Ordinal collapsing function

Let ${\displaystyle \Omega }$ stand for the first uncountable ordinal ${\displaystyle \omega _{1}}$, or, in fact, any ordinal which is an ${\displaystyle \varepsilon }$-number and guaranteed to be greater than all the countable ordinals which will be constructed (for example, the Church–Kleene ordinal is adequate for our purposes; but we will work with ${\displaystyle \omega _{1}}$ because it allows the convenient use of the word countable in the definitions).

We define a function ${\displaystyle \psi }$ (which will be non-decreasing and continuous), taking an arbitrary ordinal ${\displaystyle \alpha }$ to a countable ordinal ${\displaystyle \psi (\alpha )}$, recursively on ${\displaystyle \alpha }$, as follows:

Assume ${\displaystyle \psi (\beta )}$ has been defined for all ${\displaystyle \beta <\alpha }$, and we wish to define ${\displaystyle \psi (\alpha )}$.

Let ${\displaystyle C(\alpha )}$ be the set of ordinals generated starting from ${\displaystyle 0}$, ${\displaystyle 1}$, ${\displaystyle \omega }$ and ${\displaystyle \Omega }$ by recursively applying the following functions: ordinal addition, multiplication and exponentiation and the function ${\displaystyle \psi \upharpoonright _{\alpha }}$, i.e., the restriction of ${\displaystyle \psi }$ to ordinals ${\displaystyle \beta <\alpha }$. (Formally, we define ${\displaystyle C(\alpha )_{0}=\{0,1,\omega ,\Omega \}}$ and inductively ${\displaystyle C(\alpha )_{n+1}=C(\alpha )_{n}\cup \{\beta _{1}+\beta _{2},\beta _{1}\beta _{2},{\beta _{1}}^{\beta _{2}}:\beta _{1},\beta _{2}\in C(\alpha )_{n}\}\cup \{\psi (\beta ):\beta \in C(\alpha )_{n}\land \beta <\alpha \}}$ for all natural numbers ${\displaystyle n}$ and we let ${\displaystyle C(\alpha )}$ be the union of the ${\displaystyle C(\alpha )_{n}}$ for all ${\displaystyle n}$.)

Then ${\displaystyle \psi (\alpha )}$ is defined as the smallest ordinal not belonging to ${\displaystyle C(\alpha )}$.

In a more concise (although more obscure) way:

${\displaystyle \psi (\alpha )}$ is the smallest ordinal which cannot be expressed from ${\displaystyle 0}$, ${\displaystyle 1}$, ${\displaystyle \omega }$ and ${\displaystyle \Omega }$ using sums, products, exponentials, and the ${\displaystyle \psi }$ function itself (to previously constructed ordinals less than ${\displaystyle \alpha }$).

Here is an attempt to explain the motivation for the definition of ${\displaystyle \psi }$ in intuitive terms: since the usual operations of addition, multiplication and exponentiation are not sufficient to designate ordinals very far, we attempt to systematically create new names for ordinals by taking the first one which does not have a name yet, and whenever we run out of names, rather than invent them in an ad hoc fashion or using diagonal schemes, we seek them in the ordinals far beyond the ones we are constructing (beyond ${\displaystyle \Omega }$, that is); so we give names to uncountable ordinals and, since in the end the list of names is necessarily countable, ${\displaystyle \psi }$ will "collapse" them to countable ordinals.

To clarify how the function ${\displaystyle \psi }$ is able to produce notations for certain ordinals, we now compute its first values.

First consider ${\displaystyle C(0)}$. It contains ordinals ${\displaystyle 0,1,2,3,\omega ,\omega +1,\omega +2,\omega \cdot 2,\omega \cdot 3,\omega ^{2},\omega ^{3},\omega ^{\omega },\omega ^{\omega ^{\omega }}}$ and so on. It also contains such ordinals as ${\displaystyle \Omega ,\Omega +1,\omega ^{\Omega +1},\Omega ^{\Omega }}$. The first ordinal which it does not contain is ${\displaystyle \varepsilon _{0}}$ (which is the limit of ${\displaystyle \omega }$, ${\displaystyle \omega ^{\omega }}$, ${\displaystyle \omega ^{\omega ^{\omega }}}$ and so on — less than ${\displaystyle \Omega }$ by assumption). The upper bound of the ordinals it contains is ${\displaystyle \varepsilon _{\Omega +1}}$ (the limit of ${\displaystyle \Omega }$, ${\displaystyle \Omega ^{\Omega }}$, ${\displaystyle \Omega ^{\Omega ^{\Omega }}}$ and so on), but that is not so important. This shows that ${\displaystyle \psi (0)=\varepsilon _{0}}$.

