# Tagged Questions

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### Choice function to obtain all elements of an infinite set

I'm following a proof of "Axiom of Choice implies Well Ordering Theorem" (from a not-well-known book we use for class), and the author uses this procedure: First, he takes any non-empty set $X$ and ...
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### Proof of one instance of the Axiom of Choice from another.

I'm trying to show (i) $\implies$ (ii): (i) For any relation $R$, there exists a function $H\subseteq R$, with $\newcommand{\dom}{\mathrm{dom} \ } \dom H = \dom R$. (ii) For any set $I$ and any ...
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### Proving equivalence of Axiom of Choice

I am working on the following question concerning the axiom of choice and one of its many equivalences. Advice as to whether I am on the right track would be appreciated. As a preface, I have looked ...
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### Existence of a Set Function Axiom of Choice

I have the following problem. Let $A$ be a set and $B\neq\emptyset$ be a proper subset. Prove the existence of a function $f:A\to A$ such that $f\circ f=f$ and $\text{im}~f=B$. In the case where $A$ ...
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### Does the Cartesian product of an infinite family have all the elements we expect?

Given the axiom of choice, we know that the Cartesian product of an infinite family of non-empty sets is non-empty. However, this doesn't tell us whether the Cartesian product contains every element ...
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### Disjoint $AC$ equivalent to $AC$

I want to show that these two forms of $AC$ are equivalent: $(1)$ For each collection of nonempty sets $X$ there is a choice function. $(2)$ For each collection of pairwise disjoint nonempty sets $X$ ...
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### Axiom of choice variants [duplicate]

Is there any good book where the equivalence of AC to the statement "Any surjection has a right inverse" is proved (and maybe other equivalences)? I could do $AC \Rightarrow$ "Any surjection has a ...
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### Proof of: $X$ is finite $\iff X$ is Tarski-finite

I am self-studying Horst Herrlich, Axiom of Choice (Lecture Notes in Mathematics, Vol. 1876). In the fourth chapter, he deals with different definitions of finite set. Here is the classical one: ...
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### Is the Axiom of Choice necessary to prove $\mathbb R \approx \mathcal P(\omega)$?

I am self-studying Horst Herrlich, Axiom of Choice (Lecture Notes in Mathematics, Vol. 1876), and I'm getting quite confused about cardinal arithmetic without AC. Here (Which sets are well-orderable ...
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### Why do we need the axiom of choice in showing the non-emptiness of an infinite Cartesian product

Given $I$ a set of indexes and $X_i$ a set of topological spaces, define The Cartesian product: $\prod_{i \in I}X_i = \{ f:I \rightarrow \bigcup X_i | f(i) \in X_i \}$ I have read that we need ...
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### Is the Axiom of Choice restricted to $\Bbb R$ independent of ZF? [duplicate]

This is really a yes-no question, and I am hundred percent certain that the answer is "yes". I simply have not found it written directly anywhere. True, the Axiom of Choice is independent of ZF if we ...
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### Bernstein sets, Well-Ordering theorem vs Axiom of Choice

In the construction of Bernstein sets (see here), is it necessary to use the well-ordering theorem? Why can't you just use the Axiom of Choice to pick two points?
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### Proving Equivalence of Two Version of Axiom of Choice

I am working on an assignment that requires proving the equivalence of two versions of the axiom of choice. (1st form): For any relation $R$, there is a function $H \subseteq R$ with dom $H =$ dom ...
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### Theorem 4.15 page 22, Real and Abstract Analysis, Hewitt and Stromberg

Following is Theorem 4.15 from Real and Abstract Analysis, Hewitt and Stromberg Theorem: Every infinite set has a countably infinite subset. Proof: Let $A$ be a infinite set. We show by induction ...
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### factorial of infinite Cardinals

Let $S_A$ be set of all bijections over $A$ such that $Card(A)=\kappa$. Define foctorial as $\kappa!:=Card(S_A)$. Show that if $\kappa$ is infinite, then : $\kappa!=2^\kappa$ First, I've ...
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### If $|X|<|Y|$ then $|Y|=|Y-X|$ (with $Y$ infinite)

Like the title says, I would like to prove that if $|X|<|Y|$ then $|Y|=|Y-X|$. (with $Y$ infinite) I know I have to use the axiom of choice, but I've no idea about how to proceed. Any help is ...
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### If $A$ is D-Infinite then $|P_{\infty}(A)|=|P(A)|$

I want to prove that if $A$ is a D-Infinite set (i.e. it contains a countable subset $X$), then the set of the infinite parts of $A$, $P_{\infty}(A)$ has the same cardinality of $P(A)$. I know that ...
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### Zorich's misinterpretation of “Axiom of Choice”?

I'm reading Zorich'es "Mathematical Analysis I", Ed 4, 2004, and wonder if this is a trifle misinterpretation of "Axiom of Choice". Ch 1.4 "Supplementary Material" says: 8°. (A x i o m o f c h o i ...
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### What is wrong with the following proof saying Zorn's lemma implies Hausdorff maximum principle?

