotorsion” is a property which can be recognized by its small cotorsion epimorphic images: If G is a torsion-free group such that every torsion-free reduced homomorphic image of cardinality is cotorsion, is G necessarily cortorsion? In this note we will give some counterexamples to this problem. In fact, there is no cardinal k which is large enough to test cotorsion.

APA, Harvard, Vancouver, ISO, and other styles

37

PARKER, MATTHEW W. "SET SIZE AND THE PART–WHOLE PRINCIPLE." Review of Symbolic Logic 6, no. 4 (September 20, 2013): 589–612. http://dx.doi.org/10.1017/s1755020313000221.

Full text

Abstract:

AbstractGödel argued that Cantor’s notion of cardinal number was uniquely correct. More recent work has defended alternative “Euclidean”' theories of set size, in which Cantor’s Principle (two sets have the same size if and only if there is a one-to-one correspondence between them) is abandoned in favor of the Part–Whole Principle (if A is a proper subset of B then A is smaller than B). Here we see from simple examples, not that Euclidean theories of set size are wrong, nor merely that they are counterintuitive, but that they must be either very weak or in large part arbitrary and misleading. This limits their epistemic usefulness.

APA, Harvard, Vancouver, ISO, and other styles

38

Levy, Azriel. "Akihiro Kanamori. The higher infinite. Large cardinals in set theory from their beginnings. Perspectives in mathematical logic. Springer-Verlag, Berlin, Heidelberg, New York, etc., 1994, xxiv + 536 pp." Journal of Symbolic Logic 61, no. 1 (March 1996): 334–36. http://dx.doi.org/10.2307/2275615.

Full text

APA, Harvard, Vancouver, ISO, and other styles

39

Greene, Ernest. "A Test of the Gravity Lens Theory." Perception 27, no. 10 (October 1998): 1221–28. http://dx.doi.org/10.1068/p271221.

Full text

Abstract:

Naito and Cole [1994, in Contributions to Mathematical Psychology: Psychometrics and Methodology Eds G H Fischer and D Laming (New York: Springer)] provide a configuration which they describe as the Gravity Lens illusion. In this configuration, four small dots are presented in proximity to four large disks, and one is asked to compare the slope of an imaginary line which connects one pair of dots with the slope of a line which connects the other pair. In fact the slopes are the same, ie their axes are parallel, but because of the positioning of the large disks they appear to be at different orientations. Naito and Cole propose that the perceptual bias is analogous to the effects of gravity on the metrics of physical space, such that mental projections in the vicinity of a disk (or an open circle) are distorted just as the path of light is bent as it passes a massive body such as a star. Here we provide a simple test of this concept by having subjects judge alignments of dots which lie near tangents to a circle. Subjects were asked to project straight lines through pairs of stimulus dots, selecting and marking points in open space which were collinear with each pair. As would be predicted by the Gravity Lens theory, the locations selected by subjects were displaced from straight lines. However, the error magnitudes were substantially larger for judgments of dot pairs which had an oblique alignment, as compared with dot pairs which were aligned with a cardinal axis. This differential of effect as a function of stimulus orientation is not predicted by the gravity concept.

APA, Harvard, Vancouver, ISO, and other styles

40

Booth, David. "Hereditarily finite Finsler sets." Journal of Symbolic Logic 55, no. 2 (June 1990): 700–706. http://dx.doi.org/10.2307/2274659.

Full text

Abstract:

Hereditarily finite sets are sets which are finite, whose members are finite, the members of whose members are finite, and so on. In ZF there are but countably many such sets; they constitute Vω. Were ZF to lose its axiom of regularity, however, one could not guarantee that the number of hereditarily finite sets would remain countable.In Mostowski set theory, in which atomic sets are permissible, each atom, in isolation, would form a hereditarily finite collection. The number of hereditarily finite sets could be anything one should choose.Even in a world that did not permit the free adjunction of arbitrary, meaningless atoms, the number of hereditarily finite sets could remain large. In Finsler set theory, it is shown as Theorem 22, below, that there are uncountably many hereditarily finite sets.The reader who is interested in this paradoxical sounding fact can turn directly to §4 after grasping these introductory concepts. §3 is an exhaustive list of the smallest Finsler sets; it is hoped that this list will prove useful in checking future attempts to classify the finite Finsler sets.Finsler set theory is not a firmly axiomatized theory. It is, at its present stage, a family of theories undergoing evolution. It permits the usual mathematical operations with sets. One can employ ordinal numbers, cardinal numbers, and the usual methods of Cantorian set theory freely. But there is a somewhat different interpretation attached to the concept “set” than one is used to in Zermelo-Fraenkel set theory, ZF.

