Distributivity and an axiom of choice

1954 ◽  
Vol 19 (4) ◽  
pp. 275-277 ◽  
Author(s):  
George E. Collins
Keyword(s):  
Axiom Of Choice ◽  
Complete Distributivity ◽  
Similar Proof ◽  
Image Position

In this paper a theorem will be established which states that a particular axiom of choice is equivalent to complete distributivity of union and intersection. The theorem will be formulated and proved in the system of logic of [4]. In addition to definitions of [4], the following will be used.In terms of these definitions, the theorem can be formulated as follows.The dual of this statement, obtained by interchanging I and U, is also a theorem and has a similar proof.

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).

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.

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.

Simple Identification of Electron Diffraction Ring Patterns

1969 ◽  
Vol 27 ◽  
pp. 160-161
Author(s):  
Carolyn Nohr ◽  
Ann Ayres
Keyword(s):  
Electron Microscope ◽  
Electron Diffraction ◽  
Diffraction Ring ◽  
Image Position

Texts on electron diffraction recommend that the camera constant of the electron microscope be determine d by calibration with a standard crystalline specimen, using the equation

Atomic orientation effects in proton stopping at low velocity

1983 ◽  
Vol 41 ◽  
pp. 708-709
Author(s):  
Kin Lam
Keyword(s):  
Polycrystalline Material ◽  
Charged Particles ◽  
Target Material ◽  
Stopping Power ◽  
Low Velocity ◽  
Electronic Density ◽  
Interatomic Distances ◽  
Frame Work ◽  
Image Position ◽  
Atomic Orientation

The energy of moving ions in solid is dependent on the electronic density as well as the atomic structural properties of the target material. These factors contribute to the observable effects in polycrystalline material using the scanning ion microscope. Here we outline a method to investigate the dependence of low velocity proton stopping on interatomic distances and orientations.The interaction of charged particles with atoms in the frame work of the Fermi gas model was proposed by Lindhard. For a system of atoms, the electronic Lindhard stopping power can be generalized to the formwhere the stopping power function is defined as

A Magnetic Spectrometer for a Scanning Transmission Electron Microscope

1974 ◽  
Vol 32 ◽  
pp. 436-437
Author(s):  
A. Kosiara ◽  
J. W. Wiggins ◽  
M. Beer
Keyword(s):  
Scanning Transmission Electron Microscope ◽  
Magnetic Spectrometer ◽  
Fringe Field ◽  
Curvature Radius ◽  
Magnetic Pole ◽  
Quadrupole Field ◽  
Transmission Electron ◽  
Scanning Transmission ◽  
Under Construction ◽  
Image Position

A magnetic spectrometer to be attached to the Johns Hopkins S. T. E. M. is under construction. Its main purpose will be to investigate electron interactions with biological molecules in the energy range of 40 KeV to 100 KeV. The spectrometer is of the type described by Kerwin and by Crewe Its magnetic pole boundary is given by the equationwhere R is the electron curvature radius. In our case, R = 15 cm. The electron beam will be deflected by an angle of 90°. The distance between the electron source and the pole boundary will be 30 cm. A linear fringe field will be generated by a quadrupole field arrangement. This is accomplished by a grounded mirror plate and a 45° taper of the magnetic pole.

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