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.
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Does PFA imply that any $\aleph_1$-dense subsets of the real line are order-isomorphic via a nonexpansive bijection?
According to Theorem 6.9 in Baumgartner's survey "Applications of the Proper Forcing Axiom", under PFA, any two $\aleph_1$-dense subsets of the real line are order-isomorphic.
On the other ...
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Which "specific cases" of order types outside of $M$ could Laver mean? What are examples of undecidable statements in order theory?
Richard Laver finishes his seminal paper "On Fraïssé's order type conjecture", with:
Finally, the question arises as to how the order types outside of $M$ behave
under embeddability. For ...
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Does cocountable topology on $\mathbb{R}$ have the $2$-Markov Menger property?
References:
Applications of limited information strategies in Menger’s game by Clontz
Almost compatible functions and infinite length games by Clontz and Dow
Def. 3.7 of [1] $\mathcal{A}(\kappa)$ ...
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Simple true $\Pi^0_1$ statements independent of weak arithmetics
I originally asked this question on Math StackExchange here, but I have copied it here as I now feel it is more appropriate for this site.
There is an explicitly known 549-state Turing machine where, ...
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Higher-Order Analogues of Gödel’s First Incompleteness Theorem
I was playing with ideas around Gödel’s first incompleteness theorem which, roughly speaking, says that for every ($\omega$-)consistent, recursively axiomatizable formal system $F$ that is ...
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Interchanging limits
The following definition is by Sinclair, G.E. A finitely additive generalization of the Fichtenholz–Lichtenstein theorem. Transactions of the American Mathematical Society. 1974;193:359-74.
A function ...
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Embedding of $\omega_2+1$ with $G_\delta$-topology into Stonean space
For a compact Hausdorff space $X$, let $EX$ be the Stonean space corresponding to the Boolean algebra of regular open sets of $X$. Explicitly, if $\text{RO}(X)$ denotes regular open sets of $X$, let $...
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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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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 ...
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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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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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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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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 ...
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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 $\textbf{PA}$:...
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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}(...
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