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 […]

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 […]

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: […]

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 […]

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?

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 […]

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. […]

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 < […]

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, […]

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?

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