Cardinal characteristics of the continuum

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The subject known as cardinal characteristics of the continuum explores the rich territory---sometimes hidden from view, depending on the set-theoretic background---between the countably infinite cardinal $\aleph_0$ and the uncountable cardinality of the continuum. The subject begins with Cantor's theorem laying out the basic dichotomy that the continuum $\frak{c}=2^{\aleph_0}$ is strictly larger than $\aleph_0$, and goes on to explore the various ways that properties of $\aleph_0$ might extend to uncountable cardinals.

For example, the union of countably many measure zero subsets of $\mathbb{R}$ has measure $0$; the union of countably many meager sets is meager; every countable number of functions $f:\omega\to\omega$ is bounded by a single function under eventual domination; every countable set of reals has measure $0$. To what extent can we hope to extend such properties to uncountable collections? The various cardinal characteristics of the continuum, many of which are described below, are defined exactly to be the cardinalities where these and other similar such properties first begin to fail for uncountable collections. Each cardinal characteristic measures the extent to which a particular mathematical phenomenon extends from the countable to the uncountable, and the lesson of the subject is that there is an enormous diversity of such characteristics, exhibiting diverse combinations in various models of set theory. When the continuum is small, the characteristics are pressed together---under the continuum hypothesis, for example, they are all equal to the continuum---but in other models, the different characteristics are teased apart and seen to express fundamentally different inequivalent properties. The subject breaks into two major components: first, proving the positive relations amongs the characteristics, such as $\omega_1\leq\frak{b}\leq\frak{d}\leq\frak{c}$; and second, constructing models of set theory, generally by forcing, which reveal the range of possibility, such as a model in which $\omega_1\lt\frak{b}\lt\frak{d}\lt\frak{c}$. Thus, the philosophy of the subject naturally exhibits an unusual degree of contingency for set-theoretic truth: we understand the cardinal characteristic more deeply because we know the range of possibility for their relations to each other.

An excellent general resource on the subject is [1].

The bounding number

The bounding number $\frak{b}$ is the size of the smallest family of functions $f:\omega\to\omega$ that is not bounded with respect to eventual domination.

The dominating number

The dominating number $\frak{d}$ is the size of the smallest family of functions $f:\omega\to\omega$, such that every function is eventually dominated by a function in the family.

The covering numbers

The additivity numbers

Cichoń's diagram

${\rm cov}({\mathcal L})$ $\longrightarrow$ ${\rm non}({\mathcal K})$ $\longrightarrow$ ${\rm cof}({\mathcal K})$ $\longrightarrow$ ${\rm cof}({\mathcal L})$ $\longrightarrow$ $2^{\aleph_0}$
$ \Bigg\uparrow $ $\uparrow$ $ \uparrow$ $ \Bigg\uparrow $
${\mathfrak b} $ $\longrightarrow$ ${\mathfrak d} $
$\uparrow$ $\uparrow$
$\aleph_1$ $\longrightarrow$ ${\rm add}({\mathcal L})$ $\longrightarrow$ ${\rm add}({\mathcal K})$ $\longrightarrow$ ${\rm cov}({\mathcal K})$ $\longrightarrow$ ${\rm non}({\mathcal L})$

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  1. Blass, Andreas. Chapter 6: Cardinal characteristics of the continuum. Handbook of Set Theory , 2010. www   bibtex
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