Intereting Posts

The Gradient as a Row vs. Column Vector
Why can quotient groups only be defined for subgroups?
Solve a cubic polynomial?
Testing continuity of the function $f(x) = \lim\limits_{n \to \infty} \frac{x}{(2\sin{x})^{2n}+1} \ \text{for} \ x \in \mathbb{R}$
Quickest way to determine a polynomial with positive integer coefficients
Two circles intersection
Why aren't there more numbers like e, pi, and i? This is based on looking through the Handbook of Mathematical Functions and online.
Would it be fine to use Serge Lang's two Calculus books as textbooks for freshman as Maths major?
Prove that the only eigenvalue of a nilpotent operator is 0?
What is the integral of function $f(x) = (\sin x)/x$
Proof on showing function $f \in C^1$ on an open & convex set $U \subset \mathbb R^n$ is Lipschitz on compact subsets of $U$
Reference request: Nonlinear dynamics graduate reference
Showing that $(\mathbb{Z}/p^{a}\mathbb{Z})^*$ is a cyclic group
Which number was removed from the first $n$ naturals?
Principal ideal ring

In a previous, and quite popular, question it was discussed about whether or not $\pi$ contains all finite number combinations.

Let us assume for a moment that $\pi$ does in fact contain all finite combinations of numbers. What prevents $\pi$ from also containing all infinite sets?

It seems that at some point one would also see the first couple digits of, for example, $e$ (2.71828). But why does it need to stop there, couldn’t it contain a bunch of digits of $e$? Perhaps even an infinite number of digits of $e$?

- Are all infinities equal?
- Does the Mandelbrot fractal contain countably or uncountably many copies of itself?
- $\infty$ and $-\infty$ are to $\aleph_0$ / $\beth_0$ as “what” is to $\beth_1$?
- What good is infinity?
- Infinite square-rooting
- A question with infinity

My understanding is that $e$ could also be replaced by $\sqrt2$ or any other irrational number, so long as that irrational number contained all finite sets of number combinations. Which might imply that somewhere along the way, $e$ contains a number of digits of $\pi$. Implying this ridiculous situation where within $\pi$ we see $e$, and then within $e$ we again begin to see $\pi$ again. Then all the universe collapses into a singularity. Or maybe someone can just explain why one infinite sequence can’t contain another infinite sequence, and perhaps why we have not defined some type of super-infinity that can.

To reiterate the **primary question**: What prevents $\pi$, or other infinite irrational number that contains all finite sets of numbers, from also containing all infinite sets?

- random thought: are some infinite sets larger than other
- Types of infinity
- Use of the Reciprocal Fibonacci constant?
- Does an equation containing infinity not equal 0 or infinity exist?
- Proof that the harmonic series is < $\infty$ for a special set..
- limit as $x$ approaches infinity of $\frac{1}{x}$
- sum of irrational numbers - are there nontrivial examples?
- How big is the size of all infinities?
- Is there a “positive” definition for irrational numbers?
- Is $\aleph_0$ the minimum infinite cardinal number in $ZF$?

If the decimal expansion of $\pi$ contained the string of digits ‘$000\ldots$’ then it only has a finite number of non-zero digits (some subset of the digits before the string of $0$s starts) and so $\pi$ is a rational number. Contradiction.

There are an uncountable number of infinite digit strings, and $\pi$ contains only countably many infinite digit strings-namely all the ones that are the digits after some point in the expansion. It does contain infinite strings, just not most of them.

- Famous uses of the inclusion-exclusion principle?
- Proving a solution to a double recurrence is exhaustive
- A circle rolls along a parabola
- Universal $C^*$-algebra of countable family of self-adjoint operators have boundedly complete standard Schauder basis
- Derivatives of the Riemann zeta function at $s=0$
- difference between dot product and inner product
- Question about galois imaginary and modular arithmetic
- Why does basic algebra provide one value for $x$ when there should be two?
- A dilogarithm identity (simplification/compaction)
- What is the topological dual of a dual space with the weak* topology?
- Clarification of a remark of J. Steel on the independence of Goldbach from ZFC
- How to prove that for all $m,n\in\mathbb N$, $\ 56786730 \mid mn(m^{60}-n^{60})$?
- What is a projective ideal?
- how to prove, $f$ is onto if $f$ is continuous and satisfying $|f(x) – f(y)| ≥ |x – y|$ for all $x,y$ in $\mathbb{R}$
- Diophantine equation $a^m + b^m = c^n$ ($m, n$ coprime)