1
vote
2answers
97 views

Set Theory and Zorn Lemma

Prove that there is a set $B\subseteq P(\mathbb N)$ such that for all $n\in \mathbb N:\mathbb N- n\in B$, every finite intersection of elements in B is not empty and for all $C\subseteq\mathbb N$ such ...
2
votes
1answer
114 views

Logic: Teichmüller-Tukey Lemma and the Axiom of Choice

How can you proof that the Teichmüller-Tukey Lemma (which says that if $S$ is nonempty and of finite character, $S$ contains a maximal element with respect to the subset ordering), implies the Axiom ...
2
votes
3answers
543 views

Prove that Every Vector Space Has a Basis

My textbook extended the following proof to show that every vector space, including the infinite-dimensional case, has a basis. Condition: $S$ is a linearly independent subset of a vector space ...
3
votes
3answers
127 views

Equivalence with the axiom of choice

This is a problem from Tao's Analysis I. We are asked to show that the axiom of choice is equivalent to the statement that for any sets $A$ and $B$ for which a surjection $g:B\to A$ exists, an ...
2
votes
3answers
88 views

Cardinals of set operations without AC

Given info: $|A|=\mathfrak{c}$ , $|B|=\aleph_0$ in ZF (no axiom of choice). Prove: $|A\cup B|=\mathfrak{c}$ If $B \subset A\implies|A \backslash B|=\mathfrak{c}$? I have found several places ...
6
votes
1answer
97 views

Prove the existence of a set in the Euclidean plane

I got stuck on the following problem. Prove that there exists a subset $A$ of $\mathbb{R}^2$ such that every line in $\mathbb{R}^2$ goes exactly through two points in $A$. I know that I should apply ...
5
votes
1answer
432 views

On every infinite-dimensional Banach space there exists a discontinuous linear functional.

On every infinite-dimensional Banach space there exists a discontinuous linear functional. Assuming the axiom of choice, every vector space has a basis. With an infinite basis, I can define on a ...
7
votes
2answers
415 views

axiom of choice: cardinality of general disjoint union

I have this exercise involving the axiom of choice, but I don't understand where it's needed: Let $(X_i)_{i \in I}$ and $(Y_i)_{i \in I}$ be pairwise disjoint sets with $|X_i| = |Y_i|$. Prove, using ...
3
votes
2answers
750 views

Cardinality of union of ${{\aleph }_{0}}$ disjoint sets of cardinality $\mathfrak{c}$

I have a home work question which is: " what is the cardinality of the union of ${{\aleph }_{0}}$ disjoint sets of cardinality $\mathfrak{c}$?" I believe somehow we can get to: cardinality = ...
5
votes
3answers
860 views

Proving any infinite set has a denumerable subset with the Axiom of Choice

Derive from the axiom of choice that any infinite set contains a denumerable subset
12
votes
1answer
2k views

The cardinality of a countable union of countable sets, without the axiom of choice

One of my homework questions was to prove, from the axioms of ZF only, that a countable union of countable sets does not have cardinality $\aleph_2$. My solution shows that it does not have ...