Intereting Posts

How to explain brackets to young students
Where is $f(x) := \sum_{n=1}^\infty \frac{\langle nx\rangle}{n^2+n}$ discontinuous?
For $x, y \in \mathbb{R}$, define $x \sim y $ if $x-y \in \mathbb{Q}$. Is $\mathbb{R}/\!\!\sim$ Hausdorff?
Summation of series $\sum_{k=0}^\infty 2^k/\binom{2k+1}{k}$
What is difference between cycle, path and circuit in Graph Theory
Number of Elements Not divisble by 3 or 5 or 7
Why is this series of square root of twos equal $\pi$?
What is the degree of the zero polynomial and why is it so?
Questions about right ideals
Creating an ellipsoidal 3D surface
Why does a countably infinite dimensional space not have an uncountable chain of subspaces?
Creating a function incrementally
What do modern-day analysts actually do?
local convexity of $L_p$ spaces
The Duplication Formula for the Gamma Function by logarithmic derivatives.

Can someone provide a description of ordinals within ZFC in a rigorous way that exhibits motivation? Every description or explanation I see in the literature or on the Internet is either too formal with no motivation provided or too simple with no rigor. I’ve been truly baffled by the concept of ordinals for the last couple days.

- When does it make sense to define a generator of a set system?
- Intuition behind arc length formula
- Do Kolmogorov's axioms really need only disjointness rather than pairwise disjointness?
- Prove that if $A$ is an infinite set then $A \times 2$ is equipotent to $A$
- Modus Operandi. Formulae for Maximum and Minimum of two numbers with a + b and $|a - b|$
- If A is an infinite set and B is at most countable set, prove that A and $A \cup B$ have the same cardinality
- In (relatively) simple words: What is an inverse limit?
- What makes elementary functions elementary?
- True or False $A - C = B - C $ if and only if $A \cup C = B \cup C$
- Finite Cartesian Product of Countable sets is countable?

Well.

To understand the motivation behind ordinals first you need to understand what are well-orders and why they are useful. Luckily, that’s not hard.

Well-orders are sets where every element has a unique “next” element, and we can use proofs by [transfinite] induction. These are useful since induction is useful to describe a step-by-step construction, well-orders can be used to argue step-by-step but in a way that allows us to go to infinity *and beyond*.

Well-ordered sets are very nice. Given two well-ordered sets, one of them is isomorphic to an initial segment of the other, and the isomorphism is unique, so the initial segment is unique as well. This is of course not true when you consider just linear orders, for example $[0,1]$ and $[0,1)$ are isomorphic to initial segments of one another, but there are plenty of such initial segments.

So now that we understood the motivation for well-ordered sets, we should probably observe that there is a proper class of isomorphic well-ordered sets in any given order type (read: isomorphism class) with the exception of the empty set, since there’s only one of those.

In other words, the collection of all well-ordered sets with exactly one point is not a set. So if we want to prove something about “all well-ordered sets”, which in turn means for all possible ways to do something step-by-step, we need to resort to various tricks and all sort of annoying meta0mathematical difficulties.

But as luck would have it, the von Neumann ordinals gives us an out. They allow us to pick one set which lies in every equivalence class, with a distinguished well-order of course. And since the language of set theory has only one binary relation symbol, the easiest thing to do is to make that well-order be the $\in$ relation on that set.

So an ordinal is a transitive set, such that $\in$ is a well-order on that set. Now we can show that every well-ordered set is isomorphic to a unique von Neumann ordinal, and that the isomorphism is unique. We can show that every two ordinals are comparable, which means that one of them is an element of the other (and therefore a subset, since those are transitive sets).

And more importantly, if we proved something holds for all ordinals, and that thing is a property depending only on the order, then we effectively proved it for *every* well-ordered set.

Starting from the most basic intuition, you use ordinals when you talk about things that are in a certain order: first, second…

This particular order can be represented by a never ending chain with a starting point: for every ring of the chain, there is a next ring that didn’t occur before (the chain never loops, goes on forever, and never splits).

So, we could want to find something similar in a set theory, since orders are important for a lot of reasons, the most basic being to be able to say when one ordered set is smaller than another ordered set sharing the way in which it orders its elements. (If we don’t care about orders, we can just use cardinality to compare sets)

In ZFC, we do this: we take a starting point as an ordinal, and decide on a way to generate a “next” ordinal from any ordinal you have.

One way is to take the empty set as a starting point, and define the ordinal that comes after x as xU{x}.

In this way, you have this:

{}

{{}}

{{}, {{}}}

{{}, {{}}, {{},{{}}}}

…

To visualize better, let’s name each ordinal after a natural number. We obtain this:

{}=0

{0}=1

{0,1}=2

{0,1,2}=3

…

So, with this method, each ordinal ends up being the set of the ordinals that precede it, if we take the membership relation as the order relation (taking the subset relation works too)

Taking a step further, we could want to expand our horizons and think of an ordinal as anything containing everything that precedes it.

So, let’s consider a set containing every ordinal we obtain starting from {} and finding the next one:

ω={0,1,2,…}

There’s no problem in considering ω an ordinal too: if we do, it certainly contains every ordinal that precedes it.

In finding ω, we took a ‘transfinite step’ from the ‘finite’ ordinals.

Now, we can repeat the process again, by taking ω as a new starting point:

ω+1=ωU{ω}={0,1,2…,ω},

ω+2={0,1,2…,ω,ω+1},

…

And then we can make, again, a transfinite step and go on forever from one level of ordinals to the next.

We can’t, however, have a set of all ordinals, as that leads to Burali-Forti’s paradox (very roughly, if there is a set of all ordinals, it would be the last ordinal, but if would also have a following ordinal as any ordinal does, so it can’t be the last one).

- How far is being star compact from being countably compact？
- Is $V$ a simple $\text{End}_kV$-module?
- Polynomials irreducible over $\mathbb{Q}$ but reducible over $\mathbb{F}_p$ for every prime $p$
- Measure theoretic entropy of General Tent maps
- Integrate: $\int \frac{dx}{x \sqrt{(x+a) ^2- b^2}}$
- Commuting in Matrix Exponential
- Understanding positive definite kernel
- Calculating probability of 'at least one event occurring'
- Evaluating $\int_a^b \arccos\left(x\,/\sqrt{(a+b)x-ab\,}\,\right)\,\mathrm {d}x$ assuming $0<a<b$
- Is there an algebraic closure for the quaternions?
- How often is a sum of $k$ consecutive primes also prime?
- Explicit fractional linear transformations which rotate the Riemann sphere about the $x$-axis
- Moving a limit inside an infinite sum
- How can I prove this limit doesn't exist?
- When does independence imply conditional independence, and vice versa?