# Tagged Questions

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### Can all theorems of $\sf ZFC$ about the natural numbers be proven in $\sf ZF$?

I know a proof of Hindman's theorem that uses ultrafilters on the natural numbers, and ultimately, the axiom of choice. But the theorem itself is essentially a combinatorial property of the natural ...
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### An equivalence of AC

I have to prove the following: In $ZF^-$ the axiom of choice implies: For every set X there exist $Y \subseteq \bigcup X$ such that: Y has at most one element in common with each of X Y is maximal ...
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### Existence of a real uncountable $\aleph_{\alpha}$ without $\mathsf{AC}$

Set theory (Jech) $\text{p.}\;27:$ It is an open problem whether one can prove without the axiom of choice that there exists a regular uncountable $\aleph_{\alpha}\;($the informed guess is that ...
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### Book/Books leading up to the the axiom of choice?

I am familiar with the axioms of ZF set theory and some basic uses of them to completely formally construct more complex objects such as natural numbers etc. However I have pretty much no background ...
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### Unions and the axiom of choice.

Is the following equivalent to the axiom of choice? Let $A = \{a_i: i \in I\}$ be a collection of pairwise-disjoint non-empty sets indexed by $I$. Similarly, let $B = \{b_i : i \in I \}$. Further ...
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### Proving there is a sequence convergent to a limit point of a set without axiom of countable choice?

Often, we use a construction like this: Given a subset $A$ of a metric space and its limit point $a$, we know that for every $\epsilon > 0$ there is another point $x$ different from $a$ ...
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### surjective map and cardinality

I work in $\mathsf{ZF}$(without the axiom of choice). Let $A, B$ be sets such that $\left| A \right |$ and $\left|B \right |$ are both defined and let $f \colon A \to B$ a surjective function. Can I ...
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### Is the well-ordering in the well-ordering axiom required to be definable?

It is well-known that Axiom of Choice is equivalent to the statement that every set can be well-ordered. Now, to show that $M\models AC$, is it sufficient to show that there exists some well-order of ...
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### Axiom of choice needed? finite sets

I have to show that the following 2 quotes are equivalent: a) The set A is finite b) Every injective function from A to A is also surjective The direction a) ==> b): I've worked with induction on the ...
2answers
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### The Continuum Hypothesis & The Axiom of Choice

Does anyone here know of a reference to an analysis on a proposed relationship between The Continuum Hypothesis and The Axiom of Choice?
1answer
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### Dependent choice does not imply “the reals are well-ordered”; citation?

As silly as this sounds, I can't find a proof that the axiom of dependent choice (DC) does not imply that the reals are well-orderable. My memory is that this is a fairly early result in the history ...