# Definition of Ordinals in Set Theory in Layman Terms

I've taken a huge interest in the mathematical concept of infinity and often been contemplating the same over years. But the fundamental concept of set theory ordinals continues to evade my understanding. The questions below comprise (more or less) the gaps in my comprehension of the mathematical infinite:

1. Ordinal numbers in general (1st, 2nd, 3rd, 4th...) are entirely different from ordinal numbers in set theory, correct?
2. I understand that set theory ordinals are basically sets that contain a least element by definition. But, is it necessary for the elements of an ordinal to be strictly in order? For example, must the ordinal 4 be represented as {∅, {∅}, {∅,{∅}}, {∅, {∅}, {∅,{∅}}}....} and not as {{∅,{∅}}, ∅, {∅}, {∅, {∅}, {∅,{∅}}}....}?
3. The cardinality of ω is א‎0 (please correct me if I'm wrong), but where exactly is the position of ω along the number line. Is it א‎0th position (so to speak)?

I apologize for the naivety of the questions above (honestly, I really don't find a layman explanation of ordinals anywhere on the web. I saw a very good YouTube video though). The objective is to understand the core concept of set theory ordinals (well enough to be able to explain the same to a layman) rather than memorizing formal, mathematical definitions with little to no true comprehension of the same. Thanks in advance!

• I'm 99% confident that the video of which you speak is riddled with mathematical holes, which of course lay people cannot see. Better to leave it unmentioned... – user21820 Jul 16 '16 at 4:26
• Yes, I was kind of having the same thought myself :) – Hemnath Prabagaran Jul 16 '16 at 6:04
• I'm writing an explanation now that rigorously explains the intuition behind ordinals, but it will take a while. – user21820 Jul 16 '16 at 6:13
• Please, please do!! Mathematics is not my favourite subject, but when I came across Georg Cantor's perspective of infinity, it blew my mind! I've been a ardent follower of the set theory representation of infinity since. At this juncture, I find it extremely disappointing that virtually nowhere on the web I'm able to find a simple, core explanation for infinite ordinals. Your explanation, I'm sure, will be a huge contribution to existing, online information on transfinite ordinals. You have my utmost gratitude. – Hemnath Prabagaran Jul 16 '16 at 6:52
• When you say $n$th position, you are talking about an ordinal number $n$. Since $\aleph_0$ is not an ordinal, but rather a cardinal, talking about the $\aleph_0$th position along the ordinal line makes no sense. However $\omega$ is the $\omega$th ordinal. – Asaf Karagila Jul 16 '16 at 14:08

Counting has two purposes, namely for specifying sizes and indices. These are directly related for finite quantities, because the number of natural numbers (including $0$) less than $n$ (before the position $n$) is $n$. But in set theory, when generalizing to infinite sets these two notions become different. $\def\nn{\mathbb{N}}$ $\def\zz{\mathbb{Z}}$ $\def\qq{\mathbb{Q}}$ $\def\rr{\mathbb{R}}$ $\def\eq{\leftrightarrow}$ $\def\less{\smallsetminus}$ $\def\none{\varnothing}$