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.

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.

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).

Normal subgroups of infinite symmetric groups, with an application to stratified set theory

2009 ◽  
Vol 74 (1) ◽  
pp. 17-26 ◽  
Author(s):  
Nathan Bowler ◽  
Thomas Forster
Keyword(s):  
Set Theory ◽  
Symmetric Groups ◽  
Axiom Of Choice ◽  
Normal Subgroups ◽  
Standard Analysis ◽  
Small Index ◽  
The Axiom Of Choice ◽  
Two Stages ◽  
Infinite Set ◽  
Stratified Set

It is generally known that infinite symmetric groups have few nontrivial normal subgroups (typically only the subgroups of bounded support) and none of small index. (We will explain later exactly what we mean by small). However the standard analysis relies heavily on the axiom of choice. By dint of a lot of combinatorics we have been able to dispense—largely—with the axiom of choice. Largely, but not entirely: our result is that if X is an infinite set with ∣X∣ = ∣X × X∣ then Symm(X) has no nontrivial normal subgroups of small index. Some condition like this is needed because of the work of Sam Tarzi who showed [4] that, for any finite group G, there is a model of ZF without AC in which there is a set X with Symm(X)/FSymm(X) isomorphic to G.The proof proceeds in two stages. We consider a particularly useful class of permutations, which we call the class of flexible permutations. A permutation of X is flexible if it fixes at least ∣X∣-many points. First we show that every normal subgroup of Symm(X) (of small index) must contain every flexible permutation. This will be theorem 4. Then we show (theorem 7) that the flexible permutations generate Symm(X).

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 Power Kripke–Platek set theory and the axiom of choice

2020 ◽  
Vol 30 (1) ◽  
pp. 447-457
Author(s):  
Michael Rathjen
Keyword(s):  
Set Theory ◽  
Axiom Of Choice ◽  
Selection Mechanism ◽  
Ordinal Analysis ◽  
Inner Model ◽  
Power Set ◽  
The Axiom Of Choice ◽  
The Relationship ◽  
Set Operation

Abstract While power Kripke–Platek set theory, ${\textbf{KP}}({\mathcal{P}})$, shares many properties with ordinary Kripke–Platek set theory, ${\textbf{KP}}$, in several ways it behaves quite differently from ${\textbf{KP}}$. This is perhaps most strikingly demonstrated by a result, due to Mathias, to the effect that adding the axiom of constructibility to ${\textbf{KP}}({\mathcal{P}})$ gives rise to a much stronger theory, whereas in the case of ${\textbf{KP}}$, the constructible hierarchy provides an inner model, so that ${\textbf{KP}}$ and ${\textbf{KP}}+V=L$ have the same strength. This paper will be concerned with the relationship between ${\textbf{KP}}({\mathcal{P}})$ and ${\textbf{KP}}({\mathcal{P}})$ plus the axiom of choice or even the global axiom of choice, $\textbf{AC}_{\tiny {global}}$. Since $L$ is the standard vehicle to furnish a model in which this axiom holds, the usual argument for demonstrating that the addition of ${\textbf{AC}}$ or $\textbf{AC}_{\tiny {global}}$ to ${\textbf{KP}}({\mathcal{P}})$ does not increase proof-theoretic strength does not apply in any obvious way. Among other tools, the paper uses techniques from ordinal analysis to show that ${\textbf{KP}}({\mathcal{P}})+\textbf{AC}_{\tiny {global}}$ has the same strength as ${\textbf{KP}}({\mathcal{P}})$, thereby answering a question of Mathias. Moreover, it is shown that ${\textbf{KP}}({\mathcal{P}})+\textbf{AC}_{\tiny {global}}$ is conservative over ${\textbf{KP}}({\mathcal{P}})$ for $\varPi ^1_4$ statements of analysis. The method of ordinal analysis for theories with power set was developed in an earlier paper. The technique allows one to compute witnessing information from infinitary proofs, providing bounds for the transfinite iterations of the power set operation that are provable in a theory. As the theory ${\textbf{KP}}({\mathcal{P}})+\textbf{AC}_{\tiny {global}}$ provides a very useful tool for defining models and realizability models of other theories that are hard to construct without access to a uniform selection mechanism, it is desirable to determine its exact proof-theoretic strength. This knowledge can for instance be used to determine the strength of Feferman’s operational set theory with power set operation as well as constructive Zermelo–Fraenkel set theory with the axiom of choice.

