Questions tagged [independence-results]
This tag is for questions about proving that some statement is independent from a theory, meaning it is neither provable nor refutable from that theory. Common examples are the continuum hypothesis from the axioms of ZFC, and the axiom of choice from the axioms of ZF.
14 questions with no upvoted or accepted answers
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$\mathsf{AC}_\mathsf{WO}+\mathsf{AC}^\mathsf{WO}\Rightarrow \mathsf{AC}$?
Let
$\mathsf{AC}_\mathsf{WO}$: Every well-orderable family of non-empty sets has a choice function.
$\mathsf{AC}^\mathsf{WO}$: Every family of non-empty well-orderable sets has a choice function.
My ...
- 2,286
11
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0
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256
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Existence of a strong antichain
Call an antichain (set of pairwise incomparable elements) $A$ of a poset $P$ strong if for every $p,q \in P$ with $p \leq q$ there exists an $a\in A$ which is comparable with both $p$ and $q$.
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8
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Metrically Ramsey ultrafilters
On Thuesday I was in Kyiv and discussed with Igor Protasov the system of MathOverflow and its power in answering mathematical problems. After this discussion Igor Protasov suggested to ask on MO the ...
- 41.8k
7
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177
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Asymptotically discrete ultrafilters
Definition 1. A ultrafilter $\mathcal U$ on $\omega$ is called discrete (resp. nowhere dense) if for any injective map $f:\omega\to \mathbb R$ there is a set $U\in\mathcal U$ whose image $f(U)$ is a ...
- 41.8k
6
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375
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How bad a proper forcing of size $\aleph_1$ can be?
This question concerns proper forcings of size $\aleph_1$. In the context of
$\rm ZFC+\neg CH$, I couldn't find any counter example to the following property. Suppose $\mathbb P$ is a proper forcing ...
- 2,381
5
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212
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Questions about very fat sets
If $\kappa$ is a regular uncountable cardinal, we call a set $S\subseteq\kappa$ fat if for every $\alpha<\kappa$ and every club $C\subseteq\kappa$, there is a closed subset of $S\cap C$ of ...
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5
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175
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Selecting an almost disjoint family in a given family of sets
A family $\mathcal A$ of infinite subsets of $\omega$ is called almost disjoint if for any distinct sets $A,B\in\mathcal A$ the intersection $A\cap B$ is finite.
Let $\mathfrak a'$ be the largest ...
- 41.8k
3
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206
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Independence and truth in PA
By $\textbf{PA}$ I will mean the usual first-order Peano Arithmetic. I will denote an element of $\mathbb{N}$ by $n$, and by $[n]$ I will denote the corresponding term in the language of $\textbf{PA}$:...
- 3,318
3
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0
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157
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The "absolute" version of the Axiom Schema of Replacement in ZFC
The well-known Axiom Schema of Replacement in ZFC says that for any formula $\varphi$ of the Set Theory with free variables among $w_1,\dots,w_n,A,x,y$ the following holds:
$$\forall w_1,\dots,w_n\;\...
- 41.8k
3
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0
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79
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Is $\mathfrak q_0$ equal to the smallest cardinality of a second-countable $T_1$-space which is not a $Q$-space?
A topological space $X$ is a $Q$-space if every subset of $X$ is of type $G_\delta$.
The smallest cardinality of a metrizable separable space which is not a $Q$-space is denoted by $\mathfrak q_0$ and ...
- 41.8k
3
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122
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The existence of $T$-ultrafilters in ZFC
Looking at this MO-problem, my collegue Igor Protasov suggested to ask on Mathoverflow his old question on $T$-ultrafilters hoping that somebody on MO can solve it.
First I recall the necessary ...
- 41.8k
2
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0
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120
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Two small uncountable cardinals related to Q-sets
A subset $A$ of the real line is called a Q-set if any subset of of $A$ is of type $F_\sigma$ in $A$.
Let $\mathfrak q_0$ be the smallest cardinality of a subset $X\subset\mathbb R$ which is not a Q-...
- 41.8k
1
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289
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About Whitehead's problem
Hi I am new to proofs of consistency and independence with ZFC of some claims. I have read "The uses of set theory" by Judith Roitman, in that article it is mentioned that the Whitehead ...
- 561
0
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187
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How are incompleteness and independence proofs related?
(1) Some typical inclompleteness proofs use a kind of fixed point argument - for certain $\Phi$ you find a $\varphi$ with $\Phi(\mathrm{code}(\varphi))\leftrightarrow\varphi$.
(2) Some independence ...
- 18.4k