Every analytic set is Ramsey

1970 ◽  
Vol 35 (1) ◽  
pp. 60-64 ◽  
Author(s):  
Jack Silver
Keyword(s):  
Measurable Cardinal ◽  
Infinite Subset ◽  
The Other ◽  
Natural Numbers ◽  
Ramsey Property ◽  
Analytic Subset ◽  
Borel Set ◽  
Analytic Set ◽  
Principal Theorem ◽  
Countably Infinite

If X is a set, [Χ]ω will denote the set of countably infinite subsets of X. ω is the set of natural numbers. If S is a subset of [ω]ω, we shall say that S is Ramsey if there is some infinite subset X of ω such that either [Χ]ω ⊆ S or [Χ]ω ∩ S = 0. Dana Scott (unpublished) has asked which sets, in terms of logical complexity, are Ramsey.The principal theorem of this paper is: Every Σ11 (i.e., analytic) subset of [ω]ω is Ramsey (for the Σ, Π notations, see Addison [1]). This improves a result of Galvin-Prikry [2] to the effect that every Borel set is Ramsey. Our theorem is essentially optimal because, if the axiom of constructibility is true, then Gödel's Σ21 Π21 well-ordering of the set of reals [3], having the convenient property that the set of ω-sequences of reals enumerating initial segments is also Σ21 ∩ Π21, rather directly gives a Σ21 ∩ Π21 set which is not Ramsey. On the other hand, from the assumption that there is a measurable cardinal we shall derive the conclusion that every Σ21 (i.e., PCA) is Ramsey. Also, we shall explore the connection between Martin's axiom and the Ramsey property.

Strong and weak duality principles for fractal dimension in Euclidean space

1995 ◽  
Vol 118 (3) ◽  
pp. 393-410 ◽  
Author(s):  
Colleen D. Cutler
Keyword(s):  
Strong Duality ◽  
The Other ◽  
Duality Principle ◽  
Weak Duality ◽  
Pointwise Dimension ◽  
Analytic Subset ◽  
Analytic Set ◽  
Borel Measures ◽  
Dual Representations ◽  
Packing Dimensions

AbstractTricot [27] provided apparently dual representations of the Hausdorff and packing dimensions of any analytic subset of Euclidean d-space in terms of, respectively, the lower and upper pointwise dimension maps of the finite Borel measures on ℝd. In this paper we show that Tricot's two representations, while similar in appearance, are in fact not duals of each other, but rather the duals of two other ‘missing’ representations. The key to obtaining these missing representations lies in extended Frostman and antiFrostman lemmas, both of which we develop in this paper. This leads to the formulation of two distinct characterizations of dim (A) and Dim (A), one which we call the weak duality principle and the other the strong duality principle. In particular, the strong duality principle is concerned with the existence, for each analytic set A, of measures on A that are (almost) of the same exact dimension (Hausdorff or packing) as A. The connection with Rényi (or information) dimension and a variational principle of Cutler and Olsen[12] is also established.

On the existence of an analytic set meeting each compact set in a Borel set

1978 ◽  
Vol 84 (1) ◽  
pp. 5-10 ◽  
Author(s):  
A. R. D. Mathias ◽  
A. J. Ostaszewski ◽  
M. Talagrand
Keyword(s):  
Set Theory ◽  
Polish Space ◽  
Continuum Hypothesis ◽  
Compact Set ◽  
Analytic Subset ◽  
Borel Set ◽  
Analytic Set ◽  
Counter Example ◽  
The Axiom Of Choice ◽  
The Continuum

C. A. Rogers and J. E. Jayne have asked whether, given a Polish space and an analytic subset A of which is not a Borel set, there is always a compact subset K of such that, A ∩ K is not Borel. In this paper we give both a proof, using Martin's axiom and the negation of the continuum hypothesis, of and a counter-example, using the axiom of constructibility, to the conjecture of Rogers and Jayne, which set theory with the axiom of choice is thus powerless to decide.

