Doughnuts, floating ordinals, square brackets, and ultraflitters

2000 ◽  
Vol 65 (1) ◽  
pp. 461-473 ◽  
Author(s):  
Carlos A. Di Prisco ◽  
James M. Henle
Keyword(s):  
Axiom Of Choice ◽  
Inaccessible Cardinal ◽  
Natural Numbers ◽  
Real Numbers ◽  
Power Set ◽  
Infinite Sets ◽  
The Axiom Of Choice ◽  
Infinite Set ◽  
Infinite Sequences

In this paper, we study partition properties of the set of real numbers. The meaning of “set of real numbers” will vary, referring at times to the collection of sequences of natural numbers, ωω; the collection of infinite sets of natural numbers [ω]ω; the collection of infinite sequences of zeroes and ones, 2ω; or (ω), the power set of ω.The archetype for the relations is the property: “all sets of reals are Ramsey,” in the notation of Erdős and Hajnal, ω → (ω)ω. This states that for every partition F : [ω]ω → 2, there is an infinite set H ∈ [ω]ω such that F is constant on [H]ω. Like virtually all of the properties we will discuss, it contradicts the Axiom of Choice but is compatible with the principle of dependent choices (DC). DC will be used throughout the paper wihtout further mention.The properties discussed in this paper will vary in two respects. Some, like ω → (ω)ω, will be incompatible with the existence of an ultrafilter on ω (UF) and some will not. Some are known to be consistent relative to ZF alone, and for some, such as ω → (ω)ω, the question is still open. All properties, however, are true in Solovay's model and hence are consistent relative to Con(ZF + “there exists an inaccessible cardinal”).

Definability of measures and ultrafilters

1977 ◽  
Vol 42 (2) ◽  
pp. 179-190 ◽  
Author(s):  
David Pincus ◽  
Robert M. Solovay
Keyword(s):  
Set Theory ◽  
Closed Interval ◽  
Axiom Of Choice ◽  
Inaccessible Cardinal ◽  
Private Communication ◽  
Real Numbers ◽  
Power Set ◽  
Disjoint Sets ◽  
The Axiom Of Choice ◽  
Dependent Choice

Nonprincipal ultrafilters are harder to define in ZFC, and harder to obtain in ZF + DC, than nonprincipal measures.The function μ from P(X) to the closed interval [0, 1] is a measure on X if μ. is finitely additive on disjoint sets and μ(X) = 1. (P is the power set.) μ is nonprincipal if μ ({x}) = 0 for each x Є X. μ is an ultrafilter if Range μ= {0, 1}. The existence of nonprincipal measures and ultrafilters on any infinite X follows from the axiom of choice.Nonprincipal measures cannot necessarily be defined in ZFC. (ZF is Zermelo–Fraenkel set theory. ZFC is ZF with choice.) In ZF alone they cannot even be proved to exist. This was first established by Solovay [14] using an inaccessible cardinal. In the model of [14] no nonprincipal measure on ω is even ODR (definable from ordinal and real parameters). The HODR (hereditarily ODR) sets of this model form a model of ZF + DC (dependent choice) in which no nonprincipal measure on ω exists. Pincus [8] gave a model with the same properties making no use of an inaccessible. (This model was also known to Solovay.) The second model can be combined with ideas of A. Blass [1] to give a model of ZF + DC in which no nonprincipal measures exist on any set. Using this model one obtains a model of ZFC in which no nonprincipal measure on the set of real numbers is ODR. H. Friedman, in private communication, previously obtained such a model of ZFC by a different method. Our construction will be sketched in 4.1.

scholarly journals On Arbitrary sets andZFC

2011 ◽  
Vol 17 (3) ◽  
pp. 361-393 ◽  
Author(s):  
José Ferreirós
Keyword(s):  
Set Theory ◽  
Axiom Of Choice ◽  
Real Numbers ◽  
The Real ◽  
Formal Systems ◽  
Infinite Sets ◽  
The Axiom Of Choice ◽  
The Continuum

AbstractSet theory deals with the most fundamental existence questions in mathematics-questions which affect other areas of mathematics, from the real numbers to structures of all kinds, but which are posed as dealing with the existence of sets. Especially noteworthy are principles establishing the existence of some infinite sets, the so-called “arbitrary sets.” This paper is devoted to an analysis of the motivating goal of studying arbitrary sets, usually referred to under the labels ofquasi-combinatorialismorcombinatorial maximality. After explaining what is meant by definability and by “arbitrariness,” a first historical part discusses the strong motives why set theory was conceived as a theory of arbitrary sets, emphasizing connections with analysis and particularly with the continuum of real numbers. Judged from this perspective, the axiom of choice stands out as a most central and natural set-theoretic principle (in the sense of quasi-combinatorialism). A second part starts by considering the potential mismatch between the formal systems of mathematics and their motivating conceptions, and proceeds to offer an elementary discussion of how far the Zermelo–Fraenkel system goes in laying out principles that capture the idea of “arbitrary sets”. We argue that the theory is rather poor in this respect.