APA, Harvard, Vancouver, ISO, and other styles

41

Burgess, John P. "George Boolos. The iterative conception of set. The journal of philosophy, vol. 68 (1971), pp. 215–231. - Dana Scott. Axiomatizing set theory. Axiomatic set theory, edited by Thomas J. Jech, Proceedings of symposia in pure mathematics, vol. 13 part 2, American Mathematical Society, Providence1974, pp. 207–214. - W. N. Reinhardt. Remarks on reflection principles, large cardinals, and elementary embeddings. Axiomatic set theory, edited by Thomas J. Jech, Proceedings of symposia in pure mathematics, vol. 13 part 2, American Mathematical Society, Providence1974, pp. 189–205. - W. N. Reinhardt. Set existence principles of Shoenfield, Ackermann, and Powell. Fundament a mathematicae, vol. 84 (1974), pp. 5–34. - Hao Wang. Large sets. Logic, foundations of mathematics, and computahility theory. Part one of the proceedings of the Fifth International Congress of Logic, Methodology and Philosophy of Science, London, Ontario, Canada–1975, edited by Robert E. Butts and Jaakko Hintikka, The University of Western Ontario series in philosophy of science, vol. 9, D. Reidel Publishing Company, Dordrecht and Boston1977, pp. 309–333. - Charles Parsons. What is the iterative conception of set?Logic, foundations of mathematics, and computahility theory. Part one of the proceedings of the Fifth International Congress of Logic, Methodology and Philosophy of Science, London, Ontario, Canada–1975, edited by Robert E. Butts and Jaakko Hintikka, The University of Western Ontario series in philosophy of science, vol. 9, D. Reidel Publishing Company, Dordrecht and Boston1977, pp. 335–367." Journal of Symbolic Logic 50, no. 2 (June 1985): 544–47. http://dx.doi.org/10.2307/2274241.

Full text

APA, Harvard, Vancouver, ISO, and other styles

42

Fuson, Karen C., and Adrienne M. Fuson. "Instruction Supporting Children's Counting on for Addition and Counting up for Subtraction." Journal for Research in Mathematics Education 23, no. 1 (January 1992): 72–78. http://dx.doi.org/10.5951/jresematheduc.23.1.0072.

Full text

Abstract:

Children in the United States ordinarily invent a series of increasingly abbreviated and abstract strategies to solve addition and subtraction problems during their first 4 years in school (Carpenter & Moser, 1984; Fuson, 1988, in press–a, in press–b; Steffe & Cobb, 1988). Several studies have shown that instruction can help children learn specific strategies in this developmental sequence. Fuson (1986), Fuson and Secada (1986), and Fuson and Willis (1988) demonstrated that by the end of first grade children of all achievement levels could add and subtract single-digit sums and differences (sums to 18) by sequence counting on and sequence counting up. Sequence counting on and counting up are abbreviated counting strategies in which the number words present the addends and the sum. In both strategies the counting begins by saying the number word of the first addend. For example, to count on to add 8 + 6, a child would say, “8 (pause), 9, 10, 11, 12, 13, 14.” The same sequence of number words is used to find 14–8 by counting up, but the answer is the number of words said after the first addend word rather than the last word in the sequence. When the second addend is larger than 2 or 3, some method of keeping track of the words said for the second addend is required. In the studies above this method was one-handed finger panems that showed quantities l through 9 (the thumb is 5) so that children could hold their pencil in their writing hand all of the time. The counting-on and counting-up instruction related the counting words to objects showing the addends and the sum, thus focusing on conceptual prerequisites for these abbreviated counting procedures and enabling children to relate counting and cardinal meanings of number words (Secada, Fuson, & Hall, 1983). The countingup instruction provided interpretations of subtraction and the“–” symbol as adding on, as well as the usual take-away interpretation that leads children to count down for subtraction.

APA, Harvard, Vancouver, ISO, and other styles

43

Prisco, Carlos Augusto Di. "Robert M. Solovay, William N. Reinhardt, and Akihiro Kanamori. Strong axioms of infinity and elementary embeddings. Annals of mathematical logic, vol. 13 (1978), pp. 73–116. - Menachem Magidor. HOW large is the first strongly compact cardinal? or A study on identity crises. Annals of mathematical logic, vol. 10 (1976), pp. 33–57." Journal of Symbolic Logic 51, no. 4 (December 1986): 1066–68. http://dx.doi.org/10.2307/2273920.