Mathematical Knowledge and the Origin of Phenomenology

2021 ◽  
Vol 21 ◽  
pp. 273-294
Author(s):  
Gabriele Baratelli ◽  
Keyword(s):  
Mathematical Knowledge ◽  
Cardinal Number ◽  
The Other ◽  
Mathematical Science ◽  
Natural Numbers ◽  
Symbolic Thinking ◽  
The One

The paper is divided into two parts. In the first one, I set forth a hypothesis to explain the failure of Husserl’s project presented in the Philosophie der Arithmetik based on the principle that the entire mathematical science is grounded in the concept of cardinal number. It is argued that Husserl’s analysis of the nature of the symbols used in the decadal system forces the rejection of this principle. In the second part, I take into account Husserl’s explanation of why, albeit independent of natural numbers, the system is nonetheless correct. It is shown that its justification involves, on the one hand, a new conception of symbols and symbolic thinking, and on the other, the recognition of the question of “the formal” and formalization as pivotal to understand “the mathematical” overall.

A Characterization of the Topological Dimension

1982 ◽  
Vol 25 (4) ◽  
pp. 487-490
Author(s):  
Gerd Rodé
Keyword(s):  
Hausdorff Space ◽  
The Other ◽  
Natural Numbers ◽  
Topological Dimension ◽  
Other Hand ◽  
The One

AbstractThis paper gives a new characterization of the dimension of a normal Hausdorff space, which joins together the Eilenberg-Otto characterization and the characterization by finite coverings. The link is furnished by the notion of a system of faces of a certain type (N1,..., NK), where N1,..., NK, K are natural numbers. It is shown that a space X contains a system of faces of type (N1,..., NK) if and only if dim(X) ≥ N1 + … + NK. The two limit cases of the theorem, namely Nk = 1 for 1 ≤ k ≤ K on the one hand, and K = 1 on the other hand, give the two known results mentioned above.

scholarly journals Some dual aspects of the Poisson kernel

1990 ◽  
Vol 33 (2) ◽  
pp. 207-232 ◽  
Author(s):  
F. F. Bonsall
Keyword(s):  
Continuous Function ◽  
Borel Measure ◽  
Poisson Kernel ◽  
Infinite Subset ◽  
Open Unit ◽  
Positive Borel Measure ◽  
Positive Continuous Function ◽  
Open Unit Disc ◽  
Counting Measure ◽  
Countably Infinite

The Poisson kernel is defined for z in the open unit disc D and ζ in the unit circle ∂D. As usually employed, it is integrated with respect to the second variable and a measure on ∂D to yield a harmonic function on D. Here, we fix a σ-finite positive Borel measure m on D and integrate the Poisson kernel with respect to the first variable against a function φ in L1(m) to obtain a function Tmφ on ∂D. We ask for what measures m the range of Tm is L1(∂D), for what m the kernel of Tm is non-zero, and for what m every positive continuous function on ∂D is of the form Tmφ with φ non-negative. When m is the counting measure of a countably infinite subset {ak:k∈ℕ} of D, the function (Tmφ)(ζ) is of the form with . The main results generalize results previously obtained for sums of this form. A related mapping from Lp(m) into Lp(∂D) with 1 <p<∞ is briefly considered.

scholarly journals On the endomorphism near-ring of a free group

1980 ◽  
Vol 23 (1) ◽  
pp. 103-121 ◽  
Author(s):  
R. Warwick Zeamer
Keyword(s):  
Wreath Product ◽  
Free Group ◽  
The Other ◽  
Finite Support ◽  
Finite Product ◽  
Prime Factorization ◽  
Infinite Rank ◽  
One To One ◽  
Almost All ◽  
Countably Infinite

Suppose F is an additively written free group of countably infinite rank with basis T and let E = End(F). If we add endomorphisms pointwise on T and multiply them by map composition, E becomes a near-ring. In her paper “On Varieties of Groups and their Associated Near Rings” Hanna Neumann studied the sub-near-ring of E consisting of the endomorphisms of F of finite support, that is, those endomorphisms taking almost all of the elements of T to zero. She called this near-ring Φω. Now it happens that the ideals of Φω are in one to one correspondence with varieties of groups. Moreover this correspondence is a monoid isomorphism where the ideals of φω are multiplied pointwise. The aim of Neumann's paper was to use this isomorphism to show that any variety can be written uniquely as a finite product of primes, and it was in this near-ring theoretic context that this problem was first raised. She succeeded in showing that the left cancellation law holds for varieties (namely, U(V) = U′(V) implies U = U′) and that any variety can be written as a finite product of primes. The other cancellation law proved intractable. Later, unique prime factorization of varieties was proved by Neumann, Neumann and Neumann, in (7). A concise proof using these same wreath product techniques was also given in H. Neumann's book (6). These proofs, however, bear no relation to the original near-ring theoretic statement of the problem.

