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Results tagged with axiom-of-choice
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user 3106
An important and fundamental axiom in set theory sometimes called Zermelo's axiom of choice. It was formulated by Zermelo in 1904 and states that, given any set of mutually disjoint nonempty sets, there exists at least one set that contains exactly one element in common with each of the nonempty sets. The axiom of choice is related to the first of Hilbert's problems.
7
votes
Implicit uses of Countable or Dependent Choice
Halmos is not alone in proving the countable additivity of Lebesgue measure without explicitly mentioning (countable) choice. I have the third edition of H. L. Royden's Real Analysis in front of me. I …
8
votes
Objects which can't be defined without making choices but which end up independent of the ch...
A possible example of what you're looking for, though of a somewhat different character from the examples given so far, is [the] "algebraic closure of $\mathbb Q$." Läuchli (Auswahlaxiom in der Algeb …
4
votes
Objects which can't be defined without making choices but which end up independent of the ch...
I don't know of a way to completely formalize your question, but here is something with the same flavor that may hint at how to proceed. Blass and Gurevich defined a complexity class—or more accurate …
4
votes
Difference between ZFC and ZF+GCH
There are two distinct questions that you might be asking.
Why has the mathematical community adopted ZFC as a standard foundation and not ZF+GCH?
What mathematical and philosophical arguments can …
24
votes
Absolute Galois group, number theory and the Axiom of Choice
In the absence of the axiom of choice, it is still possible to define the "usual" algebraic closure of $\mathbb{Q}$ because you can just explicitly enumerate all polynomials with integer coefficients. …
- 82.7k
22
votes
What can be preserved in mathematics if all constructions are carried out in ZF?
I agree with Asaf Karagila that the question as literally stated is a bit too sprawling, but you might want to start with Simpson's book, Subsystems of Second-Order Arithmetic. Its goals aren't the s …
- 82.7k
6
votes
Unnecessary uses of the axiom of choice
The highly-upvoted, accepted answer (by Theo Johnson-Freyd) to another MO question, Why worry about the axiom of choice?, points out that the usual proof of the Poincaré–Birkhoff–Witt theorem assumes …
14
votes
Does the Axiom of Choice (or any other "optional" set theory axiom) have real-world conseque...
Before answering your question about the axiom of choice, let me take another set-theoretic axiom: "There exists an inaccessible cardinal." This axiom implies that ZFC is consistent. One could argue …
- 82.7k
37
votes
Axiom of choice, Banach-Tarski and reality
The other answers don't seem to have said much about why the axiom of choice is widely regarded as plausible. Let me try to address that question.
First let's dispose of some non-reasons. In respon …
- 82.7k