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Results tagged with set-theory
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user 26809
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.
3
votes
2
answers
294
views
Hyperfinite set containing the reals, with specified upper bound on internal cardinality?
Is this true? For any hyperfinite $n$ that isn't finite, there is a hyperfinite set $A$ such that $\mathbb R \subset A$ and $|A|\le n$ (that's the crucial part, of course)? Intuitively it seems righ …
- 2,555
9
votes
2
answers
865
views
Is it consistent with ZFC that some translation-invariant extension of Lebesgue measure assi...
It is consistent with ZFC (but not ZFC+CH, of course) that there is a subset $A$ of nonzero outer Lebesgue measure that has cardinality less than $c$. There will then be an extension of Lebesgue meas …
- 2,555
3
votes
Is it consistent with ZFC that some translation-invariant extension of Lebesgue measure assi...
For what it's worth, here is a slightly more elementary proof.
If there is a translation-invariant measure on $\mathbb R$ assigning a non-zero value to some set of cardinality less than $\frak c$ and …
- 2,555
9
votes
1
answer
2k
views
A set of positive measure with cardinality less than that of the continuum?
Is it consistent with ZFC that there is a subset of $[0,1]$ whose cardinality is less than that of the continuum but which has positive Lebesgue measure?
Obviously not given CH. And, given ZFC, ther …
- 2,555
1
vote
Consequences of ZF+"all subsets of reals are Lebesgue measurable"
One can multiply examples, of course. For the amusement of anyone interested, here are three more consequences of there being no nonmeasurable sets (also of the Banach-Tarski decomposition failing).
…
- 2,555
20
votes
0
answers
449
views
Hahn-Banach and the "Axiom of Probabilistic Choice"
Stipulate that the Axiom of Probabilistic Choice (APC) says that for every collection $\{ A_i : i \in I \}$ of non-empty sets, there is a function on $I$ that assigns to $i$ a finitely-additive probab …
- 2,555
10
votes
Unique existence and the axiom of choice
In A definable nonstandard model of the reals, Kanovei and Shelah surprisingly managed to prove the existence of a ZFC-definable (i.e., specifiable via an explicit ZFC construction) nonstandard model …
- 2,555
3
votes
Strength of some claims about finitely additive measures on infinite sets?
A quick note that (2) doesn't follow from ZF (assuming ZF is consistent) or even ZF+DC.
(2)+Countable Choice for Finite Sets (CC(fin)) implies that every uncountable set has a non-principal (finite …
- 2,555
6
votes
4
answers
609
views
Strength of some claims about finitely additive measures on infinite sets?
Assume ZF. Consider the claim:
(1) For any infinite set $\Omega$, there is a finitely additive probability measure $\mu:2^\Omega\to[0,1]$ with $\mu(A) = 0$ whenever $|A|<|\Omega|$.
Then (1) is impl …
- 2,555
10
votes
0
answers
755
views
Full conditional probabilities and versions of AC?
A probability is a finitely additive measure on a boolean algebra with total measure $1$.
A function $P:\scr B \times (\scr B - \{ 0 \})$ is a full conditional probability on $\scr B$ (for a boolean …