Full text

APA, Harvard, Vancouver, ISO, and other styles

44

ARGYROS, SPIROS A., JESÚS F. CASTILLO, ANTONIO S. GRANERO, MAR JIMÉNEZ, and JOSÉ P. MORENO. "COMPLEMENTATION AND EMBEDDINGS OF c0(I) IN BANACH SPACES." Proceedings of the London Mathematical Society 85, no. 3 (October 14, 2002): 742–68. http://dx.doi.org/10.1112/s0024611502013618.

Full text

Abstract:

We investigate in this paper the complementation of copies of $c_0(I)$ in some classes of Banach spaces (in the class of weakly compactly generated (WCG) Banach spaces, in the larger class $\mathcal{V}$ of Banach spaces which are subspaces of some $C(K)$ space with $K$ a Valdivia compact, and in the Banach spaces $C([1, \alpha ])$, where $\alpha$ is an ordinal) and the embedding of $c_0(I)$ in the elements of the class $\mathcal{C}$ of complemented subspaces of $C(K)$ spaces. Two of our results are as follows:(i) in a Banach space $X \in \mathcal{V}$ every copy of $c_0(I)$ with $\# I < \aleph _{\omega}$ is complemented;(ii) if $\alpha _0 = \aleph _0$, $\alpha _{n+1} = 2^{\alpha _n}$, $n \geq 0$, and $\alpha = \sup \{\alpha _n : n \geq 0\}$ there exists a WCG Banach space with an uncomplemented copy of $c_0(\alpha )$.So, under the generalized continuum hypothesis (GCH), $\aleph _{\omega}$ is the greatest cardinal $\tau$ such that every copy of $c_0(I)$ with $\# I < \tau$ is complemented in the class $\mathcal{V}$. If $T : c_0(I) \to C([1,\alpha ])$ is an isomorphism into its image, we prove that:(i) $c_0(I)$ is complemented, whenever $\| T \| ,\| T^{-1} \| < (3/2)^{\frac 12}$;(ii) there is a finite partition $\{I_1, \dots , I_k\}$ of $I$ such that each copy $T(c_0(I_k))$ is complemented.Concerning the class $\mathcal{C}$, we prove that an already known property of $C(K)$ spaces is still true for this class, namely, if $X \in \mathcal{C}$, the following are equivalent:(i) there is a weakly compact subset $W \subset X$ with ${\rm Dens}(W) = \tau$;(ii) $c_0(\tau )$ is isomorphically embedded into $X$.This yields a new characterization of a class of injective Banach spaces.2000 Mathematical Subject Classification: 46B20, 46B26.

APA, Harvard, Vancouver, ISO, and other styles

45

Todorcevic, Stevo. "Some compactifications of the integers." Mathematical Proceedings of the Cambridge Philosophical Society 112, no. 2 (September 1992): 247–54. http://dx.doi.org/10.1017/s0305004100070936.

Full text

Abstract:

