For
n
∈
ω
, the weak choice principle
RC
n
is defined as follows:
For every infinite set
X
there is an infinite subset
Y
⊆
X
with a choice function on
Y
n
:
=
{
z
⊆
Y
:
|
z
|
=
n
}
.
The choice ...principle
C
n
-
states the following:
For every infinite family of
n
-
element sets, there is an infinite subfamily
G
⊆
F
with a choice function.
The choice principles
LOC
n
-
and
WOC
n
-
are the same as
C
n
-
, but we assume that the family
F
is linearly orderable (for
LOC
n
-
) or well-orderable (for
WOC
n
-
). In the first part of this paper, for
m
,
n
∈
ω
we will give a full characterization of when the implication
RC
m
⇒
WOC
n
-
holds in
ZF
. We will prove the independence results by using suitable Fraenkel-Mostowski permutation models. In the second part, we will show some generalizations. In particular, we will show that
RC
5
⇒
LOC
5
-
and that
RC
6
⇒
C
3
-
, answering two open questions from Halbeisen and Tachtsis (Arch Math Logik 59(5):583–606, 2020). Furthermore, we will show that
RC
6
⇒
C
9
-
and that
RC
7
⇒
LOC
7
-
.
Abstract
In this article, we will first discuss the completeness of real numbers in the context of an alternate definition of the straight line as a geometric continuum. According to this definition, ...points are not regarded as the basic constituents of a line segment and a line segment is considered to be a fundamental geometric object. This definition is in particular suitable to coordinatize different points on the straight line preserving the order properties of real numbers. Geometrically fundamental nature of line segments are required in physical theories like the string theory. We will construct a new topology suitable for this alternate definition of the straight line as a geometric continuum. We will discuss the cardinality of rational numbers in the later half of the article. We will first discuss what we do in an actual process of counting and define functions well-defined on the set of all positive integers. We will follow an alternate approach that depends on the Hausdorff topology of real numbers to demonstrate that the set of positive rationals can have a greater cardinality than the set of positive integers. This approach is more consistent with an actual act of counting. We will illustrate this aspect further using well-behaved functionals of convergent functions defined on the finite dimensional Cartezian products of the set of positive integers and non-negative integers. These are similar to the partition functions in statistical physics. This article indicates that the
axiom of choice
can be a better technique to prove theorems that use second-countability. This is important for the metrization theorems and physics of spacetime.
Abstract
In the Zermelo–Fraenkel set theory (ZF), $|\textrm {fin}(A)|<2^{|A|}\leq |\textrm {Part}(A)|$ for any infinite set $A$, where $\textrm {fin}(A)$ is the set of finite subsets of $A$, ...$2^{|A|}$ is the cardinality of the power set of $A$ and $\textrm {Part}(A)$ is the set of partitions of $A$. In this paper, we show in ZF that $|\textrm {fin}(A)|<|\textrm {Part}_{\textrm {fin}}(A)|$ for any set $A$ with $|A|\geq 5$, where $\textrm {Part}_{\textrm {fin}}(A)$ is the set of partitions of $A$ whose members are finite. We also show that, without the Axiom of Choice, any relationship between $|\textrm {Part}_{\textrm {fin}}(A)|$ and $2^{|A|}$ for an arbitrary infinite set $A$ cannot be concluded.
Improving and clarifying a construction of Horowitz and Shelah, we show how to construct (in
$\mathsf {ZF}$
, i.e., without using the Axiom of Choice) maximal cofinitary groups. Among the groups we ...construct, one is definable by a formula in second-order arithmetic with only a few natural number quantifiers.
How to have more things by forgetting how to count them Karagila, Asaf; Schlicht, Philipp
Proceedings of the Royal Society. A, Mathematical, physical, and engineering sciences,
07/2020, Letnik:
476, Številka:
2239
Journal Article
Recenzirano
Odprti dostop
Cohen’s first model is a model of Zermelo–Fraenkel set theory in which there is a Dedekind-finite set of real numbers, and it is perhaps the most famous model where the Axiom of Choice fails. We ...force over this model to add a function from this Dedekind-finite set to some infinite ordinal
κ
. In the case that we force the function to be injective, it turns out that the resulting model is the same as adding
κ
Cohen reals to the ground model, and that we have just added an enumeration of the canonical Dedekind-finite set. In the case where the function is merely surjective it turns out that we do not add any reals, sets of ordinals, or collapse any Dedekind-finite sets. This motivates the question if there is any combinatorial condition on a Dedekind-finite set
A
which characterises when a forcing will preserve its Dedekind-finiteness or not add new sets of ordinals. We answer this question in the case of ‘Adding a Cohen subset’ by presenting a varied list of conditions each equivalent to the preservation of Dedekind-finiteness. For example, 2
A
is extremally disconnected, or
A
<
ω
is Dedekind-finite.
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