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Results tagged with set-theory
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user 15570
forcing, large cardinals, descriptive set theory, infinite combinatorics, cardinal characteristics, forcing axioms, ultrapowers, measures, reflection, pcf theory, models of set theory, axioms of set theory, independence, axiom of choice, continuum hypothesis, determinacy, Borel equivalence relations, Boolean-valued models, embeddings, orders, relations, transfinite recursion, set theory as a foundation of mathematics, the philosophy of set theory.
47
votes
10
answers
5k
views
What is the most "concrete-feeling" equivalent formulation of the Continuum Hypothesis that ...
There are many equivalent formulations of the Continuum Hypothesis, but I think the most standard one is that
there is no infinite cardinality lying strictly between the cardinality of the natural nu …
17
votes
2
answers
1k
views
Is it consistent with ZFC that the real line is approachable by sets with no accumulation po...
Let $P$ denote the following proposition:
There exists a set $S$ of subsets of $\mathbb{R}$ such that
$S$ is totally ordered by inclusion;
each member of $S$ has no accumulation points;
the union of …
- 708
11
votes
3
answers
933
views
Some "axiom of choice" and "dependent choice" issues
I am probably about to ask some fairly basic questions, and yet I have found it quite hard to find the answers to these.
If I understand correctly, mathematicians tend to be quite happy working with …
- 708
9
votes
1
answer
601
views
Is it a theorem of ZF that a non-empty countable Cartesian product of finite non-singleton s...
I think that, without countable choice, I can prove quite easily that for a sequence $(S_n)_{n \in \mathbb{N}}$ of sets $S_n$ with $|S_n|=2$, the Cartesian product $\,\prod_{n=1}^\infty S_n\,$ is eith …
- 708
8
votes
3
answers
690
views
Is "the purely probabilistic version of Freiling's axiom of symmetry" disprovable in ZFC?
I'm trying to pinpoint the "intuitive argument" for Freiling's Axiom of Symmetry. It's meant to be a "probabilistic" argument, so thinking about what seems to me to be the probabilistic intuition, it …
- 708
3
votes
1
answer
314
views
Can a weaker version of the Hausdorff paradox be proved without AC?
The Hausdorff paradox is an incredibly counter-intuitive consequence of the axiom of choice; it is also important for demonstrating the non-existence, under AC, of a rotation-invariant measure on the …
- 708
2
votes
Can a weaker version of the Hausdorff paradox be proved without AC?
Following the suggestion in the first comment below my question (and with the help of the second comment), I can give an example of a scenario that is "even worse" than what I requested, where $A \cup …
- 708
2
votes
1
answer
280
views
Are the jumps of a càdlàg function "summable"?
This question is motivated by the question https://math.stackexchange.com/questions/4644235/ on Math Stack Exchange. First, I need to define a notion of transfinite summability that I have not seen el …
- 708
1
vote
Are the jumps of a càdlàg function "summable"?
Anthony Quas has provided an example of a càdlàg function for which the jumps are not summable:
As in https://math.stackexchange.com/questions/10257/, for any $S \subset (0,1]$ and $(x_t)_{t \in S} \i …
- 708
1
vote
2
answers
327
views
Heuristic interpretations of the PA-unprovability of Goodstein's Theorem
I've relatively recently learned about Goodstein's Theorem and its unprovability in Peano arithmetic (the Kirby-Paris Theorem). I do not have any real knowledge of formal logic; but I think I've seen …
- 708