scholarly journals Consequences of arithmetic for set theory

1994 ◽  
Vol 59 (1) ◽  
pp. 30-40 ◽  
Author(s):  
Lorenz Halbeisen ◽  
Saharon Shelah
Keyword(s):  
Set Theory ◽  
Axiom Of Choice ◽  
Power Set ◽  
The Axiom Of Choice ◽  
Infinite Set

AbstractIn this paper, we consider certain cardinals in ZF (set theory without AC, the axiom of choice). In ZFC (set theory with AC), given any cardinals and , either ≤ or ≤ . However, in ZF this is no longer so. For a given infinite set A consider seq1-1(A), the set of all sequences of A without repetition. We compare |seq1-1(A)|, the cardinality of this set, to ||, the cardinality of the power set of A. What is provable about these two cardinals in ZF? The main result of this paper is that ZF ⊢ ∀A(| seq1-1(A)| ≠ ||), and we show that this is the best possible result. Furthermore, it is provable in ZF that if B is an infinite set, then | fin(B)| < | (B*)| even though the existence for some infinite set B* of a function ƒ from fin(B*) onto (B*) is consistent with ZF.

scholarly journals Small sets with large power sets

1973 ◽  
Vol 8 (3) ◽  
pp. 413-421 ◽  
Author(s):  
G.P. Monro
Keyword(s):  
Set Theory ◽  
Axiom Of Choice ◽  
Large Power ◽  
Present Evidence ◽  
Power Set ◽  
Infinite Sets ◽  
The Axiom Of Choice

One problem in set theory without the axiom of choice is to find a reasonable way of estimating the size of a non-well-orderable set; in this paper we present evidence which suggests that this may be very hard. Given an arbitrary fixed aleph κ we construct a model of set theory which contains a set X such that if Y ⊆ X then either Y or X - Y is finite, but such that κ can be mapped into S(S(S(X))). So in one sense X is large and in another X is one of the smallest possible infinite sets. (Here S(X) is the power set of X.)

scholarly journals What is the Size of the Hilbert Hotel's Computer?

2021 ◽  
Author(s):  
ANDRÉ LUIZ BARBOSA
Keyword(s):  
Set Theory ◽  
Elementary Mathematics ◽  
Axiom Of Choice ◽  
Natural Numbers ◽  
The Road ◽  
Infinite Sets ◽  
The Axiom Of Choice ◽  
Countably Infinite

Abstract The Hilbert's Hotel is a hotel with countably infinitely many rooms. The size of its hypothetical computer was the pretext in order to think about whether it makes sense and what would be log_2(\aleph_0). Thus, at the road of this journey, this little paper demonstrates – surprisingly – that there exist countably infinite sets strictly smaller than \mathbb{N} (the natural numbers), with very elementary mathematics, so shockingly stating the inconsistency of the Zermelo-Fraenkel Set Theory with the Axiom of Choice (ZFC)

Products of some special compact spaces and restricted forms of AC

2010 ◽  
Vol 75 (3) ◽  
pp. 996-1006 ◽  
Author(s):  
Kyriakos Keremedis ◽  
Eleftherios Tachtsis
Keyword(s):  
Set Theory ◽  
Closed Subset ◽  
Choice Function ◽  
Ordinal Number ◽  
Axiom Of Choice ◽  
Finite Support ◽  
Compact Spaces ◽  
The Axiom Of Choice ◽  
Infinite Set

AbstractWe establish the following results:1. In ZF (i.e., Zermelo-Fraenkel set theory minus the Axiom of Choice AC), for every set I and for every ordinal number α ≥ ω, the following statements are equivalent:(a) The Tychonoff product of ∣α∣ many non-empty finite discrete subsets of I is compact.(b) The union of ∣α∣ many non-empty finite subsets of I is well orderable.2. The statement: For every infinite set I, every closed subset of the Tychonoff product [0, 1]Iwhich consists offunctions with finite support is compact, is not provable in ZF set theory.3. The statement: For every set I, the principle of dependent choices relativised to I implies the Tychonoff product of countably many non-empty finite discrete subsets of I is compact, is not provable in ZF0 (i.e., ZF minus the Axiom of Regularity).4. The statement: For every set I, every ℵ0-sized family of non-empty finite subsets of I has a choice function implies the Tychonoff product of ℵ0many non-empty finite discrete subsets of I is compact, is not provable in ZF0.

