# axiom-of-choice's questions - English 1answer

334 axiom-of-choice questions.

### 11 Is $\mathbb{R}$ a $\mathbb{C}$-module without AC?

Assuming ZFC. We can make $(\mathbb{R},+)$ into a nontrivial (scalar multiplication is not identically zero) $\mathbb{C}$-module. Now my questions are? 0.Is it consistent with $ZF$ that $\mathbb{R}$ ...

### 1 The Rise and Fall of Dictators & How it Depends on Our Choice

This question is loosely inspired by the following paper of Shelah on Arrow property in which he answered a question of Gil Kalai affirmatively. Shelah, Saharon, On the Arrow property. Adv. in Appl. ...

### 6 Countable (?) dependent choice

In some circumstances I've been using a form of choice over the first uncountable ordinal knowing a priori that only a countable number of choices were going to be made (without any a priori upper ...

### 10 Kaplansky's theorem and Axiom of choice

Kaplansky in his paper titled by Projective Modules gave an important and essential theorem as follow: Theorem: Let $R$ be a ring, $M$ an $R$-module which is a direct sum of (any number of) countably ...

### -4 Are there paradoxes in ZF + (the Axiom of Choice for finite sets)? [closed]

1 answers, 377 views set-theory lo.logic axiom-of-choice
The good intuitive reasons that the Axiom of Choice (AC) for arbitrary sets-- that a function f with f(i) in S(i) for each i in {i}, for any non-empty sets {i} and all S(i), exists-- doesn't follow ...

### 14 Does every model of ZF-foundation have an extension, with no new well-founded sets, where every set is bijective with a well-founded set?

This question follows up on an issue arising in Peter LeFanu Lumsdaine's nice question: Does foundation/regularity have any categorical/structural consequences, in ZF? Let me mention first that my ...

### 3 Ascending chain of vertex-transitive graphs

For any set $X$ we set $[X]^2 = \big\{\{x,y\}: x, y\in X\text{ and } x\neq y\big\}$. Let $G=(V,E)$ be a simple, undirected graph, and suppose ${\cal V}$ is a collection of subsets of $V$ such that ...

### 12 Subset of the plane that intersects every line exactly twice

In a comment to this question, Tim Gowers remarked that using the axiom of choice, one can show that there exists a subset of the plane that intersects every line exactly twice (although it has yet to ...

### 7 Totally bounded spaces and axiom of choice

Wikipedia article on totally bounded spaces states "... the completion of a totally bounded space might not be compact in the absence of choice." Where is the axiom of choice used, and do you need it ...

### 1 Selection in a small category

0 answers, 104 views ct.category-theory axiom-of-choice
I came across the following problem. I do not know if these notions are known (I would actually be interested to know), so the names might not be the canonical ones. Given a small category $C$, a ...

### 15 Cauchy real numbers with and without modulus

In constructive mathematics there are many possible inequivalent definitions of real numbers. The greatest variety seems to be in Dedekind-style approaches: in addition to "the" Dedekind real numbers ...

### 9 Forcing, cuts, and Dedekind-finite cardinalities

Tl;dr version: there are two natural classes of cuts in the nonstandard model of arithmetic consisting of the Dedekind-finite sets (if, in fact, they constitute such a model); both these classes are ...

### 6 Consistency of a non-measurable set of reals when the continuum cannot be well-ordered

1 answers, 417 views set-theory axiom-of-choice
Can it be shown, on the assumption that $ZF$ is consistent, that there is a model of $ZF$ in which the reals cannot be well-ordered but there does exist a set of reals which is not Lebesgue measurable?...

### 34 Objects which can't be defined without making choices but which end up independent of the choice

It happens a lot of times that when one defines a new object (ring, module, space, group, algebra, morphism, whatever) out of given data, one first chooses some additional structure. And sometimes (...

### 7 Are these large cardinals properties equivalent?

Consider the three following large cardinal axioms: there exists a nontrivial elementary embedding $j:V\to V$. there exists a n.e.e. $j:V\to M$ such that $M^{j^\omega(crit(j))}\subseteq M$. there ...

### 16 Do all countable $\omega$-standard models of ZF with an amorphous set have the same inclusion relation up to isomorphism?

0 answers, 424 views lo.logic set-theory axiom-of-choice
In my recent paper with Makoto Kikuchi, J. D. Hamkins and M. Kikuchi, The inclusion relations of the countable models of set theory are all isomorphic. manuscript under review. (arÏ‡iv) we proved ...

### 2 Why doesn't choice imply global choice (in NBG)?

3 answers, 490 views set-theory axiom-of-choice
I thought ZFC proved the existence of an inductive well-ordering that is itself a set for any stage of V. NBG with only the regular AC should then prove/assert the existence of a class R of ordered ...

### 3 Choice sets and the axiom of choice

1 answers, 346 views set-theory lo.logic axiom-of-choice