**The inaccessible infinite**In math there are infinite numbers called

**cardinals**, which say how big sets are. Some are small. Some are big. Some are infinite. Some are so infinitely big that they're

**inaccessible** - very roughly, you can't reach them using operations you can define in terms of smaller cardinals.

An inaccessible cardinal is so big that if it exists, we can't prove that using the standard axioms of set theory!

The reason why is pretty interesting. Assume there's an inaccessible cardinal K. If we restrict attention to sets that we can build up using fewer than K operations, we get a whole lot of sets. Indeed, we get a set of sets that does not contain

*every* set, but which is big enough that it's "just as good" for all practical purposes.

We call such a set a

**Grothendieck universe**. It's not

**the universe** - we reserve that name for the collection of

*all* sets, which is too big to be a set. But all the usual axioms of set theory apply if we restrict attention to sets in a Grothendieck universe.

In fact, if an inaccessible cardinal exists, we can use the resulting Grothendieck universe to prove that the usual axioms of set theory are consistent! The reason is that the Grothendieck universe gives a "model" of the axioms - it obeys the axioms, so the axioms must be consistent.

However, Gödel's first incompleteness theorem says we

*can't* use the axioms of set theory to prove

*themselves* consistent.... unless they're inconsistent, in which case all bets are off.

The upshot is that we probably can't use the usual axioms of set theory to prove that it's consistent to assume there's an inaccessible cardinal. If we could, set theory would be inconsistent!

Nonetheless, bold set theorists are fascinated by inaccessible cardinals, and even much bigger cardinals. For starters, they love the infinite and its mysteries. But also, if we assume these huge infinities exist, we can prove things about arithmetic that we can't prove using the standard axioms of set theory!

I gave a very rough definition of inaccessible cardinals. It's not hard to be precise. A cardinal X is

**inaccessible** if you can't write it as a sum of fewer than X cardinals that are all smaller X, and if any cardinal Y is smaller than X, 2 to the Yth power is also smaller than X.

Well, not quite. According to this definition, 0 would be inaccessible - and so would the very smallest infinity. Neither of these can be gotten "from below". But we don't count these two cardinals as inaccessible.

https://ncatlab.org/nlab/show/inaccessible+cardinal#bigness