Proceed with care, because some infinities are larger than others

2020 ◽  
pp. 281-292
Author(s):  
Susan D'Agostino
Keyword(s):  
Number Line ◽  
Mathematics Students ◽  
Natural Numbers ◽  
Real Numbers ◽  
Countable Infinity ◽  
One To One ◽  
Infinite Set

“Proceed with care, because some infinities are larger than others” explains in detail why the infinite set of real numbers—all of the numbers on the number line—represents a far larger infinity than the infinite set of natural numbers—the counting numbers. Readers learn to distinguish between countable infinity and uncountable infinity by way of a method known as a “one-to-one correspondence.” Mathematics students and enthusiasts are encouraged to proceed with care in both mathematics and life, lest they confuse countable infinity with uncountable infinity, large with unfathomably large, or order with disorder. At the chapter’s end, readers may check their understanding by working on a problem. A solution is provided.

scholarly journals Selectively Pseudocompact Groups without Infinite Separable Pseudocompact Subsets

Axioms ◽  
2018 ◽  
Vol 7 (4) ◽  
pp. 86 ◽  
Author(s):  
Dmitri Shakhmatov ◽  
Víctor Yañez
Keyword(s):  
Topological Group ◽  
Set Theory ◽  
Axiom Of Choice ◽  
Natural Numbers ◽  
Compactness Property ◽  
Free Ultrafilter ◽  
The Axiom Of Choice

We give a “naive” (i.e., using no additional set-theoretic assumptions beyond ZFC, the Zermelo-Fraenkel axioms of set theory augmented by the Axiom of Choice) example of a Boolean topological group G without infinite separable pseudocompact subsets having the following “selective” compactness property: For each free ultrafilter p on the set N of natural numbers and every sequence ( U n ) of non-empty open subsets of G, one can choose a point x n ∈ U n for all n ∈ N in such a way that the resulting sequence ( x n ) has a p-limit in G; that is, { n ∈ N : x n ∈ V } ∈ p for every neighbourhood V of x in G. In particular, G is selectively pseudocompact (strongly pseudocompact) but not selectively sequentially pseudocompact. This answers a question of Dorantes-Aldama and the first listed author. The group G above is not pseudo- ω -bounded either. Furthermore, we show that the free precompact Boolean group of a topological sum ⨁ i ∈ I X i , where each space X i is either maximal or discrete, contains no infinite separable pseudocompact subsets.

Inconsistency of uncountable infinite sets under ZFC framework

Kybernetes ◽  
2008 ◽  
Vol 37 (3/4) ◽  
pp. 453-457 ◽  
Author(s):  
Wujia Zhu ◽  
Yi Lin ◽  
Guoping Du ◽  
Ningsheng Gong
Keyword(s):  
Natural Number ◽  
Set Theory ◽  
Design Methodology ◽  
Natural Numbers ◽  
Real Numbers ◽  
Conceptual Approach ◽  
Content Type ◽  
Infinite Sets ◽  
Axiomatic Set Theory ◽  
First Time

PurposeThe purpose is to show that all uncountable infinite sets are self‐contradictory non‐sets.Design/methodology/approachA conceptual approach is taken in the paper.FindingsGiven the fact that the set N={x|n(x)} of all natural numbers, where n(x)=df “x is a natural number” is a self‐contradicting non‐set in this paper, the authors prove that in the framework of modern axiomatic set theory ZFC, various uncountable infinite sets are either non‐existent or self‐contradicting non‐sets. Therefore, it can be astonishingly concluded that in both the naive set theory or the modern axiomatic set theory, if any of the actual infinite sets exists, it must be a self‐contradicting non‐set.Originality/valueThe first time in history, it is shown that such convenient notion as the set of all real numbers needs to be reconsidered.

scholarly journals On a cardinal equation in set theory

1972 ◽  
Vol 6 (3) ◽  
pp. 447-457 ◽  
Author(s):  
J.L. Hickman
Keyword(s):  
Set Theory ◽  
Axiom Of Choice ◽  
Linear Ordering ◽  
Finite Sets ◽  
Intrinsic Interest ◽  
The Axiom Of Choice ◽  
Infinite Set ◽  
Finiteness Criterion

We work in a Zermelo-Fraenkel set theory without the Axiom of Choice. In the appendix to his paper “Sur les ensembles finis”, Tarski proposed a finiteness criterion that we have called “C-finiteness”: a nonempty set is called “C-finite” if it cannot be partitioned into two blocks, each block being equivalent to the whole set. Despite the fact that this criterion can be shown to possess several features that are undesirable in a finiteness criterion, it has a fair amount of intrinsic interest. In Section 1 of this paper we look at a certain class of C-finite sets; in Section 2 we derive a few consequences from the negation of C-finiteness; and in Section 3 we show that not every C-infinite set necessarily possesses a linear ordering. Any unexplained notation is given in my paper, “Some definitions of finiteness”, Bull. Austral. Math. Soc. 5 (1971).

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