What is the $\text{♢-Axiom}$? (now known to be the 'Diamond Principle / Axiom')

Question

While skimming over some research papers I found this abstract

Mathematics > Operator Algebras (link here)

Large irredundant sets in operator algebras
Clayton Suguio Hida, Piotr Koszmider
(Submitted on 4 Aug 2018 (v1), last revised 2 Sep 2018 (this version, v3))

A subset X of a C*-algebra A is called irredundant if no A∈X belongs to the C*-subalgebra of A generated by X∖{A}. Separable C*-algebras cannot have uncountable irredundant sets and all members of many classes of nonseparable C*-algebras, e.g., infinite dimensional von Neumann algebras have irredundant sets of cardinality continuum. There exists a considerable literature showing that the question whether every AF commutative nonseparable C*-algebra has an uncountable irredundant set is sensitive to additional set-theoretic axioms and we investigate here the noncommutative case. Assuming ♢ (an additional axiom stronger than the continuum hypothesis) we prove that there is an AF C*-subalgebra of B(ℓ2) of density 2ω=ω1 with no nonseparable commutative C*-subalgebra and with no uncountable irredundant set. On the other hand we also prove that it is consistent that every discrete collection of operators in B(ℓ2) of cardinality continuum contains an irredundant subcollection of cardinality continuum. Other partial results and more open problems are presented.

where you'll find the following sentence fragment,

Assuming ♢ (an additional axiom stronger than the continuum hypothesis) ...

I spent 5-7 minutes searching the web trying to find out what this $\text{♢-Axiom}$ is all about, but came up empty-handed (got a couple of hits and a reference to the $\text{♢-Principle}$, but no explicit statement of it).

What is the $\text{♢-Axiom}$ (any elucidating comments/background info will be appreciated)?

Also, the abstract states that the status of the theory is sensitive to additional set-theoretic axioms, and that is certainly fascinating!

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Answer 1

The principle $\lozenge$ (diamond) is in a sense the right set-theoretic version of the continuum hypothesis, as it presents it instead as a reflection principle. Formally, it asserts that there is a diamond sequence, that is, a sequence $(A_\alpha:\alpha<\omega_1)$ such that

1. each $A_\alpha$ is a subset of $\alpha$,
2. for each $A\subseteq \omega_1$ there is an infinite $\alpha$ such that $A\cap\alpha=A_\alpha$.

Equivalently, we can require that the equality $A\cap \alpha = A_\alpha$ holds for a stationary set of $\alpha$, and we can also weaken the requirement that $A_\alpha\subseteq\alpha$ to instead require that each $A_\alpha$ is now a countable family of subsets of $\alpha$, with 2 modified to assert that stationarily often $A\cap\alpha$ is a member of the $\alpha$th family. Devlin and Kunen studied some of these variants.

The principle was introduced by Jensen in his paper on fine-structure theory, where he shows that it follows from the assumption that $V=L$. Note that, by taking $A$ a subset of $\omega$, the principle implies that any such subset appears in the list, so $|\mathcal P(\omega)|=\aleph_1$.

The principle is useful to carry out recursive constructions of objects of size $\omega_1$, particularly when we want our object $O$ to satisfy certain property $P(O,A)$ for any $A\subseteq\omega_1$. The intuitive idea is that it should be enough instead to ensure that approximations $O_\alpha$, $\alpha<\omega_1$, of the object approximately satisfy the required property, not with respect to each $A$ but instead with respect to $A_\alpha$. Jensen used the principle to build Suslin trees, and it has since found many applications in analysis, set-theoretic topology, algebra, and set theory proper.

Versions $\lozenge_{\kappa^+}$ of the axiom also exist for other cardinals, requiring the existence of an appropriate family $(A_\alpha:\alpha<\kappa^+)$ with each $A_\alpha\subseteq\alpha$ and the $A_\alpha$ predicting each $A\subseteq\kappa^+$ stationarily often. This principle implies $2^\kappa=\kappa^+$, and it is a celebrated theorem of Shelah that, other than for $\kappa=\aleph_0$, the converse holds.

For $\lozenge$ itself, it is consistent that CH holds and the principle fails, and this is related to certain uniformization principles.

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Some references:

1. MR0309729 (46 #8834). Jensen, R. Björn. The fine structure of the constructible hierarchy. With a section by Jack Silver. Ann. Math. Logic 4 (1972), 229–308; erratum, ibid. 4 (1972), 443.

2. MR0523488 (80c:03050). Devlin, Keith J. Variations on $\lozenge$. J. Symbolic Logic 44 (1979), no. 1, 51–58.

3. MR2596054 (2011m:03084). Shelah, Saharon. Diamonds. Proc. Amer. Math. Soc. 138 (2010), no. 6, 2151–2161.

4. MR2777747 (2012k:03116). Rinot, Assaf. Jensen's diamond principle and its relatives. Set theory and its applications, 125–156, Contemp. Math., 533, Amer. Math. Soc., Providence, RI, 2011.