Similarly, ${\displaystyle C(1)}$ contains the ordinals which can be formed from ${\displaystyle 0}$, ${\displaystyle 1}$, ${\displaystyle \omega }$, ${\displaystyle \Omega }$ and this time also ${\displaystyle \varepsilon _{0}}$, using addition, multiplication and exponentiation. This contains all the ordinals up to ${\displaystyle \varepsilon _{1}}$ but not the latter, so ${\displaystyle \psi (1)=\varepsilon _{1}}$. In this manner, we prove that ${\displaystyle \psi (\alpha )=\varepsilon _{\alpha }}$ inductively on ${\displaystyle \alpha }$: the proof works, however, only as long as ${\displaystyle \alpha <\varepsilon _{\alpha }}$. We therefore have:

${\displaystyle \psi (\alpha )=\varepsilon _{\alpha }=\phi _{1}(\alpha )}$ for all ${\displaystyle \alpha \leq \zeta _{0}}$, where ${\displaystyle \zeta _{0}=\phi _{2}(0)}$ is the smallest fixed point of ${\displaystyle \alpha \mapsto \varepsilon _{\alpha }}$.

(Here, the ${\displaystyle \phi }$ functions are the Veblen functions defined starting with ${\displaystyle \phi _{1}(\alpha )=\varepsilon _{\alpha }}$.)

Now ${\displaystyle \psi (\zeta _{0})=\zeta _{0}}$ but ${\displaystyle \psi (\zeta _{0}+1)}$ is no larger, since ${\displaystyle \zeta _{0}}$ cannot be constructed using finite applications of ${\displaystyle \phi _{1}\colon \alpha \mapsto \varepsilon _{\alpha }}$ and thus never belongs to a ${\displaystyle C(\alpha )}$ set for ${\displaystyle \alpha \leq \Omega }$, and the function ${\displaystyle \psi }$ remains "stuck" at ${\displaystyle \zeta _{0}}$ for some time:

${\displaystyle \psi (\alpha )=\zeta _{0}}$ for all ${\displaystyle \zeta _{0}\leq \alpha \leq \Omega }$.

Again, ${\displaystyle \psi (\Omega )=\zeta _{0}}$. However, when we come to computing ${\displaystyle \psi (\Omega +1)}$, something has changed: since ${\displaystyle \Omega }$ was ("artificially") added to all the ${\displaystyle C(\alpha )}$, we are permitted to take the value ${\displaystyle \psi (\Omega )=\zeta _{0}}$ in the process. So ${\displaystyle C(\Omega +1)}$ contains all ordinals which can be built from ${\displaystyle 0}$, ${\displaystyle 1}$, ${\displaystyle \omega }$, ${\displaystyle \Omega }$, the ${\displaystyle \phi _{1}\colon \alpha \mapsto \varepsilon _{\alpha }}$ function up to ${\displaystyle \zeta _{0}}$ and this time also ${\displaystyle \zeta _{0}}$ itself, using addition, multiplication and exponentiation. The smallest ordinal not in ${\displaystyle C(\Omega +1)}$ is ${\displaystyle \varepsilon _{\zeta _{0}+1}}$ (the smallest ${\displaystyle \varepsilon }$-number after ${\displaystyle \zeta _{0}}$).

We say that the definition ${\displaystyle \psi (\Omega )=\zeta _{0}}$ and the next values of the function ${\displaystyle \psi }$ such as ${\displaystyle \psi (\Omega +1)=\varepsilon _{\zeta _{0}+1}}$ are impredicative because they use ordinals (here, ${\displaystyle \Omega }$) greater than the ones which are being defined (here, ${\displaystyle \zeta _{0}}$).

The fact that ${\displaystyle \psi (\Omega +\alpha )}$ equals ${\displaystyle \varepsilon _{\zeta _{0}+\alpha }}$ remains true for all ${\displaystyle \alpha \leq \zeta _{1}=\phi _{2}(1)}$. (Note, in particular, that ${\displaystyle \psi (\Omega +\zeta _{0})=\varepsilon _{\zeta _{0}\cdot 2}}$: but since now the ordinal ${\displaystyle \zeta _{0}}$ has been constructed there is nothing to prevent from going beyond this). However, at ${\displaystyle \zeta _{1}=\phi _{2}(1)}$ (the first fixed point of ${\displaystyle \alpha \mapsto \varepsilon _{\alpha }}$ beyond ${\displaystyle \zeta _{0}}$), the construction stops again, because ${\displaystyle \zeta _{1}}$ cannot be constructed from smaller ordinals and ${\displaystyle \zeta _{0}}$ by finitely applying the ${\displaystyle \varepsilon }$ function. So we have ${\displaystyle \psi (\Omega \cdot 2)=\zeta _{1}}$.

