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.
77 questions
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Perfectly normal but not collectionwise normal space in ZFC
In the article A Perfectly Normal, Locally Compact, Noncollectionwise Normal Space Form $\lozenge^\ast$ by Daniels and Gruenhage (I presume "form" is a typo and it should be "from",...
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0
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228
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Can the Collatz conjecture be independent of ZFC? [closed]
It is known that the Continuum Hypothesis is independent of ZFC.
The formulation of the Collatz conjecture looks somehow more simple than that of the Continuum Hypothesis.
Is it possible that the ...
- 654
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3
answers
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What's the earliest result (outside of logic) that cannot be proven constructively?
Although mathematicians usually do not work in constructive mathematics per se, their results often are constructively valid (even if the original proof isn't).
An obvious counter-example is the law ...
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6
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When can a function defined on $[a, b] \cup [b, c]$ be constructively extended to a function defined on $[a, c]$?
Let $a, b, c \in \mathbb R$ such that $a \le b \le c$. Let $S$ be some set and $f : [a, b] \cup [b, c] \to S$ be a function. When can we find a function $g : [a, c] \to S$ that meets the following ...
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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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1
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121
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CH and the existence of a Borel partition of small cardinality
Say $\kappa$ is small if any set of cardinality $\kappa$ has outer-Lebesgue measure zero. We know that, in the Cohen model of ZFC where CH is false, there is a Borel partition of the unit interval of ...
- 231
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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
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A surjection from square onto power: Is limit Hartogs/Lindenbaum number necessary?
I am considering the construction in [Peng—Shen—Wu] in which the authors show the consistency of a set $X$ such that there is a surjection from $X^2$ onto the power set of $X$ (henceforth $\mathscr{P}(...
- 2,171
7
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1
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The existence of definable subsets of finite sets in NBG
This question is motivated by my preceding MO-question on (in)consistency of NBG theory of classes.
Let $\varphi(x,Y,C)$ be a formula of NBG with free parameters $x,Y,C$ and all quantifiers running ...
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A contradiction in the Set Theory of von Neumann–Bernays–Gödel?
Thinking on the theory NBG (of von Neumann–Bernays–Gödel) I arrived at the conclusion that it is contradictory using an argument resembling Russell's Paradox. I am sure that I made a mistake in my ...
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A strong form of the Axiom Schema of Replacement
Let us consider the following stronger version of the Axiom Schema of Replacement (let us call it the Axiom Schema of Replacement for Definable Relations):
Let $\varphi$ be any formula in the language ...
- 41.8k
13
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0
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325
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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 ...
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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
11
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3
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Why can we assume a ctm of ZFC exists in forcing
Following Kunen's book, it makes clear that countable transitive models (ctm) exist only for a finite list of axioms of ZFC. So, why can we assume a ctm of the whole ZFC axioms exists and use it as ...
- 111
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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
4
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1
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Is every first-countable Lindelof space of cardinality $<\mathfrak c$ a $Q$-space under MA?
Definition. A topological space $X$ is a $Q$-space if every subset of $X$ is of type $G_\delta$.
It is clear that every $Q$-space has countable pseudocharacter (= all singletons are $G_\delta$) and is ...
- 41.8k
5
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1
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The cardinal characteristic $\mathfrak r_{(X,f)}$ of a dynamical system
I am interested in a "dynamical" modification of the cardinals $\mathfrak r$ and $\mathfrak r_\sigma$, well-known in the theory of cardinal characteristics of the continuum.
For a compact ...
- 41.8k
7
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3
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439
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Dedekind-"finiteness" for arbitrary limit cardinals
In $\mathbf{ZF}$, it is possible for a set $A$ to be infinite but not to admit a countable set. In other words, for any $\alpha\in\omega$, there is an injection from $\alpha$ into $A$, but there is no ...
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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 ...
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0
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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 $q$.
...
- 111
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2
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788
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Arithmetic statement which is independent, and whose independence is independent, and so on?
My vague intuition is that not only it is common for a simple arithmetic proposition $p$ to be independent of ZFC, but it is common for the statement "$p$ is independent of ZFC" to be ...
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54
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1
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In the two-person Killing the Hydra game, what is the winning strategy?
My question is which player has a winning strategy in the
two-player version of the Killing the Hydra game?
In their amazing paper,
Kirby, Laurie; Paris, Jeff, Accessible independence results for ...
- 236k
3
votes
1
answer
975
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Implications of the existence of a pair of surjective functions, without Axiom of Choice
The classical Cantor-Schroder-Bernstein Theorem says that there exists a bijective function $X\leftrightarrow Y$ if and only if there exist injective functions $X\hookrightarrow Y$ and $Y\...
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8
votes
1
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490
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Natural $\Pi_1$ sentence independent of PA
Order invariant graphs and finite incompleteness by Harvey Friedman gives an example of a combinatorial/non-metamathematical $\Pi_1$ sentence that is independent of ZFC. Is there a simpler example of ...
- 2,203
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1
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325
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Is Axiom of Choice equivalent to its version for families of sets, indexed by ordinals? [duplicate]
Is Axiom of Choice equivalent to the following statement?
Axiom of Ordinal Choice: For any ordinal $\lambda$ and any indexed family of sets $(X_\alpha)_{\alpha\in\lambda}$ there exists a function $...
