In set theory we typically work with infinite sized sets of different sizes.
We call these sizes cardinalities, because they are sizes of sets rather than numbers.
Click a panel to enter it, scroll to zoom, select each object to see what it is.
Every set of finite size.
Click a row to see which cardinality it represents.
Pronounced “aleph-zero” / “beth-zero” / “omega”.
Countable infinity: the smallest infinite set.
Scroll to get closer to the right-hand end, click a dot to see which number it represents.
Every possible subset of , with cardinality up to .
The same size as the real number line, or the “continuum”.
Scroll to travel, click any constellation to see which cardinality that subset has.
Every possible subset of , with cardinality up to .
Scroll to travel, click a ringed star to go inside it.
Every possible subset of , with cardinality up to .
Scroll to travel, click a ringed star to see how big it is.
Every level of the tower at once, collected into a single set.
This is a union of all our previous power sets, rather than itself being a power set.
Think of it like , but each point is an infinite cardinality rather than a finite number.
Scroll to move through the tower, click a level to see which set it is.
There are two conventions for labelling infinite sets, and they may or may not line up.
The alephs () are every infinite size there is, with each one being the next size up. The beths () are what you get by repeatedly taking power sets.
Every beth is equal to one of the alephs. Nobody can prove which. Saying that is the continuum hypothesis, and it can be neither proved nor disproved.
These are to the infinite sizes what the first infinity, , is to the finite ones.
cannot be reached by any amount of taking power sets and unions of finite sets.
Large cardinals cannot be reached by any amount of taking power sets and unions of the infinite sets we have already come across.
Whether any of them exist cannot be settled by the axioms of set theory (ZFC). Each one is an extra axiom you either adopt or don't.
The smallest large cardinal.
Take the union of everything we have so far.
The light outside cannot be reached from inside, however long you carry on. An inaccessible cardinal is an infinite size so vast that it cannot be reached by taking unions and power sets of anything that has come before.
Scroll and click inside, or click the light.
Pick an uncountable infinite cardinal.
Colour every pair of subsets of our cardinal red or blue, however you like.
If somewhere inside there is a whole universe of the same size as our original cardinal where everything came out one colour, then our original cardinal is weakly compact.
Scroll to travel, click a constellation, or click inside the red circle to see what is really in it.
The whole circle represents κ, a measurable cardinal.
Every subset of κ is either big or small, never both. The whole is big, each single star is small, and any fewer than κ small sets combine into a small one.
Click regions to combine them.
A map from the universe into itself that isn't the identity and keeps every statement true.
How far up it agrees with the universe is how strong the cardinal is.
Click one.
Inconsistent with ZFC.
A map j from the universe into itself, not the identity, which keeps every statement true, and whose target is the whole universe rather than some part of it.
Every cardinal above measurable is graded by how much of the universe its map reaches. This one reaches all of it.
Kunen proved in 1971 that there is no such map.
by Milette Gillow