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.

AN AXIOMATIC THEORY OF WELL-ORDERINGS

2011 ◽  
Vol 4 (2) ◽  
pp. 186-204 ◽  
Author(s):  
OLIVER DEISER
Keyword(s):  
Set Theory ◽  
Axiom Of Choice ◽  
Point Of View ◽  
Axiomatic Theory ◽  
First Order ◽  
Power Set ◽  
The Axiom Of Choice

We introduce a new simple first-order framework for theories whose objects are well-orderings (lists). A system ALT (axiomatic list theory) is presented and shown to be equiconsistent with ZFC (Zermelo Fraenkel Set Theory with the Axiom of Choice). The theory sheds new light on the power set axiom and on Gödel’s axiom of constructibility. In list theory there are strong arguments favoring Gödel’s axiom, while a bare analogon of the set theoretic power set axiom looks artificial. In fact, there is a natural and attractive modification of ALT where every object is constructible and countable. In order to substantiate our foundational interest in lists, we also compare sets and lists from the perspective of finite objects, arguing that lists are, from a certain point of view, conceptually simpler than sets.

A FINITE-TO-ONE MAP FROM THE PERMUTATIONS ON A SET

2016 ◽  
Vol 95 (2) ◽  
pp. 177-182 ◽  
Author(s):  
NATTAPON SONPANOW ◽  
PIMPEN VEJJAJIVA
Keyword(s):  
Set Theory ◽  
Axiom Of Choice ◽  
Symbolic Logic ◽  
The Axiom Of Choice ◽  
The Relationship ◽  
Infinite Set

Forster [‘Finite-to-one maps’, J. Symbolic Logic68 (2003), 1251–1253] showed, in Zermelo–Fraenkel set theory, that if there is a finite-to-one map from ${\mathcal{P}}(A)$, the set of all subsets of a set $A$, onto $A$, then $A$ must be finite. If we assume the axiom of choice (AC), the cardinalities of ${\mathcal{P}}(A)$ and the set $S(A)$ of permutations on $A$ are equal for any infinite set $A$. In the absence of AC, we cannot make any conclusion about the relationship between the two cardinalities for an arbitrary infinite set. In this paper, we give a condition that makes Forster’s theorem, with ${\mathcal{P}}(A)$ replaced by $S(A)$, provable without AC.

The independence of a weak axiom of choice

1956 ◽  
Vol 21 (4) ◽  
pp. 350-366 ◽  
Author(s):  
Elliott Mendelson
Keyword(s):  
Set Theory ◽  
Continuum Hypothesis ◽  
Weak Form ◽  
Axiom Of Choice ◽  
Transfinite Induction ◽  
Exact Nature ◽  
Generalized Continuum ◽  
Power Set ◽  
The Axiom Of Choice ◽  
Weak Axiom

1. The purpose of this paper is to show that, if the axioms of a system G of set theory are consistent, then it is impossible to prove from them the following weak form of the axiom of choice: (H1) For every denumerable set x of disjoint two-element sets, there is a set y, called a choice set for x, which contains exactly one element in common with each element of x. Among the axioms of the system G, we take, with minor modifications, Axioms A, B, C of Gödel [6]. Clearly, the independence of H1 implies that of all stronger propositions, including the general axiom of choice and the generalized continuum hypothesis.The proof depends upon ideas of Fraenkel and Mostowski, and proceeds in the following manner. Let a be a denumerable set of objects Δ0, Δ1, Δ2, …, the exact nature of which will be specified later. Let μj = {Δ2j, Δ2j+1} for each j, c = {μ0, μ1, μ2, …}, and b = [the sum set of a]. By transfinite induction, construct the class Vc which is the closure of b under the power-set operation. For each j, it is possible to define a 1–1 mapping of Vc onto itself with the following properties. The mapping preserves the ε-relation, or, more precisely, .

