With ordinals, we use exponents to create really big numbers. The idea is that we can define ever-larger families of transfinite
ordinals using exponentiation. Exponentiation is defined in terms of
repeated multiplication, but it allows us to represent numbers that we
can’t express in terms of any finite sequence of multiplications.
As usual, the concept of ordinal exponentiation comes from a concept
of set exponentiation where the ordinal
αβ where α=|A| is the set of positions in
the well-ordering of Aβ; and Aβ is the
set of all ordered tuples of length β consisting of members of
A. (It should be obvious what a well-ordering on this looks like: it’s
the lexicographic ordering of the tuples based on the ordering of
elements in A.)
This shouldn’t be too surprising: the basic idea of exponentiation is, as I said, repeated multiplication, so that A2=A×A, which is the set of ordered pairs of members of A. To be a bit more formal about what we mean by repeated multiplication:
- α0 = 1
- α1 = α
- αb+1 = αβ×α
- If β is a limit ordinal, then αβ is the
limit ordinal of αγ for all γ<β.
The main use of exponentiation is to let us express ever larger
ordinals. We added ω to the ordinals as the first transfinite,
and we can talk about many extremely large numbers by using ω
and multiples of ω. But finite multiples of ω can only get us so far; then we need to start talking in exponents. Exponents are where things get
seriouly large: ωω, ωωω, and so on. Infinite exponents of ω open up whole new realms of ever larger numbers.
One thing that I consistently get screwed up is the difference between cardinal exponentiation and ordinal exponentiation. Given how closely related the ordinals and cardinals are, and given how they’re both consistent extensions of the naturals, it seems like they should behave the same in exponentiation – but they don’t. Not even close. In ordinal arithmetic, 2ω=ω; but in cardinal arithmetic, 2ℵ0 is the cardinality of the reals – at least ℵ1.
Once we start playing with ordinal exponents, we can find some interesting large objects by using powers of ω, but they’re all limited: using ω, we can’t construct any ordinal which can describe the set of positions in an uncountable set: ω only gives us the ability to find places in countably infinite sets.
But we can still create some interesting things. For example, there’s something called ε numbers. ε-numbers are fixed point limit ordinals of exponent chains; they’re the first numbers unreachable from ω; the smallest ε-number is the limit – the first number larger than anything definable using exponents of ω:
The ε numbers are the set of numbers x such that ωx=x. ε0, the first ε number is also the limit ordinal of ωωω…ω, where that stack of exponents has length ω.
Even the ε numbers are ordinals of countable sets. In general, we don’t really worry about ordinals beyond the ε numbers, because they’re are results showing that if a transfinite induction proof covers everything up to ε0, then it will also be true for all of the ordinals, including ordinals for uncountable sets.