CANONICAL MODELS FOR FRAGMENTS OF THE AXIOM OF CHOICE

2017 ◽  
Vol 82 (2) ◽  
pp. 489-509
Author(s):  
PAUL LARSON ◽  
JINDŘICH ZAPLETAL
Keyword(s):  
Equivalence Relation ◽  
Axiom Of Choice ◽  
Canonical Models ◽  
Develop Technology ◽  
Image Position ◽  
The Axiom Of Choice

AbstractWe develop technology for investigation of natural forcing extensions of the model $L\left( \mathbb{R} \right)$ which satisfy such statements as “there is an ultrafilter” or “there is a total selector for the Vitali equivalence relation”. The technology reduces many questions about ZF implications between consequences of the Axiom of Choice to natural ZFC forcing problems.

Relative consistency of an extension of Ackermann's set theory

1976 ◽  
Vol 41 (2) ◽  
pp. 465-466
Author(s):  
John Lake
Keyword(s):  
Set Theory ◽  
Relative Strength ◽  
Axiom Of Choice ◽  
Relative Consistency ◽  
Free Variables ◽  
Image Position ◽  
The Axiom Of Choice

The set theory AFC was introduced by Perlis in [2] and he noted that it both includes and is stronger than Ackermann's set theory. We shall give a relative consistency result for AFC.AFC is obtained from Ackermann's set theory (see [2]) by replacing Ackermann's set existence schema with the schema(where ϕ, ψ, are ∈-formulae, x is not in ψ, w is not in ϕ, y is y1, …, yn, z is z1, …, zm and all free variables are shown) and adding the axiom of choice for sets. Following [1], we say that λ is invisible in Rκ if λ < κ and we haveholding for every ∈-formula θ which has exactly two free variables and does not involve u or υ. The existence of a Ramsey cardinal implies the existence of cardinals λ and κ with λ invisible in Rκ, and Theorem 1.13 of [1] gives some further indications about the relative strength of the notion of invisibility.Theorem. If there are cardinals λ and κ with λ invisible in Rκ, then AFC is consistent.Proof. Suppose that λ is invisible in Rκ and we will show that 〈Rκ, Rλ, ∈〉 ⊧ AFC (Rλ being the interpretation of V, of course).

Term models for weak set theories with a universal set

1987 ◽  
Vol 52 (2) ◽  
pp. 374-387 ◽  
Author(s):  
T. E. Forster
Keyword(s):  
Boolean Algebra ◽  
Set Theory ◽  
Axiom Of Choice ◽  
The Other ◽  
Free Variables ◽  
Image Position ◽  
The Axiom Of Choice ◽  
Axiomatic Systems ◽  
The Universe

We shall be concerned here with weak axiomatic systems of set theory with a universal set. The language in which they are expressed is that of set theory—two primitive predicates, = and ϵ, and no function symbols (though some function symbols will be introduced by definitional abbreviation). All the theories will have stratified axioms only, and they will all have Ext (extensionality: (∀x)(∀y)(x = y· ↔ ·(∀z)(z ϵ x ↔ z ϵ y))). In fact, in addition to extensionality, they have only axioms saying that the universe is closed under certain set-theoretic operations, viz. all of the formand these will always include singleton, i.e., ι′x exists if x does (the iota notation for singleton, due to Russell and Whitehead, is used here to avoid confusion with {x: Φ}, set abstraction), and also x ∪ y, x ∩ y and − x (the complement of x). The system with these axioms is called NF2 in the literature (see [F]). The other axioms we consider will be those giving ⋃x, ⋂x, {y: y ⊆x} and {y: x ⊆ y}. We will frequently have occasion to bear in mind that 〈 V, ⊆ 〉 is a Boolean algebra in any theory extending NF2. There is no use of the axiom of choice at any point in this paper. Since the systems with which we will be concerned exhibit this feature of having, in addition to extensionality, only axioms stating that V is closed under certain operations, we will be very interested in terms of the theories in question. A T-term, for T such a theory, is a thing (with no free variables) built up from V or ∧ by means of the T-operations, which are of course the operations that the axioms of T say the universe is closed under.

FACTORIALS OF INFINITE CARDINALS IN ZF PART I: ZF RESULTS

2019 ◽  
Vol 85 (1) ◽  
pp. 224-243
Author(s):  
GUOZHEN SHEN ◽  
JIACHEN YUAN
Keyword(s):  
Fixed Points ◽  
Axiom Of Choice ◽  
Finite Sequences ◽  
Infinite Sets ◽  
Image Position ◽  
The Axiom Of Choice

