Aleph number

Aleph numbers are numbers that are bigger than all the numbers, probably including infinity.

• ${\mathrm{\aleph }}_0$ is the first aleph number, and it represents how many numbers exist, including ones only theorized to exist.
• ${\mathrm{\aleph }}_1$ is the second aleph number, and it represents how many ways you can list ${\mathrm{\aleph }}_0$ things in increasing order such that: there is always a first object, even if you remove some (but not all) of the objects from the line. Such an ordering is considered "well-ordered".
  • It is in fact possible to order an infinite number of objects in increasing order in different ways, for example: the list of finite numbers has no end (it "goes on forever"), but if you then add to the end a new number defined to come "immediately after" all the finite numbers (we'll call it $\omega$), the list's size won't change (${\mathrm{\aleph }}_0+1={\mathrm{\aleph }}_0$), but its order is different because it now has an end.
  • Sometimes ${\mathrm{\aleph }}_1$ is the number of sets of numbers, but other times is is less than that.
• ${\mathrm{\aleph }}_2$ is the total number of well-orders of ${\mathrm{\aleph }}_1$ things; this pattern continues until...
• ${\mathrm{\aleph }}_{\omega }$ has no previous aleph number ($\omega$ comes after all the finite numbers, which has no end), so it has to use a slightly more general definition.
• In general an aleph number is the total number of well-orders of $n$ things, for all $n$ which is either finite or a previous aleph number.
• ${\mathrm{\aleph }}_{\omega +1}$ does have a previous aleph number, so it is the total number of well-orders of ${\mathrm{\aleph }}_{\omega }$ things.
• ${\mathrm{\aleph }}_{{\omega }_{{\omega }_{{\omega }_{{\omega }_{{\omega }_{{\omega }_{{\omega }_{{\omega }_{{\omega }_{{\omega }_{{\omega }_{{\omega }_{{\omega }_{{\omega }_{{\omega }_{{\omega }_{{\omega }_{{\omega }_{{\omega }_{{\omega }_{{\omega }_{{\omega }_{{\omega }_{{\omega }_{{\omega }_{{\omega }_{{\omega }_{{\omega }_{{\omega }_{{\omega }_{{\omega }_{{\omega }_{{\omega }_{{\omega }_{\ddots }}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}$ what?
• If you choose the Axiom of choice, all infinite sizes are aleph numbers, which leads to a simpler definition: an aleph number is the smallest infinite size larger than all previous aleph numbers.