The same reasoning shows that ${\displaystyle \psi (\Omega +\alpha )=\phi _{2}(\alpha )}$ for all ${\displaystyle \alpha \leq \phi _{3}(0)}$, where ${\displaystyle \phi _{2}}$ enumerates the fixed points of ${\displaystyle \phi _{1}\colon \alpha \mapsto \varepsilon _{\alpha }}$ and ${\displaystyle \phi _{3}(0)}$ is the first fixed point of ${\displaystyle \phi _{2}}$. We then have ${\displaystyle \psi (\Omega ^{2})=\phi _{3}(0)}$.

Again, we can see that ${\displaystyle \psi (\Omega ^{\alpha })=\phi _{1+\alpha }(0)}$ for some time: this remains true until the first fixed point ${\displaystyle \Gamma _{0}}$ of ${\displaystyle \alpha \mapsto \phi _{\alpha }(0)}$, which is the Feferman–Schütte ordinal. Thus, ${\displaystyle \psi (\Omega ^{\Omega })=\Gamma _{0}}$ is the Feferman–Schütte ordinal.

We have ${\displaystyle \psi (\Omega ^{\Omega }+\Omega ^{\alpha })=\phi _{\Gamma _{0}+\alpha }(0)}$ for all ${\displaystyle \alpha \leq \Gamma _{1}}$ where ${\displaystyle \Gamma _{1}}$ is the next fixed point of ${\displaystyle \alpha \mapsto \phi _{\alpha }(0)}$. So, if ${\displaystyle \alpha \mapsto \Gamma _{\alpha }}$ enumerates the fixed points in question (which can also be noted ${\displaystyle \phi (1,0,\alpha )}$ using the many-valued Veblen functions) we have ${\displaystyle \psi (\Omega ^{\Omega }(1+\alpha ))=\Gamma _{\alpha }}$, until the first fixed point ${\displaystyle \phi (1,1,0)}$ of the ${\displaystyle \alpha \mapsto \Gamma _{\alpha }}$ itself, which will be ${\displaystyle \psi (\Omega ^{\Omega +1})}$ (and the first fixed point ${\displaystyle \phi (2,0,0)}$ of the ${\displaystyle \alpha \mapsto \phi (1,\alpha ,0)}$ functions will be ${\displaystyle \psi (\Omega ^{\Omega \cdot 2})}$). In this manner:

• ${\displaystyle \psi (\Omega ^{\Omega ^{2}})}$ is the Ackermann ordinal (the range of the notation ${\displaystyle \phi (\alpha ,\beta ,\gamma )}$ defined predicatively),
• ${\displaystyle \psi (\Omega ^{\Omega ^{\omega }})}$ is the "small" Veblen ordinal (the range of the notations ${\displaystyle \phi (\cdot )}$ predicatively using finitely many variables),
• ${\displaystyle \psi (\Omega ^{\Omega ^{\Omega }})}$ is the "large" Veblen ordinal (the range of the notations ${\displaystyle \phi (\cdot )}$ predicatively using transfinitely-but-predicatively-many variables),
• the limit ${\displaystyle \psi (\varepsilon _{\Omega +1})}$ of ${\displaystyle \psi (\Omega )}$, ${\displaystyle \psi (\Omega ^{\Omega })}$, ${\displaystyle \psi (\Omega ^{\Omega ^{\Omega }})}$, etc., is the Bachmann–Howard ordinal: after this our function ${\displaystyle \psi }$ is constant, and we can go no further with the definition we have given.

We now explain more systematically how the ${\displaystyle \psi }$ function defines notations for ordinals up to the Bachmann–Howard ordinal.

