basic constructions:
strong axioms
further
In classical mathematics, a countable ordinal is the order type? of a well-ordered set whose cardinality is countable (i.e., either finite or denumerable).
Countable ordinals play an important role in recursion theory, computability theory, and in proof theory, where they are used for example to measure the consistency strength of formal theories via ordinal analysis. They also play a role in philosophy of mathematics, for example to discuss what it means for a notion to be “predicative”.
As indicated above, countable ordinals can be defined structurally as countable well-ordered sets (or “wosets”). As is the case with any ordered set, a woset can be regarded as a coalgebra over the covariant power-set functor,
A coalgebra map between wosets, aka a simulation, amounts to an inclusion of the domain as an initial segment of the codomain.
The category of countable wosets and simulations, thus ordered by inclusion, is a preorder and in fact equivalent to a woset. In material set theory, this woset is often identified with the first uncountable ordinal (well-ordered set under the membership relation), often denoted . Structurally, we can define up to isomorphism as a skeleton of the category of countable ordinals, and so the study of countable ordinals becomes a study of the order structure of .
This has no top element (of course not: otherwise would be a countable ordinal), and much of the interest and fun of the subject lies in playing the childhood game of seeing who can name the bigger number, or in this case the bigger countable ordinal. A key fact is that is countably cocomplete, and therefore behaves like a self-contained universe with respect to countably cocontinuous operations. There is quite a lot of scope in what one can build up to using such operations. After playing the larger ordinal game for a while, and becoming impressed by the sizes of ordinals one can define – while recognizing at the same time that one is “never getting anywhere” relative to itself – it is hard not to become awestruck by the staggering immensity of .
Anyone who is unimpressed by the size of large countable ordinals has almost surely never seriously played with them. For that matter, people who are unimpressed by large finite numbers have probably never played around with them either, or at least not all that imaginatively! (And interestingly, one can use large countable ordinals reflectively to construct enormous integers, or one can use large uncountable ordinals reflectively to construct enormous countable ordinals.)
Here we discuss the use of fixed point theory in describing the lower stages of the Veblen hierarchies of countable ordinals, which will lead up to the Feferman-Schütte ordinal. We freely allow ourselves to reason classically and use the principle of excluded middle.
The ordinal is countably cocomplete, and we are primarily interested in functors or order-preserving maps that preserve colimits of countable (finite or denumerable) diagrams – or countable chains, without loss of generality. We’ll call such maps continuous, for short. If for all , then we say that is inflationary.
Preservation of colimits of finite nonempty chains comes for free, so we can ignore that. The colimit of the empty chain is the bottom element , and we will make a point of including preservation of that.
The first result is a baby form of a general theorem due to Adámek on initial algebras of endofunctors.
Suppose is inflationary and continuous. Then every has an upper bound that is a fixed point under .
For any , the colimit of exists, and since preserves countable colimits, is a fixed point of .
If is order-preserving and satisfies for all limit (i.e., non-successor) ordinals , then is continuous.
The converse of this result is trivially true.
For the case , the set is empty and thus has as its supremum, so .
Suppose that is the colimit of a chain with countably infinite image. Then this chain is cofinal in the set , and similarly is cofinal in . Therefore the colimit of this chain is the same as the supremum of , which is .
Suppose is inflationary and continuous. Define to be the function that takes each to the least fixed point of that is greater than for all . Then is also inflationary and continuous.
By definition, whenever , so is order-preserving. That is inflationary is proved by induction: given that for all , we have i.e. for all , whence . To prove is continuous, let be a non-successor; by continuity of we have
so that this supremum is in fact a fixed point of , and thus the least fixed point of greater than every for . It therefore follows by definition of that
and so, by the preceding lemma, is continuous.
The Veblen hierarchy is a family of functions , one for each , defined as follows.
This is inflationary, and continuous by the lemma.
By induction and the preceding proposition, is inflationary and continuous at successor cardinals , and also at limit cardinals since countable colimits of inflationary and continuous maps are also inflationary and continuous.
It is worth contemplating the growth of the first few . For , we have if is a successor, and if is a limit ordinal.
This means that next we have , , , …, , and so on. The first fixed point past is at .
Next, we have , , , …, , and so on. The first fixed point past is the same as the first fixed point of (a countably iterated exponential stack), typically denoted as .
