Tagged Questions
4
votes
2answers
59 views
Are there versions of the axiom of choice that restrict the size of the factors?
One formulation of the axiom of choice is that an arbitrary product of nonempty sets must be nonempty. The axiom of countable choice AC$_\omega$ is known to be strictly weaker than AC, but still ...
9
votes
1answer
81 views
Proof of a basic $AC_\omega$ equivalence
On Wikipedia it is mentioned that "... in order to prove that every accumulation point $x$ of a set $S\subseteq \mathbf R$ is the limit of some sequence of elements of $S\setminus \{x\}$, one uses (a ...
4
votes
1answer
110 views
Books on Axiom of Dependent Choices?
Are there any books about the axiom of dependent choice? There are books about the axiom of choice, e.g. Herrlich or Jech. But I can't seem to find any on the axiom of dependent choices.
As far as I ...
3
votes
2answers
80 views
Cohen and the axiom of choice
The wikipedia article on Paul Cohen mentions that:
Cohen is noted for developing a mathematical technique called forcing,
which he used to prove that neither the continuum hypothesis (CH), nor
...
8
votes
1answer
309 views
Constructing a subset not in $\mathcal{B}(\mathbb{R})$ explicitly
While reading David Williams's "Probability with Martingales", the following statement caught my fancy:
Every subset of $\mathbb{R}$ which we meet in everyday use is an element of Borel ...
30
votes
2answers
1k views
Is there an explicit isomorphism between $L^\infty[0,1]$ and $\ell^\infty$?
Is there an explicit isomorphism between $L^\infty[0,1]$ and
$\ell^\infty$?
In some sense, this is a follow-up to my answer to this question where the non-isomorphism between the spaces $L^r$ ...
20
votes
5answers
685 views
Axiom of choice - to use or not to use
I was wondering if there are examples of results in mathematics that were first proven using axiom of choice and later someone found a proof of the result without using the axiom of choice.