Recall that if ${\displaystyle \delta }$ is an ordinal which is a power of ${\displaystyle \omega }$ (for example ${\displaystyle \omega }$ itself, or ${\displaystyle \varepsilon _{0}}$, or ${\displaystyle \Omega }$), any ordinal ${\displaystyle \alpha }$ can be uniquely expressed in the form ${\displaystyle \delta ^{\beta _{1}}\gamma _{1}+\ldots +\delta ^{\beta _{k}}\gamma _{k}}$, where ${\displaystyle k}$ is a natural number, ${\displaystyle \gamma _{1},\ldots ,\gamma _{k}}$ are non-zero ordinals less than ${\displaystyle \delta }$, and ${\displaystyle \beta _{1}>\beta _{2}>\cdots >\beta _{k}}$ are ordinal numbers (we allow ${\displaystyle \beta _{k}=0}$). This "base ${\displaystyle \delta }$ representation" is an obvious generalization of the Cantor normal form (which is the case ${\displaystyle \delta =\omega }$). Of course, it may quite well be that the expression is uninteresting, i.e., ${\displaystyle \alpha =\delta ^{\alpha }}$, but in any other case the ${\displaystyle \beta _{i}}$ must all be less than ${\displaystyle \alpha }$; it may also be the case that the expression is trivial (i.e., ${\displaystyle \alpha <\delta }$, in which case ${\displaystyle k\leq 1}$ and ${\displaystyle \gamma _{1}=\alpha }$).

If ${\displaystyle \alpha }$ is an ordinal less than ${\displaystyle \varepsilon _{\Omega +1}}$, then its base ${\displaystyle \Omega }$ representation has coefficients ${\displaystyle \gamma _{i}<\Omega }$ (by definition) and exponents ${\displaystyle \beta _{i}<\alpha }$ (because of the assumption ${\displaystyle \alpha <\varepsilon _{\Omega +1}}$): hence one can rewrite these exponents in base ${\displaystyle \Omega }$ and repeat the operation until the process terminates (any decreasing sequence of ordinals is finite). We call the resulting expression the iterated base ${\displaystyle \Omega }$ representation of ${\displaystyle \alpha }$ and the various coefficients involved (including as exponents) the pieces of the representation (they are all ${\displaystyle <\Omega }$), or, for short, the ${\displaystyle \Omega }$-pieces of ${\displaystyle \alpha }$.

• The function ${\displaystyle \psi }$ is non-decreasing and continuous (this is more or less obvious from its definition).
• If ${\displaystyle \psi (\alpha )=\psi (\beta )}$ with ${\displaystyle \beta <\alpha }$ then necessarily ${\displaystyle C(\alpha )=C(\beta )}$. Indeed, no ordinal ${\displaystyle \beta '}$ with ${\displaystyle \beta \leq \beta '<\alpha }$ can belong to ${\displaystyle C(\alpha )}$ (otherwise its image by ${\displaystyle \psi }$, which is ${\displaystyle \psi (\alpha )}$ would belong to ${\displaystyle C(\alpha )}$ — impossible); so ${\displaystyle C(\beta )}$ is closed by everything under which ${\displaystyle C(\alpha )}$ is the closure, so they are equal.
• Any value ${\displaystyle \gamma =\psi (\alpha )}$ taken by ${\displaystyle \psi }$ is an ${\displaystyle \varepsilon }$-number (i.e., a fixed point of ${\displaystyle \beta \mapsto \omega ^{\beta }}$). Indeed, if it were not, then by writing it in Cantor normal form, it could be expressed using sums, products and exponentiation from elements less than it, hence in ${\displaystyle C(\alpha )}$, so it would be in ${\displaystyle C(\alpha )}$, a contradiction.
• Lemma: Assume ${\displaystyle \delta }$ is an ${\displaystyle \varepsilon }$-number and ${\displaystyle \alpha }$ an ordinal such that ${\displaystyle \psi (\beta )<\delta }$ for all ${\displaystyle \beta <\alpha }$: then the ${\displaystyle \Omega }$-pieces (defined above) of any element of ${\displaystyle C(\alpha )}$ are less than ${\displaystyle \delta }$. Indeed, let ${\displaystyle C'}$ be the set of ordinals all of whose ${\displaystyle \Omega }$-pieces are less than ${\displaystyle \delta }$. Then ${\displaystyle C'}$ is closed under addition, multiplication and exponentiation (because ${\displaystyle \delta }$ is an ${\displaystyle \varepsilon }$-number, so ordinals less than it are closed under addition, multiplication and exponentiation). And ${\displaystyle C'}$ also contains every ${\displaystyle \psi (\beta )}$ for ${\displaystyle \beta <\alpha }$ by assumption, and it contains ${\displaystyle 0}$, ${\displaystyle 1}$, ${\displaystyle \omega }$, ${\displaystyle \Omega }$. So ${\displaystyle C'\supseteq C(\alpha )}$, which was to be shown.
• Under the hypothesis of the previous lemma, ${\displaystyle \psi (\alpha )\leq \delta }$ (indeed, the lemma shows that ${\displaystyle \delta \not \in C(\alpha )}$).
• Any ${\displaystyle \varepsilon }$-number less than some element in the range of ${\displaystyle \psi }$ is itself in the range of ${\displaystyle \psi }$ (that is, ${\displaystyle \psi }$ omits no ${\displaystyle \varepsilon }$-number). Indeed: if ${\displaystyle \delta }$ is an ${\displaystyle \varepsilon }$-number not greater than the range of ${\displaystyle \psi }$, let ${\displaystyle \alpha }$ be the least upper bound of the ${\displaystyle \beta }$ such that ${\displaystyle \psi (\beta )<\delta }$: then by the above we have ${\displaystyle \psi (\alpha )\leq \delta }$, but ${\displaystyle \psi (\alpha )<\delta }$ would contradict the fact that ${\displaystyle \alpha }$ is the least upper bound — so ${\displaystyle \psi (\alpha )=\delta }$.
• Whenever ${\displaystyle \psi (\alpha )=\delta }$, the set ${\displaystyle C(\alpha )}$ consists exactly of those ordinals ${\displaystyle \gamma }$ (less than ${\displaystyle \varepsilon _{\Omega +1}}$) all of whose ${\displaystyle \Omega }$-pieces are less than ${\displaystyle \delta }$. Indeed, we know that all ordinals less than ${\displaystyle \delta }$, hence all ordinals (less than ${\displaystyle \varepsilon _{\Omega +1}}$) whose ${\displaystyle \Omega }$-pieces are less than ${\displaystyle \delta }$, are in ${\displaystyle C(\alpha )}$. Conversely, if we assume ${\displaystyle \psi (\beta )<\delta }$ for all ${\displaystyle \beta <\alpha }$ (in other words if ${\displaystyle \alpha }$ is the least possible with ${\displaystyle \psi (\alpha )=\delta }$), the lemma gives the desired property. On the other hand, if ${\displaystyle \psi (\alpha )=\psi (\beta )}$ for some ${\displaystyle \beta <\alpha }$, then we have already remarked ${\displaystyle C(\alpha )=C(\beta )}$ and we can replace ${\displaystyle \alpha }$ by the least possible with ${\displaystyle \psi (\alpha )=\delta }$.

