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Results tagged with set-theory
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user 41274
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.
9
votes
1
answer
1k
views
Uncountable disjoint closed coverings of $[0,1]$
It is well known that the unit interval $[0,1]$ cannot be decomposed as a countable union of pairwise disjoint closed (nonempty) subsets. See for instance this math.stackexchange question. The proof u …
- 44.4k
13
votes
1
answer
859
views
Ultrafilter theorem and translation invariant measures
The usual Vitali construction of a non-Lebesgue measurable set generalizes to a proof that there are no (non-trivial) translation invariant measures on $\mathcal P\mathbb R$.
On the other hand, there …
- 41.8k
3
votes
1
answer
416
views
Representation of meager sets in Cohen extensions
Let $M$ be a transitive model of ZFC and $c\in {}^\omega2$ a Cohen real over $M$. Let $A$ be a meager Borel subset of $^\omega2$ in $M[c]$. I would like to prove that there exists a meager Borel set $ …
CommunityBot
- 1
6
votes
1
answer
374
views
Iteration of random reals
Consider two random reals $x, y$ over a transitive model $V$ of ZFC. More specifically, if $\mathcal C^V={}^\omega2$ is the Cantor space, composing the canonical homeomorphism with the projections $\m …
CommunityBot
- 1
2
votes
1
answer
118
views
Generic sections of non-null sets are non-null
Consider the Cantor space $\mathcal C={}^\omega2$ with the usual product measure, and let $r$ be a random real (over a transitive model $V$ of ZFC). Let $B\subset \mathcal C^V\times\mathcal C^V$ a Bor …
- 24.8k
3
votes
1
answer
303
views
Invariants of category in Polish spaces
Consider the invariants appearing in the Cichoń's diagram: $add(\mathcal I)$, $cov(\mathcal I)$, $non(\mathcal I)$, $cof(\mathcal I)$, where $\mathcal I$ is either the ideal of null sets for the Lebes …
- 18.5k
6
votes
2
answers
768
views
A question on rank-to-rank embeddings
Consider a non-trivial elementary embedding $j:V_\lambda\to V_\lambda$ and, for each $A\subset V_\lambda$, set $j(A)=\bigcup_{\delta<\lambda}j(A\cap V_\delta)$.
In Implications between strong large c …
CommunityBot
- 1
4
votes
1
answer
293
views
The GCH in a reverse Easton support iteration
I am trying to understand the proof that the GCH can first fail at a weakly compact cardinal. We assume the GCH and that there exists a weakly compact cardinal $\kappa$, and we construct a reverse Eas …
- 236k
7
votes
1
answer
255
views
Generic filters of inverse limits
Maybe this doubt is silly, but I do not understand the final step of the proof of Lemma 5.2 in Hamkins' paper Fragile measurability, Journal of Symbolic Logic 59 (1994) 262-282.
There, $\mathbb P_\la …
- 18.6k
10
votes
1
answer
450
views
Erdős cardinals and $0^\sharp$
It is well-known that if the Erdős cardinal $\kappa(\omega_1)$ exists, then $0^\sharp$ exists, but what if $\kappa(\lambda)$ exists for a limit ordinal $\omega_1^L\leq \lambda<\omega_1$? Does this sti …
- 5,112
4
votes
1
answer
520
views
Iterated ultrapowers of L
If there exists a measurable cardinal, we can generate a sequence of iterated ultrapowers $\{Ult_U^\alpha(V)\}_{\alpha\in ON}$. If $0^\sharp$ exists, i.e. if there exists an elementary embedding $j:L\ …
- 1,688
10
votes
1
answer
343
views
On the definition of the $\alpha$-iterable cardinals
I am reading the paper Ramsey-like cardinals II by Victoria Gitman and Philip Welch (Journal of Symbolic Logic, vol. 76, no. 2. pp. 541-560, 2011) and maybe I am missing something.
According to the d …
- 2,240
11
votes
1
answer
832
views
Erdős cardinals and ineffable cardinals
In Cantor's Attic it is stated that an $\omega$-Erdős cardinal is a stationary limit of ineffable cardinals, and Jech book is given as a reference, but I cannot find this result in that book. I have s …
- 1,688
7
votes
1
answer
576
views
Does OCA imply $2^{\aleph_0}=\aleph_2$?
Is it known whether Todorcevic's Open Coloring Axiom implies $2^{\aleph_0}=\aleph_2$?
The only consistency proofs for OCA that I know are the following:
1) PFA implies OCA (and also $2^{\aleph_0}=\a …
- 11.5k
3
votes
2
answers
419
views
Is not SH + not CH consistent?
I guess that $\lnot$SH + $\lnot$CH is consistent, but I have not found this question discussed anywhere. Is there any relatively simple model of $\lnot$SH + $\lnot$CH?
- 7,105