We pause to note that is sometimes considered the first reasonable thing to call a “large countable ordinal”. An ordinal analysis first carried by by Gerhard Gentzen asserts, roughly speaking, that the consistency of Peano arithmetic is provable from primitive recursive arithmetic (which is weaker than PA) augmented by the principle of induction up to .
Pressing on, we have , , (the first fixed point of past ), and so on. The first fixed point past of is the limit of , sometimes indicated by an infinite descending stack of subscripts . This ordinal is of size that far, far surpasses any human powers of visualization.
And clearly we are only just getting started. Each successive dwarfs its predecessor to an indescribable degree. To speak nothing of , and so on.
Despite such colossal sizes, it is possible to effectively notate such ordinals using nothing more than finite binary trees, all the way up to the Feferman-Schütte ordinal, which seems to be the first of the “large countable ordinals” that the experts consider big enough to honor with human names. (There are others much further down the pike, such as the Bachmann-Howard ordinal or the considerably larger, almost godlike Church-Kleene ordinal.)
Before introducing this, we prove the following lemmas.
If , then every value is a fixed point of .
This is clear if . Suppose is a fixed point of whenever . If is a successor , then we have
and if is a limit, then
which completes the proof.
For all , we have .
By induction. Suppose for all . Since , we have
since is by the preceding lemma a fixed point of . Since is both a limit ordinal and an upper bound for every , it follows that .
Notice that by definition, is continuous in the argument , since we defined at limit ordinals . Therefore the function is inflationary (by lemma ) and continuous. Applying proposition , it has a least fixed point. This fixed point is called the Feferman-Schütte ordinal, denoted by .
If is less than , then there exist , both less than , such that .
If is not a limit ordinal, then and hence . Obviously cannot be a value of for , since all such values are limit ordinals.
If is a limit ordinal and , then there is some for which is not a fixed point of . Otherwise we would have for all , and this would force (by definition of ). And in that case
so that is a fixed point of , contradicting .
Thus, for a limit ordinal, take to be the least ordinal such that . Note that is a successor, , because we have for any , and this would imply if were a limit.
We have for some (for if for all , we could infer is a fixed point of ). Let be the least ordinal (less than ) such that . Then in fact this last inequality is an equality. For suppose ; then by definition of , we have
and we also have that by definition of . Since is by definition the least -fixpoint greater than , we immediately infer , as required. If on the other hand is a limit, we have
for all , and we again infer , as required.
This theorem gives a canonical finite binary tree representation of any , by recursion. The binary tree of consists of just as root, with no child-pair. Given that every less than has been assigned a binary tree , we define to have the child-pair where
is the least ordinal less than such that , and
is the least ordinal less than such that .
The resulting binary tree is finite for every , since it has binary branching and there are no infinite branches or paths up the tree (since that would yield an infinite descent with no least element, contradicting well-orderedness).
This finite binary tree notation is one of many possible ordinal notations for naming countable ordinals. It is more expressive the Cantor normal form that provides a notation for ordinals less than , where one writes uniquely in the form
with and , and continues recursively.
However, Cantor normal form has the virtue of uniqueness. For the binary operation , it is possible to have two distinct pairs , which are equivalent in the sense that
we counteracted this by using the pair which was least in its equivalence class with respect to lexicographic order.
Of course, having reached as the least fixed point of , there is nothing stopping one from starting all over again with a new Veblen hierarchy, defining and defining analogously to how we defined .
In one or another way we can continue extending ordinal notation schemes, but all such schemes are recursive by definition, and at some point we cannot go any further:
A countable ordinal is recursive if there is a computable relation on (or a recursive subset of ) that well-orders , so that the resulting woset is isomorphic to .
There are countably many computable relations on , and so the supremum of the recursive ordinals in exists. It is called the Church-Kleene ordinal. It is the first non-recursive ordinal.
For a light-hearted introduction with many useful references, see
A very readable account of countable ordinals up to the Feferman-Schütte ordinal is
An application of coalgebra theory to ordinal notation systems is described here:
Ordinal analysis as measuring consistency strengths of theories is well-described in this paper:
Last revised on November 7, 2018 at 12:24:52. See the history of this page for a list of all contributions to it.