Using the facts above, we can define a (canonical) ordinal notation for every ${\displaystyle \gamma }$ less than the Bachmann–Howard ordinal. We do this by induction on ${\displaystyle \gamma }$.

If ${\displaystyle \gamma }$ is less than ${\displaystyle \varepsilon _{0}}$, we use the iterated Cantor normal form of ${\displaystyle \gamma }$. Otherwise, there exists a largest ${\displaystyle \varepsilon }$-number ${\displaystyle \delta }$ less or equal to ${\displaystyle \gamma }$ (this is because the set of ${\displaystyle \varepsilon }$-numbers is closed): if ${\displaystyle \delta <\gamma }$ then by induction we have defined a notation for ${\displaystyle \delta }$ and the base ${\displaystyle \delta }$ representation of ${\displaystyle \gamma }$ gives one for ${\displaystyle \gamma }$, so we are finished.

It remains to deal with the case where ${\displaystyle \gamma =\delta }$ is an ${\displaystyle \varepsilon }$-number: we have argued that, in this case, we can write ${\displaystyle \delta =\psi (\alpha )}$ for some (possibly uncountable) ordinal ${\displaystyle \alpha <\varepsilon _{\Omega +1}}$: let ${\displaystyle \alpha }$ be the greatest possible such ordinal (which exists since ${\displaystyle \psi }$ is continuous). We use the iterated base ${\displaystyle \Omega }$ representation of ${\displaystyle \alpha }$: it remains to show that every piece of this representation is less than ${\displaystyle \delta }$ (so we have already defined a notation for it). If this is not the case then, by the properties we have shown, ${\displaystyle C(\alpha )}$ does not contain ${\displaystyle \alpha }$; but then ${\displaystyle C(\alpha +1)=C(\alpha )}$ (they are closed under the same operations, since the value of ${\displaystyle \psi }$ at ${\displaystyle \alpha }$ can never be taken), so ${\displaystyle \psi (\alpha +1)=\psi (\alpha )=\delta }$, contradicting the maximality of ${\displaystyle \alpha }$.

