## Personal Math Scribbles

PS: I do not guarantee that my proofs are absolutely correct. They are probably not. These notes here serve to be my personal stash of some of my exercises and revision that I do not wish to discard. Feel free to provide comments.

# A brief revisit of Gödel's Incompleteness Theorems

06 Mar 2020  0

Gödel’s Incompleteness Theorems, particularly the second of the two, stated that any system that contains the arithmetic, i.e. discusses about statements regarding the natural numbers, cannot demonstrate its own consistency. The first of the two states that there will always be statements about the natural numbers that will not be provable by whatever axiomatic system that we construct to be consistent. One may continue adding more and more statements about the natural numbers, but it will never reach a state of completeness.

This sounds rather discouraging, especially when there are so many unsolved number theoretic problems that are out in the wild. And many more can easily sprout. Does there exist some identifiable pattern amongst numbers divisible by 7, just as those divisible...

# Weierstrass Continuity

08 Apr 2019  0

We know that there are certain definitions and results in mathematics that lead to unintuitive consequences, some mild, which just requires a bit of tweak in perspective, and others straight up mind-boggling. I came across one of the former today, See MathSE. which initially shoke me so much I felt like a truck hit me and had almost sent me down a very badly-timed existential...

# Primitive Element Theorem

12 Mar 2019  0

## Preliminary

First, let’s have some preliminaries to the Primitive Element Theorem.

### Separable Extension

We say that the extension $K$ of a field $F$ We usually write $K/F$ for $K$ being an extension of $F$. is separable if it is

• algebraic: i.e. each of its elements is a root of some polynomial of $F[x]$; and
• each $\alpha \in K$ is...

# Uncountability of (0, 1)

26 Oct 2018  0

The uncountability of $(0, 1)$ is an important result that demonstrates that $\mathbb{R}$, the set of real numbers, has a greater cardinality, i.e. “bigger size” than that of $\mathbb{N}$, the natural numbers. In other words, the result shows to us that $\mathbb{N}$ is countable while $\mathbb{R}$ is uncountable.

Georg Cantor was the first to have proven this result, and consequently showing to us that there are different “degrees” of infinity....

# Axiom of Choice

22 Sep 2017  0

In PMATH351 F17, I was (re)introduced to the Axiom of Choice. The following is my understanding and interpretation, or at least my attempt at interpreting the axiom, and other statements that are equivalent to it.

#### Axiom of Choice (AC)

$$\forall \{A_i\}_{i \in I} \; A_i \neq \emptyset \quad \prod_{i \in I} A_i \neq \emptyset$$

which says that for a family of non-empty sets, the Cartesian product of all such...