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Results tagged with independence-results
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user 8133
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.
9
votes
Accepted
Arithmetic statement which is independent, and whose independence is independent, and so on?
If we fix things to avoid Will Sawin's observation, then the answer is yes under any reasonable interpretation I can think of.
For example, consider the following: let $J(p)$ be the sentence "If $\mat …
- 21.1k
3
votes
Wondering if the following set-theoretic assertion is known to be consistent w/ ZFC
Well, by Easton's theorem, we immediately get the following:
It is consistent that $2^{\aleph_0}=\aleph_{17}$ (say) and $2^\kappa=\kappa^+$ for every regular cardinal $\kappa\ge\aleph_{17}$.
In …
- 21.1k
7
votes
Examples of independent $\Sigma_4^1$ statements
As a starting point, think about the sentence "There is a nonconstructible real." This is $\Sigma^1_3$ and clearly not downwards-absolute. However, it is upwards-absolute.
To get the desired situatio …
- 21.1k
14
votes
Independence of the countable axiom of choice
I'm going to play fast and loose with details here, but the outline is correct. To answer your new questions: no, there is no short proof. And this only shows one direction of the indpenedence of AC, …
- 21.1k
5
votes
Accepted
Intuition behind Pincus' "injectively bounded statements"
(Note: this isn't something I really know, so this might be wildly off base.)
To start with, let's look at a weaker transfer principle: the Jech-Sochor Embedding Theorem.
Jech-Sochor says that sentenc …
- 21.1k
19
votes
Why can we assume a ctm of ZFC exists in forcing
Expositionally, forcing is (usually) easier to understand with a c.t.m. This does indeed lead to somewhat different results, such as
$(*)\quad$ If there is a countable transitive model of $\mathsf{ZF …
- 21.1k
9
votes
Is Axiom of Choice equivalent to its version for families of sets, indexed by ordinals?
It is strictly weaker than choice. This is explained in Asaf Karagila's answer at MSE: the $L(\mathbb{R})$ of $L$ + $\aleph_1$-many Cohen generics witnesses this.
(There the principle is phrased for w …
- 21.1k