scholarly journals La hipótesis generalizada del continuo (HGC) y su relación con el axioma de elección (AE)

1989 ◽  
Vol 21 (62) ◽  
pp. 55-66
Author(s):  
José Alfredo Amor
Keyword(s):  
Set Theory ◽  
Continuum Hypothesis ◽  
Cardinal Number ◽  
Ordinal Number ◽  
Axiom Of Choice ◽  
Relative Consistency ◽  
Power Set ◽  
The Many ◽  
Definition Of ◽  
The Axiom Of Choice

The so called Generalized Continuum Hypothesis (GCH) is the sentence: "If A is an infinile set whose cardinal number is K and 2K denotes the cardinal number of the set P(A) of subsets of A (the power set of A), and K + denotes the succesor cardinal of K, then 2K = K +". The Continuum Hypothesis (CH) asserts the particular case K = o. It is clear that GCH implies CH. Another equivalent version of GCH, is the sentence: 'Any subset of the set of subsets of a given infinite set is or of cardinality less or equal than the cardinality of the given set, or of the cardinality of all the set of subsets". Gödel in 1939, and Cohen in 1963, settled the relative consistency of the Axiom of Choice (AC) and of its negation not-AC, respectively, with respecllo the Zermelo-Fraenkel set theory (ZF). On the other hand, Gödel in 1939, and Cohen in 1963 settled too, the relative consistency of GCH , CH and of its negations not-GCH, not-CH, respectively, with respect to the Zermelo-Fraenkel set theory with the Axiom of Choice (ZF + AC or ZFC). From these results we know that GCH and AC are undecidable sentences in ZF set theory and indeed, the most famous undecidable sentences in ZF; but, which is the relation between them? From the above results, in the theory ZF + AC is not demonstrated GCH; it is clear then that AC doesn't imply GCH in ZF theory, Bul does GCH implies AC in ZF theory? The answer is yes! or equivalently, there is no model of ZF +GCH + not-AC. A very easy proof can be given if we have an adecuate definition of cardinal number of a set, that doesn't depend of AC but depending from the Regularity Axiom, which asserls that aIl sets have a range, which is an ordinal number associated with its constructive complexity. We define the cardinal number of A, denoted |A|, as foIlows: |A|= { The least ordinal number equipotent with A, if A is well orderable The set of all sets equipotent with A and of minimum range, in other case. It is clear that without AC, may be not ordinal cardinals and all cardinals are ordinal cardinals if all sets are well orderable (AC). Now we formulate: GCH*: For all ordinal cardinal I<, 2K = I< + In the paper is demonstrated that this formulation GCH* is implied by the traditional one, and indeed equivalent to it. Lemma, The power set of any well orderable set is well orderable if and only if AC. This is one of the many equivalents of AC in ZF,due lo Rubin, 1960. Proposition. In ZF is a theorem: GCH* implies AC. Supose GCH*. Let A be a well orderable set; then |A| = K an ordinal cardinal, so A is equipotent with K and then P~A) is equipotent with P(K); therefore |P(A)I|= |P(K)| = 2K = K+. But then |P(A)|= K+ and P(A) 'is equipotent with K+ and K+ is an ordinal cardinal; therefore P(A) is well orderable with the well order induced by means of the bijection, from the well order of K+. Corolary: In ZF are theorems: GCH impIies AC and GCH is equivalent to GCH*. We see from this proof, that GCH asserts that the cardinal number of the power set of a well orderable set A is an ordinal, which is equivalent to AC, but GCH asserts also that that ordinal cardinal is |A|+ , the ordinal cardinal succesor of the ordinal cardinal of the well orderable set A.

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