AbstractFor a set x, let ${\cal S}\left( x \right)$ be the set of all permutations of x. We prove in ZF (without the axiom of choice) several results concerning this notion, among which are the following:(1) For all sets x such that ${\cal S}\left( x \right)$ is Dedekind infinite, $\left| {{{\cal S}_{{\rm{fin}}}}\left( x \right)} \right| < \left| {{\cal S}\left( x \right)} \right|$ and there are no finite-to-one functions from ${\cal S}\left( x \right)$ into ${{\cal S}_{{\rm{fin}}}}\left( x \right)$, where ${{\cal S}_{{\rm{fin}}}}\left( x \right)$ denotes the set of all permutations of x which move only finitely many elements.(2) For all sets x such that ${\cal S}\left( x \right)$ is Dedekind infinite, $\left| {{\rm{seq}}\left( x \right)} \right| < \left| {{\cal S}\left( x \right)} \right|$ and there are no finite-to-one functions from ${\cal S}\left( x \right)$ into seq (x), where seq (x) denotes the set of all finite sequences of elements of x.(3) For all infinite sets x such that there exists a permutation of x without fixed points, there are no finite-to-one functions from ${\cal S}\left( x \right)$ into x.(4) For all sets x, $|{[x]^2}| < \left| {{\cal S}\left( x \right)} \right|$.

Partitions of products

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

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

Subgroups of a free group and the axiom of choice

1985 ◽  
Vol 50 (2) ◽  
pp. 458-467 ◽  
Author(s):  
Paul E. Howard
Keyword(s):  
Free Group ◽  
Cardinal Number ◽  
Cyclic Permutation ◽  
Axiom Of Choice ◽  
Embedding Theorem ◽  
Open Problems ◽  
Power Set ◽  
Partial List ◽  
Image Position ◽  
The Axiom Of Choice

Nielsen [7] has proved that every subgroup of a free group of finite rank is free. The theorem was later strengthened by Schreier [8] by eliminating the finiteness restriction on the rank. Several proofs of this theorem (known as the Nielsen-Schreier theorem, henceforth denoted by NS) have appeared since Schreier's 1927 paper (see [1] and [2]). All proofs of NS use the axiom of choice (AC) and it is natural to ask whether NS is equivalent to AC. Läuchli has given a partial answer to this question by proving [6] that the negation of NS is consistent with ZFA (Zermelo-Fraenkel set theory weakened to permit the existence of atoms). By the Jech-Sochor embedding theorem (see [3] and [4]) ZFA can be replaced by ZF. Some form of AC, therefore, is needed to prove NS. The main purpose of this paper is to give a further answer to this question.In §2 we prove that NS implies ACffin (the axiom of choice for sets of finite sets). In §3 we show that a strengthened version of NS implies AC and in §4 we give a partial list of open problems.Let y be a set; ∣y∣ denotes the cardinal number of y and (y) is the power set of y. If p is a permutation of y and t ∈ y, the p-orbit of t is the set {pn(t): n is an integer}. Ifwe call p a cyclic permutation of y. If f is a function with domain y and x ⊆ y, f″x denotes the set {f(t):t ∈ x}. If A is a subset of a group (G, °) (sometimes (G, °) will be denoted by G) then A−1 = {x−1:x ∈ A} and [A] denotes the subgroup of G generated by A.

scholarly journals ITERATING SYMMETRIC EXTENSIONS

2019 ◽  
Vol 84 (1) ◽  
pp. 123-159 ◽  
Author(s):  
ASAF KARAGILA
Keyword(s):  
Axiom Of Choice ◽  
Generic Extension ◽  
Ground Model ◽  
Intermediate Model ◽  
Symmetric Extension ◽  
Consistency Results ◽  
New Models ◽  
Image Position ◽  
The Axiom Of Choice

AbstractThe notion of a symmetric extension extends the usual notion of forcing by identifying a particular class of names which forms an intermediate model of $ZF$ between the ground model and the generic extension, and often the axiom of choice fails in these models. Symmetric extensions are generally used to prove choiceless consistency results. We develop a framework for iterating symmetric extensions in order to construct new models of $ZF$. We show how to obtain some well-known and lesser-known results using this framework. Specifically, we discuss Kinna–Wagner principles and obtain some results related to their failure.

The Axiom of Choice Machine

2018 ◽  
Author(s):  
Alexander R. Pruss
Keyword(s):  
Set Theory ◽  
Laws Of Nature ◽  
Axiom Of Choice ◽  
Dutch Book ◽  
The Axiom Of Choice

This is a mainly technical chapter concerning the causal embodiment of the Axiom of Choice from set theory. The Axiom of Choice powered a construction of an infinite fair lottery in Chapter 4 and a die-rolling strategy in Chapter 5. For those applications to work, there has to be a causally implementable (though perhaps not compatible with our laws of nature) way to implement the Axiom of Choice—and, for our purposes, it is ideal if that involves infinite causal histories, so the causal finitist can reject it. Such a construction is offered. Moreover, other paradoxes involving the Axiom of Choice are given, including two Dutch Book paradoxes connected with the Banach–Tarski paradox. Again, all this is argued to provide evidence for causal finitism.

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