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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answer
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$\omega_2$-sequence of Suslin trees
Is it possible to have an $\omega_2$-length sequence of ($\omega_1$-)Suslin trees such that if one builds the product of finitely many trees in that sequence, one ends up with a Suslin tree again?
...
- 2,180
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vote
1
answer
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Wondering if the following set-theoretic assertion is known to be consistent w/ ZFC
I'm wondering about the following (I've written a couple set theory papers but don't consider myself a set theorist, so please keep this in mind when answering):
Is it consistent w/ ZFC that there ...
- 27
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votes
1
answer
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Should axiomatic set theory be translated into graph theory?
Recently I saw the abstract of a paper by Nash-Williams: ``Should axiomatic set theory be translated into graph theory?''. The abstract, taken from Mathscinet says the following:
The author ...
- 32.1k
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4
answers
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How undecidable is the spectral gap?
Nature just published a paper by Cubitt, Perez-Garcia and Wolf titled Undecidability of the Spectral Gap, there is an extended version on arxiv which is 146 pages long. Here is from the abstract:"Many ...
- 1,731
32
votes
1
answer
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Bidi: A new cardinal characteristic of the continuum?
This question assumes familiarity with combinatorial cardinal
characteristics of the continuum.
Identify an infinite set $a\subseteq\mathbb{N}$ with its increasing
enumeration. Thus, for each natural ...
- 3,104
10
votes
2
answers
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Independence of the countable axiom of choice
How does one proove that the Countable axiom of choice is not provable in ZF?Is there any brief proof?Does the Independence of the countable axiom of choice implies the independence of the axiom of ...
- 101
9
votes
2
answers
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Relationship between fragments of the axiom of choice and the dependent choice principles
The dependent choice principle ${\rm DC}_\kappa$ states that if $S$ is a nonempty set and $R$ is a binary relation such that for every $s\in S^{\lt\kappa}$, there is $x\in S$ with $sRx$, then there ...
- 2,240
7
votes
1
answer
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On Consistency of an Existence
Let $\omega \leq \kappa <2^{\omega}$ , $\omega \leq\lambda \leq \kappa$ and $D(\kappa, \lambda)$ be the statement:
For all $ \mathfrak{B} \subseteq \mathbf{P}(\omega)$ with $|\mathfrak{B}|=\kappa$...
- 2,381
16
votes
1
answer
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Kaplansky's conjecture and Martin's axiom
Recall Kaplansky's conjecture which states that every algebra homomorphism from the Banach algebra C(X) (where X is a compact Hausdorff topological space) into any other Banach algebra, is ...
- 32.1k
9
votes
1
answer
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Intuition behind Pincus' "injectively bounded statements"
In
David Pincus, Zermelo-Fraenkel Consistency Results by Fraenkel-Mostowski Methods,
The Journal of Symbolic Logic, Vol. 37, No. 4 (Dec., 1972), pp. 721-743
Pincus introduces the notion of ...
- 35.4k
3
votes
3
answers
477
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Undecidability and holomorphic functions (Reference request)
The goal of this question is to recall a certain mathematical fact -not in my field- that I was once briefly told and that I have fogotten, and also to collect similar results.
The fact, I think, ...
- 23.3k
22
votes
3
answers
2k
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Nice algebraic statements independent from ZF + V=L (constructibility)
Background and motivation
I've always been fascinated about algebraic statements independent from ZFC set theory. One such fascinating example comes from considering $\rm{Ext}^1_\mathbb{Z}(A,\mathbb{Z}...
user1437
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votes
3
answers
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Spectra of elements of a Banach algebra and the role played by the Hahn-Banach Theorem.
This problem was posed on Math StackExchange some time ago, but it did not garner any solutions there. I think that it is interesting enough to be posed here on Math Overflow, so here it goes.
Let $ \...