Suppose that K = {K0, K1} is a partition of some finite combinatorial power [S]r of a set S. Let X(K) be the compact subspace of the Tychonoff cube [0, 1]S consisting of those functions ƒ whose supports Sƒ = {ƒ > 0} are 0-homogeneous i.e., [Sƒ]r ⊆ K0. It can be said that almost every example of a space witnessing the distinction between various chain conditions is a variation of X(K) (see [2]). This comes from the fact that the subject is closely related to the subject of Partition Calculus. (See [11] for an explanation of this point.) To have examples which are even more topologically interesting one usually tries to make them small, i.e., as closely related to the unit interval as possible. The operation X(K) has a considerable drawback in that respect. For example, if we want X(K) to be ccc this becomes equivalent to the fact that the poset of all finite 0-homogeneous sets is ccc which amounts to the fact that K1 must be very small. Hence K0 is big in the sense that there exist large 0-homogeneous sets. This usually results in X(K) having large size and having points of large character. One attempt to solve this problem was given by van Douwen in [5] by going to the subspace Xm(K) of X(K) consisting of those ƒ for which Sƒ are maximal 0-homogeneous subsets of S. Unfortunately, while Xm(K) usually does have small character it is almost never compact. This might have been the reason for his question ([2], p. 207) whether the Continuum Hypothesis implies that the class of all first countable compacta distinguishes the standard chain conditions that lie between ‘ccc’ and ‘separable’. In this paper we solve this problem completely. Moreover, we shall not go beyond the usual axioms of set theory in constructing the examples. The sequence of examples will start with a compact space of small character whose chain condition is not productive and it will end with a compact space of size c and small character which is ccc in a strong sense but which fails to have calibre θ for some regular uncountable cardinal θ, i.e., it fails to have the property of Shanin. Note that one cannot go further and show that, for example, compact spaces of small character distinguish between ‘the property of Shanin’ and ‘separable’. This follows from an old result of Efimov [3] that, under CH, first-countable spaces of calibre ℵ1 are separable. The combinatorics behind our examples have been developed in a series of papers that deal with the subject of forcing axioms in general and Martin's axiom in particular ([10, 11, 12, 13, 14]). Martin's axiom was originally invented in connection with the Souslin Problem, i.e., to show that certain compact ccc spaces must be separable (see [8]). A result of the aforementioned study of MA showed that this axiom is nothing more than the statement that all compact ccc spaces of π-weight < c must be separable (see [12]). An analysis of the fact that MA implies SH, due to Hajnal and Juhasz[6] (see also [4], §43 for a definite result in that direction due to Shapirovskii), revealed that MA implies that every compact ccc space X with the property χ(X)+ < c must be separable. This result explains why the compact ccc non-separable spaces X that we construct in this paper have the property that χ(x, X) < c for all x m X rather than the stronger property χ(X) < c or even χ(X) = ℵ0. Note that our examples show, answering a question from [6], that the Hajnal–Juhasz result is sharp in the sense that the assumption χ(X)+ < c cannot be weakened to χ(X) < c. The first example to show this was constructed by Bell [1] using a consequence of MA rather than just ZFC for its construction. Another feature of our examples is that they all are remainders of certain compactifications of the integers. This is of independent interest in certain constructions of weak P-points in compact F-spaces. An explanation of this can be found in [1] and [9] where the first examples of ccc non-separable remainders were constructed and used.

APA, Harvard, Vancouver, ISO, and other styles

46

Drápal, Aleš. "Richard Laver. The left distributive law and the freeness of an algebra of elementary embeddings. Advances in mathematics, vol. 91 (1992), pp. 209–231. - Richard Laver. A division algorithm for the free left distributive algebra. Logic Colloquium '90, ASL summer meeting in Helsinki, edited by J. Oikkonen and J. Väänänen, Lecture notes in logic, no. 2, Springer-Verlag, Berlin, Heidelberg, New York, etc., 1993, pp. 155–162. - Richard Laver. On the algebra of elementary embeddings of a rank into itself. Advances in mathematics, vol. 110 (1995), pp. 334–346. - Richard Laver. Braid group actions on left distributive structures, and well orderings in the braid groups. Journal of pure and applied algebra, vol. 108 (1996), pp. 81–98. - Patrick Dehornoy. An alternative proof of Laver's results on the algebra generated by an elementary embedding. Set theory of the continuum, edited by H. Judah, W. Just, and H. Woodin, Mathematics Sciences Research Institute publications, vol. 26, Springer-Verlag, New York, Berlin, Heidelberg, etc., 1992, pp. 27–33. - Patrick Dehornoy. Braid groups and left distributive operations. Transactions of the American Mathematical Society, vol. 345 (1994), pp. 115–150. - Patrick Dehornoy. A normal form for the free left distributive law. International journal of algebra and computation, vol. 4 (1994), pp. 499–528. - Patrick Dehornoy. From large cardinals to braids via distributive algebra. Journal of knot theory and its ramifications, vol. 4 (1995), pp. 33–79. - J. R. Steel. The well-foundedness of the Mitchell order. The journal of symbolic logic, vol. 58 (1993), pp. 931–940. - Randall Dougherty. Critical points in an algebra of elementary embeddings. Annals of pure and applied logic, vol. 65 (1993), pp. 211–241. - Randall Dougherty. Critical points in an algebra of elementary embeddings, II. Logic: from foundations to applications, European logic colloquium, edited by Wilfrid Hodges, Martin Hyland, Charles Steinhorn, and John Truss, Clarendon Press, Oxford University Press, Oxford, New York, etc., 1996, pp. 103–136. - Randall Dougherty and Thomas Jech. Finite left-distributive algebras and embedding algebras. Advances in mathematics, vol. 130 (1997), pp. 201–241." Bulletin of Symbolic Logic 8, no. 4 (December 2002): 555–60. http://dx.doi.org/10.2178/bsl/1182353941.

