Articles of axiom of choice

Infinite dimensional vector space. Linearly independent subsets and spanning subsets

This question is a follow up to this question: The dimension of the real continuous functions as a vector space over $\mathbb{R}$ is not countable? I realized that the answer I accepted used an implicit assumption: that if $V$ is a vector space and $S$ is a spanning subset and $L$ is a linearly independent […]

Context for Russell's Infinite Sock Pair Example

I wanted to verify the following considerations on the context of Russell’s infinite sock pair conundrum. The conundrum pointed out that a rule for choosing from pairs of shoes is possible a-priori. For indistinguishable socks, such rule is not possible a-priori, and has to be assumed. Thus, in Set Theory there are two major avenues […]

Does the principle of schematic dependent choice follow from ZFCU?

Let ZFCU be ZFC modified in the usual way to allow for urelements but without an axiom stating that there is a set of all urelements. Let the principle of Schematic Dependent Choice (SDC) be: $\forall x\exists y\phi(x, y) \to \forall x\exists f(\mbox{dom}(f) = \omega \wedge f(0) = x \wedge \forall n\in\omega: \phi(f(n), f(n+1)))$ Question: […]

Hall's theorem vs Axiom of Choice?

From Wikipedia Let $S$ be a family of finite sets, where the family may contain an infinite number of sets and the individual sets may be repeated multiple times. A transversal for $S$ is a set $T$ and a bijection $f$ from $T$ to $S$ such that for all $t$ in $T$, $t$ is a […]

Isomorphic Free Groups and the Axiom of Choice

When I read about free group, the proof which concerns about two free groups $F(X)$ and $F(Y)$ are isomorphic only if $\operatorname{card}(X) = \operatorname{card}(Y)$ has a sentence going as follows: $|M(X \cup X^{-1})|=|X \cup X^{-1}|=|X|$, using the axiom of choice. Can someone give me more hint about this question or some references?

Dense subset of the plane that intersects every rational line at precisely one point?

It seems there should exist a non-measurable bijection $f: \mathbb{R}\rightarrow \mathbb{R}$. And thus we can obtain a non-measurable graph on $\mathbb{R}^2$ which intersects every horizontal or vertical line at precisely one point. Does anyone know how to “construct” such a map? Can it be further made into a automorphsim (w.r.t the addtive group or field […]

Can a vector subspace have a unique complement in absence of choice?

Let $V$ be a vector space (not necessarily finite dimensional and over some arbitrary field), and $W$ a proper non-zero subspace. If we assume existence of bases, it is easy to show that $W$ can be complemented, in a necessarily non-unique way. The non-uniqueness has been proved in Unique complement of a subspace, for example. […]

$F$ is a free abelian group on a set $X$ , $H \subseteq F$ is a free abelian group on $Y$, then $|Y| \leq |X|$

I am confused by the proof a proposition: $F$ is a free abelian group on a set $X$ and $H$ is a subgroup of $F$, then $H$ is free abelian on a set $Y$, where $|Y| \leq |X|.$ The proof is: Let $X$ be well-ordered in some fashion, say as $\{ x_{\alpha} | \alpha < […]

Solovay's Model and Choice

Reference; Foundation for analysis without axiom of choice? Please let me know if I’m misunderstanding something and I hope you explain this with relatively easy words. I am eager to learn, but I have no idea what forcing is like, so i couldn’t understand this properly from past posts. Solovay’s model says that “In ZF, […]

Can one construct an algebraic closure of fields like $\mathbb{F}_p(T)$ without Zorn's lemma?

I have heard that an algebraic closure of $\mathbb{Q}$ can be constructed without Zorn’s lemma and so can an algebraic closure of a finite field $\mathbb{F}_p$. What about $\mathbb{F}_p(T)$? Do there exist fields for which it is not possible to conclude that that there is an algebraic closure without Zorn’s lemma?