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votes
6
answers
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Formal proof of Con(ZFC) => Con(ZFC + not CH) in ZFC
Is it possible to prove $Con(ZFC) \rightarrow Con(ZFC + \neg CH)$ purely within ZFC? To prove this (using forcing) one seems to need a countable transitive model of ZFC. The texts I am reading avoid ...
- 365
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votes
3
answers
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Is the Axiom of Union independent of the rest of ZF?
Short version: Is the axiom of union independent of the rest of axioms of ZF?
NO) Tourlakis (2003) says in p. 177 that the axiom of union can be derived from the rest of ZF if an appropriate version ...
- 243
8
votes
3
answers
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Natural statements independent from true $\Pi^0_2$ sentences
I am looking for sentences in the language of first order arithmetic ($0,1,+,\cdot,\leq$) which are independent from $\Pi^0_2$ consequences of true arithmetic $\Pi^0_2\text{-}\mathsf{Th}(\mathbb{N})$. ...
- 5,502
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3
answers
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Is there an "undecided" assertion of which a proof that it's not undecidable is known?
Just a curiosity:
Is there an assertion of which a proof (formalizable, say, in ZFC) is not known but a proof that it's not undecidable (in ZFC) is known?
Edit: after the comments, I think the ...
- 23.3k
44
votes
3
answers
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"Simpler" statements equivalent to Con(PA) or Con(ZFC)?
Given any reasonable formal system F (e.g., Peano Arithmetic or ZFC), we all know that one can construct a Turing machine that runs forever iff F is consistent, by enumerating the theorems of F and ...
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4
answers
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Is every p-point ultrafilter Ramsey?
A non-principal ultrafilter $\mathcal{U}$ on $\omega$ is a p-point (or weakly selective) iff for every partition $\omega = \bigsqcup _{n < \omega} Z_n$ into null sets, i.e each $Z_n \not \in \...
- 3,982
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3
answers
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Reference Request: Independence of the ultrafilter lemma from ZF
I'm looking for references for the following facts concerning the ultrafilter lemma (~ "there exist non-principal ultrafilters"):
The ultrafilter lemma is independent of ZF.
ZF + the ultrafilter ...
- 1,347
7
votes
2
answers
844
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Statements forced by one condition of a poset, but not the whole thing
In order to get the relative consistency of some statement, it suffices to find a notion of forcing, and a condition $p$ in that forcing, such that $p$ forces the desired statement. It seems to be ...
- 3,982
11
votes
5
answers
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Minimal subset of axioms for ZFC
Hello all, one may look for "minimal system of axioms" for ZFC (or any other
theory) in the following (unusual) sense : say that a subset S of ZFC is
"sufficient" if there is an explicit procedure ...
- 3,595
9
votes
2
answers
1k
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Given a cardinal k, what's the biggest dense linear order with a dense subset of size k?
It's not hard to show that for any cardinal $\kappa$, there is no dense linear order without endpoints (DLO) of size greater than $2^{\kappa}$ that has a dense subset of size $\kappa$. But one can ...
- 3,982
3
votes
2
answers
688
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A problem about posets similar to Suslin's problem
Suslin's problem is:
Given a complete dense linear order without endpoints, if it has the ccc must it be isomorphic to $\mathbb{R}$?
The answer is that it's independent of ZFC. The related ...
- 3,982
2
votes
2
answers
892
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Is the existence of a well-ordering on R independent of ZF?
I am reasonably certain this is the case, but can't find a reference that actually states this, although the Wikipedia article states something close.
- 118k
4
votes
3
answers
418
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Is there a formula phi s.t. phi and not-phi have a stronger consistency?
Let Σ be an axiom system. Can there be a formula φ, s.t.
Con(Σ) does not imply Con(Σ + φ) AND
Con(Σ) does not imply Con(Σ + not φ)
If yes, can you give me ...
- 205
297
votes
34
answers
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What are some reasonable-sounding statements that are independent of ZFC?
Every now and then, somebody will tell me about a question. When I start thinking about it, they say, "actually, it's undecidable in ZFC."
For example, suppose $A$ is an abelian group such ...