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We present a Zermelo–Fraenkel ($\textbf {ZF}$) consistency result regarding bi-orderability of groups. A classical consequence of the ultrafilter lemma is that a group is bi-orderable if and only if it is locally bi-orderable. We show that there exists a model of $\textbf {ZF}$ plus dependent choice in which there is a group which is locally free (ergo locally bi-orderable) and not bi-orderable, and the group can be given a total order. The model also includes a torsion-free abelian group which is not bi-orderable but can be given a total order.
For every $n\in \omega \setminus \{0,1\}$
we introduce the
following weak choice principle:
$\operatorname {nC}_{<\aleph _0}^-:$
For every infinite family$\mathcal {F}$
of finite sets of size at least n there is an infinite subfamily$\mathcal {G}\subseteq \mathcal {F}$
with a selection function$f:\mathcal {G}\to \left [\bigcup \mathcal {G}\right ]^n$
such that$f(F)\in [F]^n$
for all$F\in \mathcal {G}$
.
Moreover, we consider the following choice principle:
$\operatorname {KWF}^-:$
For every infinite family$\mathcal {F}$
of finite sets of size at least$2$
there is an infinite subfamily$\mathcal {G}\subseteq \mathcal {F}$
with a Kinna–Wagner selection function. That is, there is a function$g\colon \mathcal {G}\to \mathcal {P}\left (\bigcup \mathcal {G}\right )$
with$\emptyset \not =f(F)\subsetneq F$
for every$F\in \mathcal {G}$
.
We will discuss the relations between these two choice principles and their relations to other well-known weak choice principles. Moreover, we will discuss what happens when we replace
$\mathcal {F}$
by a linearly ordered or a well-ordered family.
Henle, Mathias, and Woodin proved in [21] that, provided that
${\omega }{\rightarrow }({\omega })^{{\omega }}$
holds in a model M of ZF, then forcing with
$([{\omega }]^{{\omega }},{\subseteq }^*)$
over M adds no new sets of ordinals, thus earning the name a “barren” extension. Moreover, under an additional assumption, they proved that this generic extension preserves all strong partition cardinals. This forcing thus produces a model
$M[\mathcal {U}]$
, where
$\mathcal {U}$
is a Ramsey ultrafilter, with many properties of the original model M. This begged the question of how important the Ramseyness of
$\mathcal {U}$
is for these results. In this paper, we show that several classes of
$\sigma $
-closed forcings which generate non-Ramsey ultrafilters have the same properties. Such ultrafilters include Milliken–Taylor ultrafilters, a class of rapid p-points of Laflamme, k-arrow p-points of Baumgartner and Taylor, and extensions to a class of ultrafilters constructed by Dobrinen, Mijares, and Trujillo. Furthermore, the class of Boolean algebras
$\mathcal {P}({\omega }^{{\alpha }})/{\mathrm {Fin}}^{\otimes {\alpha }}$
,
$2\le {\alpha }<{\omega }_1$
, forcing non-p-points also produce barren extensions.
We show that Dependent Choice is a sufficient choice principle for developing the basic theory of proper forcing, and for deriving generic absoluteness for the Chang model in the presence of large cardinals, even with respect to
$\mathsf {DC}$
-preserving symmetric submodels of forcing extensions. Hence,
$\mathsf {ZF}+\mathsf {DC}$
not only provides the right framework for developing classical analysis, but is also the right base theory over which to safeguard truth in analysis from the independence phenomenon in the presence of large cardinals. We also investigate some basic consequences of the Proper Forcing Axiom in
$\mathsf {ZF}$
, and formulate a natural question about the generic absoluteness of the Proper Forcing Axiom in
$\mathsf {ZF}+\mathsf {DC}$
and
$\mathsf {ZFC}$
. Our results confirm
$\mathsf {ZF} + \mathsf {DC}$
as a natural foundation for a significant portion of “classical mathematics” and provide support to the idea of this theory being also a natural foundation for a large part of set theory.
For 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, $|{[x]^2}| < \left| {{\cal S}\left( x \right)} \right|$.
For a set x, let ${\cal S}\left( x \right)$ be the set of all permutations of x. We prove by the method of permutation models that the following statements are consistent with ZF:
(1) There is an infinite set x such that $|\wp \left( x \right)| < |{\cal S}\left( x \right)| < |se{q^{1 - 1}}\left( x \right)| < |seq\left( x \right)|$, where $\wp \left( x \right)$ is the power set of x, seq (x) is the set of all finite sequences of elements of x, and seq1-1 (x) is the set of all finite sequences of elements of x without repetition.
(2) There is a Dedekind infinite set x such that $|{\cal S}\left( x \right)| < |{[x]^3}|$ and such that there exists a surjection from x onto ${\cal S}\left( x \right)$.
(3) There is an infinite set x such that there is a finite-to-one function from ${\cal S}\left( x \right)$ into x.