I am reading 'Topology' by J.R. Munkres's first chapter on set theory. In the exercises 5-7 on page 72 he asks the reader to show that Zorn's lemma implies Hausdorff maximum principle via the ...
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### Does defining the closure of a set as the intersection of all closed set that contain it requires the axiom of choice?

Given a set $S$, the closure of $S$ is sometimes defined as the intersection of all the closed sets that contain it. This type of argument is pervasive in mathematics when one want to construct the ...
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### Confusion regarding one formulation of the Axiom of Choice.

One formulation of the Axiom of Choice is: The Cartesian product of non-empty sets is always non-empty. Cartesian product is defined as making "every possible pair" between elements of two sets. ...
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### Why can't you pick socks using coin flips?

I'm teaching myself axiomatic set theory and I'm having some trouble getting my head around the axiom of choice. I (think I) understand what the axiom says, but I don't get why it is so 'contentious', ...
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### Do we need AC to prove Principle of Dependent Choices

For any nonempty set $X$ and any entire binary relation $R$ on $X$, there is a sequence $(x_n)$ in $X$ such that $x_nRx_{n+1}$ for each $n \in \mathbb{N}$. (Here an entire binary relation on $X$ is ...
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### Assuming the axiom of choice, how to find explicit group structure of a given set

Let us assume the axiom of choice. This is equivalent to every nonempty set having group structure. My question is, given some nonempty set, can we define the binary operator in a constructive way ...
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### Cardinality of union of pairwise disjoint elements needs choice?

If there is an indexed family $(i\mapsto A_i)_{i\in I}$ of pairwise disjoint sets $A_i$, why do we need choice to show that $$\left|\textstyle{\bigcup_{i\in I}A_i}\right| = \sum_{i\in I}|A_i|?$$ It ...
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### Zorn's lemma problem

Let $n$ be a positive natural number. Prove using Zorn's lemma that there is a set A of points in the plane that satisfies: 1. Any line in the plane does not contain $n+1$ points of A. 2. For every ...
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### Jech's axiom of choice, problem 2 first chapter

I have a question stemming from Jech's book on the axiom of choice., chapter 1 exercise 2. We are asked to show that a family of sets of natural numbers has a choice function. Now the version of the ...
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### Plausibility argument for Zorn's Lemma

In "Mathematical Physics" by Robert Geroch, the following 'plausibility argument' is given for Zorn's Lemma [If every totally ordered subset of a partially ordered set $S$ is bounded above, $S$ has a ...
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### How should one think about results that depend on AC?

I just encountered this: "(Theorem of A. H. Stone) Every metric space is paracompact... Existing proofs of this require the axiom of choice... It has been shown that neither ZF theory nor ZF ...
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### The Axiom of choice

I'm a little lost with this proof: If every set is equipotent to an ordinal, then we have the axiom of choice And I want to know if someone can help or maybe give me a hint of how to proceed.
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### $\aleph(X) < \aleph(\mathcal{P}(\mathcal{P}(\mathcal{P}(X))))$

Prove: $\aleph(X) < \aleph(\mathcal{P}(\mathcal{P}(\mathcal{P}(X))))$ With $W(X)=\{\langle A,R\rangle: A \in \mathcal{P} (X),R \in \mathcal{P}(X \times X)$ and $R$ wellorders $A \}$ And ...
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### What does the Axiom of Choice have to do with right inversibility?

I have encountered an exercise that asks to prove that, these two statements are equivalent: every surjective function has a right inverse. Axiom of choice. Definition: Given a function $f$, we ...
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### well ordering principle implies Zorn's Lemma

In here: Proving that well ordering principle implies Zorn's Lemma. I asked how to finish a proof of this statement. After a few helpful remarks, I think I have managed to finish it. What do you ...
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### Proving that well ordering principle implies Zorn's Lemma.

I am trying to prove that well ordering principle implies Zorn's Lemma. I think that I'm close but don't quite know to make the last step of my proof. Here is what I wrote so far: Given that on every ...
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### The union of a countable set of countable sets is countable

Here is the proof provided in my lecture notes: Let $A = \{B_n | n < \omega =\mathbb{N}\}.$ Assume each $B_n$ is countable. For each $n < \omega,$ let $E_n$ be set of all bijections between ...
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### Do we need the axiom of choice in here?

Axiom of choice: Given $\mathbb{F}$ is a set of non-empty sets. Then, there is a function $f$ with $\text{Dom}(f)=\mathbb{F}$ such that, for every $A \in \mathbb{F}$, $f(A) \in A$. The ...
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### Proving Zermelo's Theorem implies the Axiom of Choice

I thought it would be fun to try and prove this. It turned out to be pretty simply and maybe too simple so I was wondering if anybody could verify if this proof is correct. Suppose we have a family ...
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