This post is partially about opinions and partially about more precise mathematical questions. Most of this post is not as formal as a precise mathematical question. However, I hope that most readers will understand this post and the nature of the question. I will first try to explain what I wou...

I am taking a course in mathematics that covers countability. The trick with the uncountability of the real line is that no matter how many times you divide up an interval, there would still be a real number inside of that interval so that even the smallest interval contains "more than infinite" ...

I am wondering about this and have wondered about it for a short while. Usually physics is modeled using things based off of the Real Number Line $\mathbb{R}$, which is uncountable. (E.g. we may use powers of $\mathbb{R}$, we may use $\mathbb{C}$, we may use "manifolds" which essentially glue tog...

Are there any interesting theorems outside of set theory that use ordinals in their proofs? The only example I know of is Goodstein's theorem, and I haven't been able to find anything else. In other (more vague) words, what is the use of ordinals? (Other than Goodstein.) Theorems that use the w...

I want to know If there are solutions to $ \sum_{n = 0}^{\infty} a_n \space n^m = m^2 $ I assume this is not possible If we want it to hold for all real $m > 1/2 $. But i wonder about that equation If we only want it to hold for integer $ m > 0 $ ? Is there a solution Then ? Do we have uniq...

Turing's development of computability which led to the theoretical basis of computing. As a personal note, I take pride in dealing with models of ZF without the axiom of choice and all sort of strange consistency results. The only way an amorphous sets and D-finite combinatorics could be utilize...

This can be proved in several ways. The easiest I know is to use Cohen's second model (you can find it in Jech The Axiom of Choice in Chapter 5) in which there is a sequence $\langle A_n\mid n<\omega\rangle$ such that: $A_n\subseteq\mathcal P(\Bbb R)$. $A_n$ has two elements. If $n\neq m$, the...

The customary formulation of the Axiom of Infinity within Zermelo-Fraenkel set theory asserts the existence of an inductive set: a set $ I$ with $\varnothing\in I$ such that $x\in I$ implies $x\cup\{x\}\in I$. Since the intersection of any nonempty set of inductive sets is itself inductive...

A set $X$ is called Dedekind-infinite if there is a injective, non-surjective map $X\to X$ and Kuratowski-infinite if $\mathcal P(X)$ is not generated by $\{\emptyset\}\cup\{\{x\}|x\in X\}$ as a sub-semilattice with respect to $\cup$. In ZF without the axiom of infinity, any Dedekind-infinite se...

The idea behind Scott's trick of turning the equivalence classes into rather complicated sets is merely to allow working with the partial order of cardinalities within the theory with ease. In the presence of AC, we can always pick a canonical example for each cardinality, namely the initial ord...

According to this book: The Axiom of Choice is the most controversial axiom in the entire history of mathematics. Yet it remains a crucial assumption not only in set theory but equally in modern algebra, analysis, mathematical logic, and topology (often under the name Zorn's Lemma). I am no...

I saw this video about the busy beaver function and looked at the applications section of the Wikipedia article about the busy beaver function. I concluded that there is zero practical or even theoretical value in searching these numbers S(n) : n > 5. Now in the video the person says, that peopl...

Let, $V$ be a set of all homogeneous polynomial of degree $d$ in $n$-variables over a field $F$. Then dimension of $V_F$ is (A) $\left(\begin{matrix}n\\d\end{matrix}\right)$ (B) $\left(\begin{matrix}d\\n\end{matrix}\right)$ (C) $\left(\begin{matrix}n+d-1\\d-1\end{matrix}\right)$...

In Herbert Wilfs' gfology, the generating function is defined "formally" as If $\displaystyle f = \sum_{i \geq 0} a_i x^i$, and $\displaystyle g = \sum_{i \geq 0} b_i x^i $, we define $\displaystyle f+g := \sum_{i \geq 0} (a_i + b_i) x^i$ $\displaystyle fg := \sum_{i \geq 0}(\sum...