All Questions
10 questions
18
votes
2
answers
1k
views
What notable theorems cannot be automatically proven without choice using Shoenfield absoluteness?
There have been a couple of recent questions, here and here, regarding the role of the axiom of choice in real-analytic results with applicability to general relativity. This lead me to look at some ...
- 12.5k
23
votes
2
answers
1k
views
Statements in differential geometry independent from ZFC
It is well known that some problems in functional analysis and in general topology are independent from ZFC: to name a few, Kaplansky's conjecture, the existence of outer automorphisms of the Calkin ...
34
votes
2
answers
3k
views
Ur-elemental surprises
For most of my (mathematical) life, I believed that there was really no essential difference between set theory without urelements and set theory with urelements. However, while that may be true in ...
- 20.5k
64
votes
15
answers
7k
views
Unnecessary uses of the axiom of choice
What examples are there of habitual but unnecessary uses of the axiom of
choice, in any area of mathematics except topology?
I'm interested in standard proofs that use the axiom of choice, but where
...
12
votes
6
answers
750
views
Conditions equivalent to finiteness
We've all probably come across some conditions that naturally imply finiteness, or are equivalent to it. For ZFC examples:
A set $X$ can be ordered in such a way that the ordering is well-founded and ...
26
votes
9
answers
8k
views
Why are proofs so valuable, although we do not know that our axiom system is consistent? [closed]
As a person who has been spending significant time to learn mathematics, I have to admit that I sometimes find the fact uncovered by Godel very upsetting: we never can know that our axiom system is ...
143
votes
12
answers
30k
views
Solutions to the Continuum Hypothesis
Related MO questions: What is the general opinion on the Generalized Continuum Hypothesis? ; Completion of ZFC ; Complete resolutions of GCH How far wrong could the Continuum Hypothesis be? When was ...
- 24.7k
96
votes
16
answers
34k
views
Most 'unintuitive' application of the Axiom of Choice?
It is well-known that the axiom of choice is equivalent to many other assumptions, such as the well-ordering principle, Tychonoff's theorem, and the fact that every vector space has a basis. Even ...
10
votes
5
answers
5k
views
Examples of inductive proofs that can be generalized by transfinite induction
Hello. I am currently searching for some nice examples of proofs by induction in the finite case, that can be generalized to the infinite case using transfinite induction (and dont become trivial ...
298
votes
34
answers
53k
views
What are some reasonable-sounding statements that are independent of ZFC?
Every now and then, somebody will tell me about a question. When I start thinking about it, they say, "actually, it's undecidable in ZFC."
For example, suppose $A$ is an abelian group such ...