In set theory, sets containing an infinite number things are relatively easy to describe. For example, “All the counting numbers” is a set with an infinite number of things in it.

Many sets with an infinite number of things have a one to one correspondence to each other, meaning that we can describe a function that takes elements of one set as an input, gives elements of the other set as an output, and spans both sets - no element is skipped on either side.

“All the even counting numbers” has a one to one correspondence with “All the counting numbers”. You can look into Hilbert’s Hotel for a good demonstration of how this works.

Not all sets with an infinite number of things correspond with the set of all counting numbers, because some are fundamentally bigger. This difference in size doesn’t happen just once (e.g. there are countably infinite sets, and uncountably infinite sets, and that’s all we need to know), there are actually an infinite number of sets of progressively bigger infinite numbers of elements.

Because this is a confusing mess, we needed a way to keep track of how infinitely big each infinitely big set is, and the aleph cardinalities are the preferred way to do that. Any set with cardinality of aleph zero (aka “aleph null”) has a one to one correspondence with any other set with cardinality aleph zero. The same is true for every other aleph cardinality. Two sets of cardinality aleph thirty seven have a one to one correspondence with each other.

Anyways, busy beaver(tree(aleph omega)) is the biggest number.