Full text

APA, Harvard, Vancouver, ISO, and other styles

47

ASPERÓ, DAVID, and ASAF KARAGILA. "DEPENDENT CHOICE, PROPERNESS, AND GENERIC ABSOLUTENESS." Review of Symbolic Logic, July 2, 2020, 1–25. http://dx.doi.org/10.1017/s1755020320000143.

Full text

Abstract:

Abstract We show that Dependent Choice is a sufficient choice principle for developing the basic theory of proper forcing, and for deriving generic absoluteness for the Chang model in the presence of large cardinals, even with respect to $\mathsf {DC}$ -preserving symmetric submodels of forcing extensions. Hence, $\mathsf {ZF}+\mathsf {DC}$ not only provides the right framework for developing classical analysis, but is also the right base theory over which to safeguard truth in analysis from the independence phenomenon in the presence of large cardinals. We also investigate some basic consequences of the Proper Forcing Axiom in $\mathsf {ZF}$ , and formulate a natural question about the generic absoluteness of the Proper Forcing Axiom in $\mathsf {ZF}+\mathsf {DC}$ and $\mathsf {ZFC}$ . Our results confirm $\mathsf {ZF} + \mathsf {DC}$ as a natural foundation for a significant portion of “classical mathematics” and provide support to the idea of this theory being also a natural foundation for a large part of set theory.

APA, Harvard, Vancouver, ISO, and other styles

48

Kania, Tomasz, and Jarosław Swaczyna. "Large cardinals and continuity of coordinate functionals of filter bases in Banach spaces." Bulletin of the London Mathematical Society, October 13, 2020. http://dx.doi.org/10.1112/blms.12415.

Full text

APA, Harvard, Vancouver, ISO, and other styles

49

Ndiaye Berankova, Jana. "The Immanence of Truths and the Absolutely Infinite in Spinoza, Cantor, and Badiou." Filozofski vestnik 41, no. 2 (December 31, 2020). http://dx.doi.org/10.3986/fv.41.2.13.

Full text

Abstract:

The following article compares the notion of the absolute in the work of Georg Cantor and in Alain Badiou’s third volume of Being and Event: The Immanence of Truths and proposes an interpretation of mathematical concepts used in the book. By describing the absolute as a universe or a place in line with the mathematical theory of large cardinals, Badiou avoided some of the paradoxes related to Cantor’s notion of the “absolutely infinite” or the set of all that is thinkable in mathematics W: namely the idea that W would be a potential infinity. The article provides an elucidation of the putative criticism of the statement “mathematics is ontology” which Badiou presented at the conference Thinking the Infinite in Prague. It emphasizes the role that philosophical decision plays in the construction of Badiou’s system of mathematical ontology and portrays the relationship between philosophy and mathematics on the basis of an inductive not deductive reasoning.

APA, Harvard, Vancouver, ISO, and other styles

50

Gła̧b, Szymon, and Jacek Marchwicki. "Cardinal Functions of Purely Atomic Measures." Results in Mathematics 75, no. 4 (August 29, 2020). http://dx.doi.org/10.1007/s00025-020-01260-x.

Full text

Abstract:

AbstractLet $$\mu $$ μ be a purely atomic finite measure. Without loss of generality we may assume that $$\mu $$ μ is defined on $${\mathbb {N}}$$ N , and the atoms with smaller indexes have larger masses, that is $$\mu (\{k\})\ge \mu (\{k+1\})$$ μ ( { k } ) ≥ μ ( { k + 1 } ) for $$k\in {\mathbb {N}}$$ k ∈ N . By $$f_\mu :[0,\infty )\rightarrow \{0,1,2,\dots ,\omega ,{\mathfrak {c}}\}$$ f μ : [ 0 , ∞ ) → { 0 , 1 , 2 , ⋯ , ω , c } we denote its cardinal function $$f_{\mu }(t)=\vert \{A\subset {\mathbb {N}}:\mu (A)=t\}\vert $$ f μ ( t ) = | { A ⊂ N : μ ( A ) = t } | . We study the problem for which sets $$R\subset \{0,1,2,\dots ,\omega ,{\mathfrak {c}}\}$$ R ⊂ { 0 , 1 , 2 , ⋯ , ω , c } there is a measure $$\mu $$ μ such that $$R=\text {rng}(f_\mu )$$ R = rng ( f μ ) . We are also interested in the set-theoretic and topological properties of the set of $$\mu $$ μ -values which are obtained uniquely.

APA, Harvard, Vancouver, ISO, and other styles

We offer discounts on all premium plans for authors whose works are included in thematic literature selections. Contact us to get a unique promo code!