We show that the Axiom of Countable Choice is necessary and sufficient to prove that the existence of a Borel measure on a pseudometric space such that the measure of open balls is positive and ...finite implies separability of the space. In this way a negative answer to an open problem formulated in Górka (Am Math Mon 128:84–86, 2020) is given. Moreover, we study existence of maximal
δ
-separated sets in metric and pseudometric spaces from the point of view the Axiom of Choice and its weaker forms.
We prove that it is relatively consistent with \mathsf {ZF} (i.e., Zermelo-Fraenkel set theory without the Axiom of Choice ( \mathsf {AC})) that the Axiom of Countable Choice ( \mathsf {AC}^{\aleph ..._{0}}) is true, but the Urysohn Lemma ( \mathsf {UL}), and hence the Tietze Extension Theorem ( \mathsf {TET}), is false. This settles the corresponding open problem in P. Howard and J. E. Rubin Consequences of the Axiom of Choice , Mathematical Surveys and Monographs, Vol. 59, American Mathematical Society, Providence, RI, 1998. We also prove that in Läuchli's permutation model of \mathsf {ZFA} + \neg \mathsf {UL}, \mathsf {AC}^{\aleph _{0}} is false. This fills the gap in information in the above monograph of Howard and Rubin.
Extension of Positive Operators Ilina, K. Yu; Kusraeva, Z. A.
Siberian mathematical journal,
03/2020, Letnik:
61, Številka:
2
Journal Article
Recenzirano
The main result states that if
E
is a separable Fréchet lattice and
F
is a (locally solid) topological vector lattice with the
σ
-interpolation property then each positive linear operator
T
0
from a ...majorizing subspace
G
⊂
E
into
F
admits extension to a continuous positive linear operator
T
from
E
into
F
. This fact is proved by using only the axiom of countable choice.
This paper proves the approximate intermediate value theorem, constructively and from notably weak hypotheses: from pointwise rather than uniform continuity, without assuming that reals are presented ...with rational approximants, and without using countable choice. The theorem is that if a pointwise continuous function has both a negative and a positive value, then it has values arbitrarily close to 0. The proof builds on the usual classical proof by bisection, which repeatedly selects the left or right half of an interval; the algorithm here selects an interval of half the size in a continuous way, interpolating between those two possibilities.
In this note we show that the Axiom of Countable Choice is equivalent to two statements from the theory of pseudometric spaces: the first of them is a well-known characterization of uniform ...continuity for functions between (pseudo)metric spaces, and the second declares that sequentially compact pseudometric spaces are
UC
—meaning that all real valued, continuous functions defined on these spaces are necessarily uniformly continuous.
An ordered topological vector space has the countable dominated extension property if any linear operator ranging in this space, defined on a subspace of a separable metrizable topological vector ...space, and dominated there by a continuous sublinear operator admits extension to the entire space with preservation of linearity and domination. Our main result is that the strong
-interpolation property is a necessary and sufficient condition for a sequentially complete topological vector space ordered by a closed normal reproducing cone to have the countable dominated extension property. Moreover, this fact can be proved in Zermelo–Fraenkel set theory with the axiom of countable choice.
Kripke recently suggested viewing the intuitionistic continuum as an expansionin timeof a definite classical continuum. We prove the classical consistency of a three-sorted intuitionistic formal ...system IC, simultaneously extending Kleene's intuitionistic analysis I and a negative copy C◦ of the classically correct part of I, with an "end of time" axiom ET asserting that no choice sequence can be guaranteed not to be pointwise equal to a definite (classical or lawlike) sequence. "Not every sequence is pointwise equal to a definite sequence" is independent of IC. The proofs are by𝒞realizability interpretations based on classical 𝜔-models ℳ = (𝜔, 𝒞) of C◦.
A metric space is
Totally Bounded
(also called preCompact) if it has a finite
ε
-net for every
ε
> 0 and it is
preLindelöf
if it has a countable
ε
-net for every
ε
> 0. Using the
Axiom of Countable ...Choice
(
CC
), one can prove that a metric space is topologically equivalent to a Totally Bounded metric space if and only if it is a preLindelöf space if and only if it is a Lindelöf space. In the absence of
CC
, it is not clear anymore what should the definition of preLindelöfness be. There are two distinguished options. One says that a metric space
X
is:
preLindelöf if, for every
ε
> 0, there is a countable cover of
X
by open balls of radius
??
(Keremedis, Math. Log. Quart.
49
, 179–186
2003
);
Quasi Totally Bounded if, for every
ε
> 0, there is a countable subset
A
of
X
such that the open balls with centers in
A
and radius
ε
cover
X
.
As we will see these two notions are distinct and both can be seen as a good generalization of Total Boundedness. In this paper we investigate the choice-free relations between the classes of preLindelöf spaces and Quasi Totally Bounded spaces, and other related classes, namely the Lindelöf spaces. Although it follows directly from the definitions that every pseudometric Lindelöf space is preLindelöf, the same is not true for Quasi Totally Bounded spaces. Generalizing results and techniques used by Horst Herrlich in
8
, it is proven that
every pseudometric Lindelöf space is Quasi Totally Bounded
iff Countable Choice holds in general or fails even for families of subsets of
R
(Theorem 3.5).
Programming languages with countable nondeterministic choice are
computationally interesting since countable nondeterminism arises when modeling
fairness for concurrent systems. Because countable ...choice introduces
non-continuous behaviour, it is well-known that developing semantic models for
programming languages with countable nondeterminism is challenging. We present
a step-indexed logical relations model of a higher-order functional programming
language with countable nondeterminism and demonstrate how it can be used to
reason about contextually defined may- and must-equivalence. In earlier
step-indexed models, the indices have been drawn from {\omega}. Here the
step-indexed relations for must-equivalence are indexed over an ordinal greater
than {\omega}.
The Peirce translation Escardó, Martín; Oliva, Paulo
Annals of pure and applied logic,
June 2012, 2012-06-00, Letnik:
163, Številka:
6
Journal Article
Recenzirano
Odprti dostop
We develop applications of selection functions to proof theory and computational extraction of witnesses from proofs in classical analysis. The main novelty is a translation of minimal logic plus ...Peirce law into minimal logic, which we refer to as Peirce translation, as it eliminates uses of Peirce law. When combined with modified realizability, this translation applies to full classical analysis, i.e. Peano arithmetic in the language of finite types extended with countable choice and dependent choice. A fundamental step in the interpretation is the realizability of a strengthening of the double-negation shift via the iterated product of selection functions. In a separate paper, we have shown that such a product of selection functions is equivalent, over system T, to modified bar recursion.
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