The standard topological representation of a Boolean algebra via the clopen sets of a Stone space requires a nonconstructive choice principle, equivalent to the Boolean Prime Ideal Theorem. In this article, we describe a choice-free topological representation of Boolean algebras. This representation uses a subclass of the spectral spaces that Stone used in his representation of distributive lattices via compact open sets. It also takes advantage of Tarski’s observation that the regular open sets of any topological space form a Boolean algebra. We prove without choice principles that any Boolean algebra arises from a special spectral space X via the compact regular open sets of X; these sets may also be described as those that are both compact open in X and regular open in the upset topology of the specialization order of X, allowing one to apply to an arbitrary Boolean algebra simple reasoning about regular opens of a separative poset. Our representation is therefore a mix of Stone and Tarski, with the two connected by Vietoris: the relevant spectral spaces also arise as the hyperspace of nonempty closed sets of a Stone space endowed with the upper Vietoris topology. This connection makes clear the relation between our point-set topological approach to choice-free Stone duality, which may be called the hyperspace approach, and a point-free approach to choice-free Stone duality using Stone locales. Unlike Stone’s representation of Boolean algebras via Stone spaces, our choice-free topological representation of Boolean algebras does not show that every Boolean algebra can be represented as a field of sets; but like Stone’s representation, it provides the benefit of a topological perspective on Boolean algebras, only now without choice. In addition to representation, we establish a choice-free dual equivalence between the category of Boolean algebras with Boolean homomorphisms and a subcategory of the category of spectral spaces with spectral maps. We show how this duality can be used to prove some basic facts about Boolean algebras.
The HOD Dichotomy Theorem states that if there is an extendible cardinal, δ, then either HOD is “close” to V (in the sense that it correctly computes successors of singular cardinals greater than δ) or HOD is “far” from V (in the sense that all regular cardinals greater than or equal to δ are measurable in HOD). The question is whether the future will lead to the first or the second side of the dichotomy. Is HOD “close” to V, or “far” from V? There is a program aimed at establishing the first alternative—the “close” side of the HOD Dichotomy. This is the program of inner model theory. In recent years the third author has provided evidence that there is an ultimate inner model—Ultimate-L—and he has isolated a natural conjecture associated with the model—the Ultimate-L Conjecture. This conjecture implies that (assuming the existence of an extendible cardinal) that the first alternative holds—HOD is “close” to V. This is the future in which pattern prevails. In this paper we introduce a very different program, one aimed at establishing the second alternative—the “far” side of the HOD Dichotomy. This is the program of large cardinals beyond choice. Kunen famously showed that if AC holds then there cannot be a Reinhardt cardinal. It has remained open whether Reinhardt cardinals are consistent in ZF alone. It turns out that there is an entire hierarchy of choiceless large cardinals of which Reinhardt cardinals are only the beginning, and, surprisingly, this hierarchy appears to be highly ordered and amenable to systematic investigation, as we shall show in this paper. The point is that if these choiceless large cardinals are consistent then the Ultimate-L Conjecture must fail. This is the future where chaos prevails.
Completeness and other forms of Zorn’s Lemma are sometimes invoked for semantic proofs of conservation in relatively elementary mathematical contexts in which the corresponding syntactical conservation would suffice. We now show how a fairly general syntactical conservation theorem that covers plenty of the semantic approaches follows from an utmost versatile criterion for conservation given by Scott in 1974.
To this end we work with multi-conclusion entailment relations as extending single-conclusion entailment relations. In a nutshell, the additional axioms with disjunctions in positive position can be eliminated by reducing them to the corresponding disjunction elimination rules, which in turn prove admissible in all known mathematical instances. In deduction terms this means to fold up branchings of proof trees by way of properties of the relevant mathematical structures.
Applications include the syntactical counterparts of the theorems or lemmas known under the names of Artin–Schreier, Krull–Lindenbaum, and Szpilrajn. Related work has been done before on individual instances, e.g., in locale theory, dynamical algebra, formal topology and proof analysis.
Let B be a Boolean algebra and G a group of automorphisms of B. Define an equivalence relation ∼ on B by letting x ∼ y if there are x1, x2,…,xn, y1, y2, …yn in B such that x is the disjoint union of the xi, y is the disjoint union of the yi, and for each i there is a member of G taking xi to yi. The equivalence classes under ∼ are called equidecomposability types. Addition of equidecomposability types is given by (x) + (y) = (x V y) provided x ∧ y = 0. An example is given of a complete Boolean algebra B and a group G of automorphisms of B with X, Y ∊ B such that (X) + (X) = (Y) + (Y) but (X) ≠ (Y), answering a question of Wagon (see [5 p. 231 problem 14]). Moreover B may be taken to be the algebra of Borel subsets of Cantor space modulo sets of the first category. It is also remarked that in this case equidecomposability types do not form a weak cardinal algebra.
The main result of this paper is to prove that a generalization of the Principle of Dependent Choices, introduced by A. Levy [2; see also 1, Chapter 8], is equivalent to a form of Zorn's Lemma.
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