Articles of elementary set theory

$\Gamma \subset \mathbb{R}^{+}$ is uncountable. Can we choose a sequence from $\Gamma$ of which the sum is $\infty$

If $\Gamma$ is a set of uncountably many different positive real numbers, can we choose a sequence of pairwise different positive numbers from $\Gamma$, say $\{a_n\}$, such that $\sum a_n = \infty$ ? I am considering such problem and I think the answer should be yes. But what’s the rigorous way to prove it? Please […]

If $(W,<)$ is a well-ordered set and $f : W \rightarrow W$ is an increasing function, then $f(x) ≥ x$ for each $x \in W$

I could use a hand understanding a proof from Jech’s Set Theory. Firstly, note that Jech defines that an increasing function $f : P \rightarrow Q$ is a function that preserves strict inequalities (where $P$ and $Q$ are partially ordered sets). We then have Lemma 2.4. If $(W,<)$ is a well-ordered set and $f : […]

how to express the set of natural numbers in ZFC

I know that it is possible to represent each natural number using the concept of successor. But how do you assign meanings into the representation? I mean, when presented as a set, it’s just a set unless we agree on some definition that some form of sets refer to natural numbers. Or is it something […]

Do we get predicative ordinals above $\Gamma_0$ if we use hyperexponentiation?

I am trying to understand the Veblen hierarchy but I still find it confusing. The Feferman–Schütte ordinal, $\Gamma_0$, can be described as the set of all ordinals that can be written as finite expressions, starting from zero, using only the Veblen hierarchy and addition. The Veblen hierarchy starts with the function $\omega^\alpha$ and uses recursion […]

injection $\mathbb{N}\times\mathbb{N}\to\mathbb{N}$

Today a friend of mine told me a nice fact, but we couldn’t prove it. The fact is that there is an injection $\mathbb{N}\times\mathbb{N}\to\mathbb{N}$ defined by the fomula $(m,n)\mapsto (m+n)^{\max\{m,n\}}$, where $\mathbb{N}$ denotes the natural numbers. How to prove that this map is injective? It should be elementary. We might be overlooking something trivial. Thanks! […]

Are injectivity and surjectivity dual?

Are injectivity and surjectivity dual in some sense? Their set-theoretic definitions are quite different. In particular, the injectivity is a property of a function’s graph, while surjectivity is a relationship between the range of the function and its codomain.

Example of $\sigma$-algebra

I understood the definition of a $\sigma$-algebra, that its elements are closed under complementation and countable union, but as I am not very good at maths, I could not visualize or understand the intuition behind the meaning of “closed under complementation and countable union”. If we consider the set X to be a finite set, […]

How to solve probability when sample space is infinite?

I came up with a random problem yesterday: Suppose that in a random trial, each point $(x,y)$ where $x,y \in \mathbb{R}$ and $0 \leq x,y \leq 1$ is assigned a value of $0$ with 50% chance and a value of $1$ with 50% chance. What is a chance that there is a ‘continuous path’ that […]

I need to disprove an alternate definition of an ordered pair. Why is $\langle a,b\rangle = \{a,\{b\}\}$ incorrect?

So we know that the an ordered pair $(a,b) = (c,d)$ if and only if $a = c$ and $b = d$. And we know the Kuratowski definition of an ordered pair is: $(a,b) = \{\{a\},\{a,b\}\}$ The proof for the Kuratowski definition is in the wikipedia link. Now, why is this alternate definition, $(a,b) […]

Can there exist an uncountable planar graph?

I’m currently revising a course on graph theory that I took earlier this year. While thinking about planar graphs, I noticed that a finite planar graph corresponds to a (finite) polygonisation of the Euclidean plane (or whichever surface you’re working with). By considering, for example a full triangulation of the plane, you can find an […]