Intereting Posts

Left topological zero-divisors in Banach algebras.
How to efficiently solve a series of similar matrix equations using the LU decomposition
Number of different necklaces using $m$ red and $n$ white pebbles
What will be the one's digit of the remainder in: $\left|5555^{2222} + 2222^{5555}\right|\div 7=?$
Finding the “square root” of a permutation
Recurrence Relation for the nth Cantor Set
What exactly are eigen-things?
Complex Permutation
Derangement problem!
Is ${\rm conv}({\rm ext}((C(X))_1))$ dense in $(C(X))_1$?
How to round 0.4999… ? Is it 0 or 1?
Stability for higher dimensional dynamical systems
Understanding quotient groups
Existence of a power series converging non-uniformly to a continuous function
Finding the probability density function of $Y=e^X$, where $X$ is standard normal

The ordinal number $\omega^2$ can be visualized as $\omega$-many copies of $\omega$. Likewise, the ordinal number $\omega^3$ can be visualized as $\omega^2$-many copies of $\omega$, arranged as appropriately (= lexicographically) ordered rows in the cube $\omega^3$ (rows being sequences of cells parallel to the x-axis):

It “only” takes the ability to visualize higher-dimensional (hyper-)cubes to visualize arbitrary countable ordinals of the form $\omega^n$ as $\omega^{n-1}$ appropriately ordered copies of $\omega$.

- Calculation of the $s$-energy of the Middle Third Cantor Set
- Box Dimension Example
- Prove that this condition is true on an Zariski open set
- Let $S:U\rightarrow V, \ T:V\rightarrow W$ and if $S$ and $T$ are both injective/surjective, is $TS$ injective/surjective?
- Transcendence degree of $K$
- Looking for Cover's hubris-busting ${\mathbb R}^{N\gg3}$ counterexamples

But this ability comes to an end when thinking of the countable ordinal number $\omega^\omega$, since the $|\omega|$-dimensional hypercube $\omega^{|\omega|}$ consists of uncountably many cells and cannot represent a countable ordinal.^{1}

While (presumably) the countable ordinal number $\omega^\omega$ cannot be visualized as a countable number of copies of $\omega$ (however carefully arranged) – can it be visualized in a similarly easy

visualway, only “slightly” advanced? (For example, by replacing $\omega$ by some other (higher) ordinal number?)

If there *were* such a “similarly easy way”: Up to which limit would it lead? Up to $\epsilon_0$? And which “similarly easy way” would come beyond *that* limit?

^{1 Do $|\omega|$-dimensional vector spaces – e.g. Hilbert spaces – pose such serious problems?}

- $\operatorname{height} \mathfrak{p} + \dim A / \mathfrak{p} = \dim A$
- I want to know why $\omega \neq \omega+1$.
- Countable compact spaces as ordinals
- Prove that for every countable ordinal $\alpha$, $\alpha\times[0,1)$ is order isomoprhic to $[0,1)$
- Godel's pairing function and proving c = c*c for aleph cardinals
- Basic examples of ordinals
- How to prove the Milner-Rado Paradox?
- Uses of ordinals
- what simple extensions can be naturally embedded into Conway's algebraic closure of $\mathbb{F}_2$?
- How to think about ordinal exponentiation?

$\omega^\omega$ *can* be visualized, in what I think is a fairly nice way in a static 2D image featured on the wikipedia page for ordinal number:

Also, if you’re willing to allow dynamic visualizations, then Stephen Brooks’s transfinite number line goes well past $\epsilon_0$ (to $\Gamma_0$), as well as providing a more linear (if colorful) look at $\omega^\omega$:

It is wrong to think of $\omega^\omega$ as a hypercube of *actually* infinite dimension, which would correspond to the set of **all** infinite sequences of natural numbers, which is uncountable. One may think – if at all – of $\omega^\omega$ only as a hypercube of *potentially* infinite dimension, e.g. as $\bigcup_{n<\omega}\omega^n$, i.e. the functions $f:\omega \rightarrow \omega$ with finite support.

Especially: Other than $\omega^n$ – which is a well-ordered set of disjoint copies of $\omega$ – the ordinal $\omega^\omega$ cannot be thought of this way, even though the hypercube or the spiral representation might suggest it.

- A Geometric Proof of $\zeta(2)=\frac{\pi^2}6$? (and other integer inputs for the Zeta)
- An “elementary” approach to complex exponents?
- Irreducible polynomials have distinct roots?
- Understanding implicit differentiation with concepts like “function” and “lambda abstraction.”
- Is $\{0\}$ a field?
- Show that a finite group with certain automorphism is abelian
- proof of $1^4+2^4+…+n^4=\frac{n(n+1)(2n+1)(3n^2+3n-1)}{30}$
- Find Total number of ways out of N Number taking K numbers every M interval
- Does May's version of groupoid Seifert-van Kampen need path connectivity as a hypothesis?
- Fermat Last Theorem for non Integer Exponents
- Solution of a Lambert W function
- finding the multiplicative inverse in a field
- Group of order $p^2$ is commutative with prime $p$
- Calculate $\lim_{t\to\infty}\frac 1t\log\int_0^1 \cosh(tf(x))\mathrm d x$
- How to prove $\inf(S)=-\sup(-S)$?