All Questions
Tagged with set-theory boolean-algebras
11 questions with no upvoted or accepted answers
12
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406
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An internal notion of freeness for complete Boolean algebras
Background and Definition
Gaifman and Hales showed that there are no infinite free complete Boolean algebras.
But let a complete Boolean algebra $B$ be internally free if there is a set $X\subseteq B$ ...
- 121
10
votes
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759
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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 ...
10
votes
0
answers
514
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Existence of a regular subposet which collapses everything except the top cardinal
Suppose $\delta$ is an inaccessible cardinal, and $\mathbb{P}$ is the Levy Collapse $\text{Col}(\kappa, \delta)$ which adds a surjection from $\kappa \to \delta$ (for some regular $\kappa < \delta$)...
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9
votes
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356
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Direct limits of $\sigma$-centered forcing notions
It is quite well known that
Any FS (finite support) iteration of length $<\mathfrak{c}^+$ of $\sigma$-centered posets is $\sigma$-centered (see e.g. here).
Now consider the following question: ...
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7
votes
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answers
182
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rigidity of $\mathcal P(\omega_1) / NS$ under MA
In Woodin's book, Lemma 5.100 asserts that if $MA_{\omega_1}$ holds and there is an $\omega_2$-saturated ideal $I$ on $\omega_1$, then $\mathcal P(\omega_1)/I$ is a rigid boolean algebra, meaning it ...
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6
votes
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166
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Rigid boolean inclusions?
A boolean algebra $B$ is rigid if it has no nontrivial automorphisms and atomless if it has no minimal nonzero elements. $A \subseteq B$ is a complete boolean inclusion if $B$ is complete and $A$ is a ...
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4
votes
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answers
139
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Commutativity of a diagram between complete embeddings
Suppose $\mathbb{P}_0$, $\mathbb{P}_1$ and $\mathbb{P}_2$ are separative posets such that $\mathbb{P}_2$ projects into $\mathbb{P}_1$ and $\mathbb{P}_1$ projects into $\mathbb{P}_0$, i.e. there are ...
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4
votes
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207
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What algebraic identities does the iteration of forcing operation satisfy?
Let $G$ be the set of all formulas $\phi(x)$ in the language of such that $ZFC\vdash\exists x\phi(x)$ exists, $ZFC\vdash\phi(x)\rightarrow``x\,\textrm{is a complete Boolean algebra}"$, $ZFC\vdash``\...
3
votes
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answers
152
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Boolean Algebra of size $2^{<\kappa}$ without an $\aleph_1$-complete ultrafilter
For this post we work only with cardinals that live below the first measurable.
Assume that $\kappa$ is singular and $\kappa<2^{<\kappa}<2^\kappa$.
Question: Is it possible to have a Boolean ...
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2
votes
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answers
240
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3 questions around the Stone space of the free $\sigma$-algebra on $\omega_1$ free generators
During my studies, I came across several different Stone spaces, e.g.:
(i) The Cantor cube $X=\{0,1\}^{\omega_1}$, which is the Stone space of the free Boolean algebra on $\omega_1$ free generators;
...
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2
votes
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answers
147
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C.c.-ness of a forcing notion based on an atomless complete Boolean algebra
Given $\mathbb{B} = \langle B, \wedge, \vee, \neg, 0, 1 \rangle$ an atomless complete Boolean algebra that has a $< \mkern-4mu \kappa$-closed dense subset and is $\kappa^+$-c.c., we define a ...
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