Maximal subsemigroups of some semigroups of order-preserving mappings on a countably infinite set

2017 ◽  
Vol 29 (4) ◽  
Author(s):  
Tiwadee Musunthia ◽  
Jörg Koppitz
Keyword(s):  
Natural Numbers ◽  
Maximal Subsemigroups ◽  
Countably Infinite ◽  
Infinite Set

AbstractIn this paper, we study the maximal subsemigroups of several semigroups of order-preserving transformations on the natural numbers and the integers, respectively. We determine all maximal subsemigroups of the monoid of all order-preserving injections on the set of natural numbers as well as on the set of integers. Further, we give all maximal subsemigroups of the monoid of all bijections on the integers. For the monoid of all order-preserving transformations on the natural numbers, we classify also all its maximal subsemigroups, containing a particular set of transformations.

scholarly journals On closed unbounded sets consisting of former regulars

1999 ◽  
Vol 64 (1) ◽  
pp. 1-12 ◽  
Author(s):  
Moti Gitik
Keyword(s):  
Measurable Cardinal ◽  
The Other ◽  
Other Hand

AbstractA method of iteration of Prikry type forcing notions as well as a forcing for adding clubs is presented. It is applied to construct a model with a measurable cardinal containing a club of former regulars, starting with ο(κ) = κ + 1. On the other hand, it is shown that the strength of above is at leastο(κ) = κ.

Does V equal L?

1993 ◽  
Vol 58 (1) ◽  
pp. 15-41 ◽  
Author(s):  
Penelope Maddy
Keyword(s):  
Detailed Analysis ◽  
Measurable Cardinal ◽  
Simple Approach ◽  
The Other ◽  
Near Unanimity ◽  
Rational Evaluation

Does V = L? Is the Axiom of Constructibility true? Most people with an opinion would answer no. But on what grounds? Despite the near unanimity with which V = L is declared false, the literature reveals no clear consensus on what counts as evidence against the hypothesis and no detailed analysis of why the facts of the sort cited constitute evidence one way or another. Unable to produce a well-developed argument one way or the other, some observers despair, retreating to unattractive fall-back positions, e.g., that the decision on whether or not V = L is a matter of personal aesthetics. I would prefer to avoid such conclusions, if possible. If we are to believe that L is not V, as so many would urge, then there ought to be good reasons for this belief, reasons that can be stated clearly and subjected to rational evaluation. Though no complete argument has been presented, the literature does contain a number of varied argument fragments, and it is worth asking whether some of these might be developed into a persuasive case.One particularly simple approach would be to note that the existence of a measurable cardinal (MC) implies that V ≠ L,1and to argue that there is a measurable cardinal. The drawback to this approach is that its implying V ≠ L cannot then be counted as evidence in favor of MC, as it often is. Indeed, there seems to have been considerable sentiment against V = L even before the proof of its negation from MC,2and this sentiment must either be accounted for as reasonable or explained away as an aberration of some kind.

Theories incomparable with respect to relative interpretability

1962 ◽  
Vol 27 (2) ◽  
pp. 195-211 ◽  
Author(s):  
Richard Montague
Keyword(s):  
Positive Integer ◽  
The Other ◽  
Natural Numbers ◽  
Infinite Set

The present paper concerns the relation of relative interpretability introduced in [8], and arises from a question posed by Tarski: are there two finitely axiomatizable subtheories of the arithmetic of natural numbers neither of which is relatively interpretable in the other? The question was answered affirmatively (without proof) in [3], and the answer was generalized in [4]: for any positive integer n, there exist n finitely axiomatizable subtheories of arithmetic such that no one of them is relatively interpretable in the union of the remainder. A further generalization was announced in [5] and is proved here: there is an infinite set of finitely axiomatizable subtheories of arithmetic such that no one of them is relatively interpretable in the union of the remainder. Several lemmas concerning the existence of self-referential and mutually referential formulas are given in Section 1, and will perhaps be of interest on their own account.

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