Intereting Posts

Fitting of Closed Curve in the Polar Coordinate.
Which of the following polynomials are subspaces of $\mathbb{P}_n$ for an appropriate value of n?
How many sets can we get by taking interiors and closures?
Prove that the set $ \{\sin(x),\cos(x),\sin(2x),\cos(2x)\}$ is linearly independent.
Proving Cauchy condensation test
A “Cantor-Schroder-Bernstein” theorem for partially-ordered-sets
Riemann Zeta function – number of zeros
Finding an unknown angle
How can Zeno's dichotomy paradox be disproved using mathematics?
Prob. 1, Chap. 6, in Baby Rudin: If $f(x_0)=1$ and $f(x)=0$ for all $x \neq x_0$, then $\int f\ \mathrm{d}\alpha=0$
A subset whose sum of elements is divisible by $n$
Isomorphisms (and non-isomorphisms) of holomorphic degree $1$ line bundles on $\mathbb{CP}^1$ and elliptic curves
Inverting the modular $J$ function
Difference between root, zero and solution.
How to define the operation of division apart from the inverse of multiplication?

Is there any meaningful definition to afford for $\aleph_r$ (as in cardinality) where $r\in\mathbb{R}^+$? $r\in\mathbb{C}$? What about $\aleph_{\aleph_0}$? Can we iterate this? $\aleph_{\aleph_{\aleph_{\cdots}}}$

I may be throwing in bunch of rather naive/basic questions, for I haven’t learnt much about infinite cardinalities. If I am referring to bunch of stuff abundantly dealt in established areas, please kindly point out.

- Cardinality and infinite sets: naturals, integers, rationals, bijections
- Can we distinguish $\aleph_0$ from $\aleph_1$ in Nature?
- Can sets of cardinality $\aleph_1$ have nonzero measure?
- What are Aleph numbers intuitively?
- Why is the Continuum Hypothesis (not) true?
- Cardinality != Density?

- Proving $A \cap C = B \cap C$, but $ A \neq B$
- A question about cardinal arithmetics without the Axiom of Choice
- Is the axiom of choice needed to show that $a^2=a$?
- How to show if $A \subseteq B$, then $A \cap B = A$?
- The cartesian product $\mathbb{N} \times \mathbb{N}$ is countable
- Proving all rational numbers including negatives are countable
- Why is $|Y^{\emptyset}|=1$ but $|\emptyset^Y|=0$ where $Y\neq \emptyset$
- $A ⊂ B$ if and only if $A − B = ∅$
- Where is the flaw in my Continuum Hypothesis Proof?
- Proof of $A\cap(B\cup C) = (A\cap B)\cup(A\cap C)$

The $\aleph$ numbers are well-ordered. This means that every non-empty set has a minimal element. Furthermore, they are *linearly* ordered.

This means that any indexing imposed on them should at least have these two properties. The real numbers are not well-ordered (consider the subset $(0,1)$, or even $\mathbb R$ itself) and the complex numbers are not even ordered in any natural sense.

The idea behind having a well-ordering is to say what is the next cardinality. Given a set, we can easily tell what is the least $\aleph$ which is larger. In the natural numbers (and their generalization, the ordinals) we have a successor function which does that, so it is a good ground to use when indexing cardinalities. We don’t have a nice successor function for the real numbers, or for any dense ordering for a matter of fact.

It is possible to have $\aleph_{\aleph_0}$, but there is a minor problem here. $\aleph_0$ is the notation discussing *size*, whereas $\aleph_\alpha$ is the cardinal that the cardinals below it have order type $\alpha$. So we write $\aleph_\omega$, where $\omega$ is the least infinite ordinal. This is a *limit* cardinal, which means that it is not a successor of any cardinal — but there are smaller $\aleph$’s nonetheless.

This of course can be reiterated, but we need to use the *ordinal* form, rather the cardinal form. Namely, every $\aleph$ number is also an ordinal. $\aleph_\alpha$ is the actually the ordinal $\omega_\alpha$, where these ordinals are defined recursively as ordinals which are $(1)$ infinite; and $(2)$ do not have an injection into any smaller ordinal. The least is $\aleph_0$ and it is the cardinality of the natural numbers, which is the ordinal $\omega$.

So if we wish to iterate, $\aleph_0\to\aleph_{\aleph_0}\to\aleph_{\aleph_{\aleph_0}}\to\dots$ we actually need to do it as following: $$\aleph_0\to\aleph_{\omega}\to\aleph_{\omega_\omega}\to\ldots$$

That been said, without the axiom of choice it is consistent to have sets whose cardinality is *not* an $\aleph$ number, namely sets which cannot be well-ordered. It is consistent that there is a collection of sets which is ordered (by inclusion) like the real numbers, and no two sets have the same cardinality (there is no bijection between two distinct sets).

**Some threads which may interest the reader:**

- "Homomorphism" from set of sequences to cardinals?
- Non-aleph infinite cardinals
- What are Aleph numbers intuitively?

- Positive definiteness of Fubini-Study metric
- Eigenvalues in terms of trace and determinant for matrices larger than 2 X 2
- Seeing quotient groups
- Expand $\binom{xy}{n}$ in terms of $\binom{x}{k}$'s and $\binom{y}{k}$'s
- The integral of a closed form along a closed curve is proportional to its winding number
- Is the quotient map a homotopy equivalence?
- Why does a meromorphic function in the (extended) complex plane have finitely many poles?
- Probability of dice sum just greater than 100
- FLOSS tool to visualize 2- and 3-space matrix transformations
- Derivative of a determinant
- How many tries to get at least k successes?
- Possible to solve this differential equation?
- Combinatorial Proof Of ${n \choose k}={n-1\choose {k-1}}+{n-1\choose k}$
- Can a limit of an integral be moved inside the integral?
- Indefinite Integral of $\sqrt{\sin x}$