- 41.8k
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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
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1
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What is the smallest cardinality of a base of an ultrafilter on $\omega$ related to an almost disjoint family of cardinality $\mathfrak c$?
Let $(A_\alpha)_{\alpha\in\mathfrak c}$ be an almost disjoint family of infinite subsets of $\omega$. The almost disjointness of the family means that $A_\alpha\cap A_\beta$ is finite for any ordinals ...
- 41.8k
4
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1
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What is the smallest cardinality of a maximal ultrafamily of infinite subsets of $\omega$?
A family $\mathcal U$ of infinite subsets of $\omega$ is called an ultrafamily if for any sets $U,V\in\mathcal U$ one of the sets $U\setminus V$, $U\cap V$ or $V\setminus U$ is finite.
By the ...
- 41.8k
5
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A topologically transitive dynamical system without dense orbits
By a dynamical system I understand a pair $(K,G)$ consisting a compact Hausdorff space and a subgroup $G$ of the homeomorphism group of $K$.
We say that a dynamical system $(K,G)$
$\bullet$ is ...
- 41.8k
11
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1
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Is $\mathfrak j_{2:1}=\mathfrak{j}_{2:2}$ in ZFC?
A function $f:\omega\to\omega$ is called
$\bullet$ 2-to-1 if $|f^{-1}(y)|\le 2$ for any $y\in\omega$;
$\bullet$ almost injective if the set $\{y\in \omega:|f^{-1}(y)|>1\}$ is finite.
Let us ...
- 41.8k
3
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0
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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 ...
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1
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628
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A new cardinal characteristic (related to partitions)?
In this post I will discuss some cardinal characteristic of the continuum, related to partitions of $\omega$ and would like to know if it is equal to some known cardinal characteristic.
By a partition ...
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3
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1
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140
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Is $\mathfrak p=\omega_1$ equivalent to the existence of a Hausdorff gap without infinite pseudointersection?
Given two set $A,B$ we write $A\subset^* B$ if the complement $A\setminus B$ is infinite.
A Hausdorff gap is a transfinite family $\langle A_\alpha,B_\alpha\rangle_{\alpha\in\omega_1}$ of infinite ...
- 41.8k
6
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0
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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
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1
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626
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Can $H_{\omega_1}$ and $H_{\omega_2}$ be in bi-interpretation synonymy?
This question concerns the possibility of the bi-interpretation synonymy of the structure
$\langle
H_{\omega_1},\in\rangle$, consisting of the hereditarily countable sets, and the structure $\langle ...
- 236k
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1
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512
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Smallest size of a non-measurable set of reals
The question is pretty much the title. I'm wondering if anything is known about the smallest size $\kappa$ of a non-measurable subset of the real numbers (regarding the Lebesgue measure). Since we ...
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Complexity of a proper class of extendibles
If consistent, is existence of a proper class of extendible cardinals provably equivalent to a $Σ^V_5$ statement?
Recall that in ZFC, a cardinal $κ$ is extendible iff for every $λ>κ$ there is an ...
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Examples of independent $\Sigma_4^1$ statements
As in the title, I'm looking for examples of $\Sigma^1_4$ (preferably complete) sentences which are independent from ZFC in both ways, namely given a model $V$ we can extend it to $V'$ where such a ...
4
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1
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Is there a universally meager air space?
Let $\mathcal P$ be a family of nonempty subsets of a topological space $X$. A subset $D\subset X$ is called $\mathcal P$-generic if for any $P\in\mathcal P$ the intersection $P\cap D$ is not empty.
A ...
- 41.8k
9
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1
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482
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Relationship between AC, WO, and Zorn's lemma in ZF-Powerset
In regular ZF, AC, WO, and Zorn's Lemma are equivalent, but every proof I know (of the implication AC -> WO and AC -> Zorn) uses the axiom of choice on the powerset of X (where X is the Set which is ...
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4
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1
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K-analytic spaces whose any compact subset is countable
A regular topological space $X$ is called
$\bullet$ analytic if $X$ is a continuous image of a Polish space;
$\bullet$ $K$-analytic if $X$ is the image of a Polish space $P$ under an upper ...
- 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
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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
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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 ...
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1
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432
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Proving Conditional Independence
Each of the scalar random variables, $ Y $, $ X $, $ U $, and $ V $, is continuous and possibly has $ \mathbb{R} $ as its support. The random variable, $Z$, could be vector valued, but continuous.
I ...
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Is it still an open problem whether $\mathbb{R}^\omega$ is normal in the box topology?
On page 205 of his Topology textbook, James Munkres made an interesting remark:
It is not known whether $\mathbb{R}^\omega$ is normal in the box topology. Mary-Ellen Rudin has shown that the answer ...
- 4,587
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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 ...
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401
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Variants of reflection principle
This question concerns two definitions of the reflection principle. One of them known to be a consequence of the other one. I would like to understand if the reverse is true.
Let us state the first ...
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2
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477
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Experiments physically performable in a finite amount of time whose results are independent of ZFC [closed]
In On independence and large cardinal strength of physical statements we see that their are physical statements which are independent of ZFC, and even strong cardinal axioms. There were many answers, ...
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2
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799
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What things does ZFC not know if it knows?
The statement "ZFC $\vdash 0=1$" is independent of ZFC due to Goedel's second incompleteness theorem. That got me wondering, for what other statements $\phi$ is "ZFC $\vdash \phi$" independent of ZFC?
...
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