the First Inaccessible Cardinal
(this number was submitted by @hyperoperationfractallisation)
If you've spent any amount of time around set theory, you've probably seen increasingly large infinities, ℵ0, ℵ1, and so on, marching upwards in a tidy sequence. It's tempting to think that all infinite sizes behave roughly the same way: you take a set, build something bigger from it, and keep going forever.
But at some point, something strange happens.
Eventually, you encounter cardinals that cannot be obtained from smaller cardinals using the usual operations of taking successors, unions, and power sets. This is where inaccessible cardinals enter the picture.
I suppose it's best to start simple.
We begin with ℵ0, the size of the natural numbers. From there, we can build larger and larger infinities in a few standard ways.
1. You can take the next size up (successor cardinals like ℵ1, ℵ2, ...)
ℵ1 is the smallest cardinal strictly bigger than ℵ0. Then ℵ2 is the smallest cardinal strictly bigger than ℵ1. And so on.
2. Take unions of smaller sets
Suppose you have a whole collection of sets, A0, A1, A2, and so on, where each is "small" (say, countable). If you take the union of all of these sets, you get a countable union of countable sets, which is still countable. But if you take ℵ1 many sets, each of size less than ℵ1, then their union can reach size ℵ1.
More generally, by taking unions of many smaller sets, you can sometimes build larger cardinals.
3. Take power sets (which are always strictly bigger).
Given a set A, we write P(A) for the set of all subsets of A. For example, if A = {1,2}, then P(A) = {∅, {1}, {2}, {1,2}}. So the size doubles (from 2 to 4). For finite sets, this doubles the exponent. A set of size n has a power set of size 2^n.
If |A| = ℵ0, then |P(A)| = 2^ℵ0, which is the size of the real numbers. And crucially, 2^ℵ0 > ℵ0. So power sets always give you a strictly larger infinity.
Using these operations, we can climb higher and higher through the infinite landscape.
So a natural question is to ask if every infinite size is reachable this way.
A Cardinal You Can't Reach
An inaccessible cardinal is, roughly speaking, a size so large it cannot be obtained from smaller cardinals using these operations applied only below κ.
More precisely, an inaccessible cardinal κ has 3 properties:
Uncountable: which means that it is bigger than ℵ0
Regular: κ cannot be written as the union of fewer than κ sets, each of size less than κ
Strong limit: This means that even taking power sets of smaller cardinals never catches up to it. More formally, for every λ < κ, we have 2^λ < κ
Taken together, this means that κ is out of reach of all the standard ways we construct larger infinities.
Why is it called "Inaccessible"?
The name is quite literal. If you imagine building cardinals step-by-step from below, an inaccessible cardinal is one you simply can't arrive at using those steps. No construction using fewer than κ many stages, each involving only smaller cardinals, can reach κ.
It's like climbing a staircase where, at some point, there's a platform floating above you with no steps leading to it.
Here's a fun fact. You cannot prove that inaccessible cardinals exist using the standard axioms of set theory (ZFC).
To assume one exists, you have to add a new axiom. That's why inaccessible cardinals are called the first "large cardinal" concept.
Inaccessible cardinals are the entry point into a whole hierarchy of stronger and stronger assumptions about infinity.
In a sense, it's the first point where infinity stops feeling incremental and starts feeling foundational.
