2

What are a few applications of $\beth_3$ other than $\beth_3=\mathcal{P}(\mathcal{P}(\mathcal{P}(\mathbb{N})))$ or any other variations of generalized identities that can easily be found on Wikipedia? This is not a homework problem; I am merely studying independently.

To explain what I mean by “applications,” here is an application of each beth number before $\beth_3$ other than the power set definition:

  • $\beth_0=\aleph_0$ is the basis of the beth numbers, and $\beth_0$ equals the cardinality of the natural numbers and also equals the cardinality of the integers. For concreteness, you could say that a series sums together this many terms.
  • $\beth_1$ is the cardinality of the real numbers or any interval subset of the real line. You could loosely say that an integral adds together this many terms.
  • While there are infinitely many functions that take a subset of the real line and return a subset of the real line (denoted $\mathbb{R^R}$), the number of them is defined an equals $\beth_2$.
gen-ℤ ready to perish
  • 6,610
  • 6
    It's not true that $\beth_1$ is the cardinality of any subset of the real line - It's an upper bound for the cardinality of any subset of the real line. There are always finite/countable subsets of the real line and - should $\operatorname{CH}$ fail - there may be many more cardinalities which are realized by sets of real numbers. – Stefan Mesken Jul 30 '17 at 11:01
  • @StefanMesken Ah, you are absolutely right. What I meant to communicate was that a continuous interval like $(0,1]$ or $[-4,2]$ has cardinality $\beth_1$. I hadn't stopped and realized that $\mathbb{N}\subset\mathbb{R}$. Thank you for catching that error! – gen-ℤ ready to perish Jul 30 '17 at 16:58

1 Answers1

1

Perhaps not an interesting application. According to this answer, the number of topologies (or nonhomeomorphic topologies) over an infinite set $X$ is $2^{2^{|X|}}$. So the number of topologies (or nonhomeomorphic topologies) over $\mathbb{R}^k$, $\mathbb{C}$, $\mathbb{Q}_p$, $\mathbb{C}_p$, $\mathbb{R}^\mathbb{N}$, $\ell_p$ with $1\le p\le\infty$, set of measurable functions over $E$ (a measurable subset of $\mathbb{R}$ of positive measure), $L^p(E)$ for measurable $E$ with $1\le p\le\infty$, $C^k(\Omega)$ with open $\Omega\subset\mathbb{R}^k$ and $0\le k\le\infty$, set of holomorphic functions over an open subset of $\mathbb{C}$, ... is $\beth_3$.

Jianing Song
  • 1,707