Note: Actually, we have defined canonical notations not just for ordinals below the Bachmann–Howard ordinal but also for certain uncountable ordinals, namely those whose ${\displaystyle \Omega }$-pieces are less than the Bachmann–Howard ordinal (viz.: write them in iterated base ${\displaystyle \Omega }$ representation and use the canonical representation for every piece). This canonical notation is used for arguments of the ${\displaystyle \psi }$ function (which may be uncountable).

For ordinals less than ${\displaystyle \varepsilon _{0}=\psi (0)}$, the canonical ordinal notation defined coincides with the iterated Cantor normal form (by definition).

For ordinals less than ${\displaystyle \varepsilon _{1}=\psi (1)}$, the notation coincides with iterated base ${\displaystyle \varepsilon _{0}}$ notation (the pieces being themselves written in iterated Cantor normal form): e.g., ${\displaystyle \omega ^{\omega ^{\varepsilon _{0}+\omega }}}$ will be written ${\displaystyle {\varepsilon _{0}}^{\omega ^{\omega }}}$, or, more accurately, ${\displaystyle \psi (0)^{\omega ^{\omega }}}$. For ordinals less than ${\displaystyle \varepsilon _{2}=\psi (2)}$, we similarly write in iterated base ${\displaystyle \varepsilon _{1}}$ and then write the pieces in iterated base ${\displaystyle \varepsilon _{0}}$ (and write the pieces of that in iterated Cantor normal form): so ${\displaystyle \omega ^{\omega ^{\varepsilon _{1}+\varepsilon _{0}+1}}}$ is written ${\displaystyle {\varepsilon _{1}}^{\varepsilon _{0}\omega }}$, or, more accurately, ${\displaystyle \psi (1)^{\psi (0)\,\omega }}$. Thus, up to ${\displaystyle \zeta _{0}=\psi (\Omega )}$, we always use the largest possible ${\displaystyle \varepsilon }$-number base which gives a non-trivial representation.

Beyond this, we may need to express ordinals beyond ${\displaystyle \Omega }$: this is always done in iterated ${\displaystyle \Omega }$-base, and the pieces themselves need to be expressed using the largest possible ${\displaystyle \varepsilon }$-number base which gives a non-trivial representation.

Note that while ${\displaystyle \psi (\varepsilon _{\Omega +1})}$ is equal to the Bachmann–Howard ordinal, this is not a "canonical notation" in the sense we have defined (canonical notations are defined only for ordinals less than the Bachmann–Howard ordinal).

The notations thus defined have the property that whenever they nest ${\displaystyle \psi }$ functions, the arguments of the "inner" ${\displaystyle \psi }$ function are always less than those of the "outer" one (this is a consequence of the fact that the ${\displaystyle \Omega }$-pieces of ${\displaystyle \alpha }$, where ${\displaystyle \alpha }$ is the largest possible such that ${\displaystyle \psi (\alpha )=\delta }$ for some ${\displaystyle \varepsilon }$-number ${\displaystyle \delta }$, are all less than ${\displaystyle \delta }$, as we have shown above). For example, ${\displaystyle \psi (\psi (\Omega )+1)}$ does not occur as a notation: it is a well-defined expression (and it is equal to ${\displaystyle \psi (\Omega )=\zeta _{0}}$ since ${\displaystyle \psi }$ is constant between ${\displaystyle \zeta _{0}}$ and ${\displaystyle \Omega }$), but it is not a notation produced by the inductive algorithm we have outlined.

Canonicalness can be checked recursively: an expression is canonical if and only if it is either the iterated Cantor normal form of an ordinal less than ${\displaystyle \varepsilon _{0}}$, or an iterated base ${\displaystyle \delta }$ representation all of whose pieces are canonical, for some ${\displaystyle \delta =\psi (\alpha )}$ where ${\displaystyle \alpha }$ is itself written in iterated base ${\displaystyle \Omega }$ representation all of whose pieces are canonical and less than ${\displaystyle \delta }$. The order is checked by lexicographic verification at all levels (keeping in mind that ${\displaystyle \Omega }$ is greater than any expression obtained by ${\displaystyle \psi }$, and for canonical values the greater ${\displaystyle \psi }$ always trumps the lesser or even arbitrary sums, products and exponentials of the lesser).

For example, ${\displaystyle \psi (\Omega ^{\omega +1}\,\psi (\Omega )+\psi (\Omega ^{\omega })^{\psi (\Omega ^{2})}42)^{\psi (1729)\,\omega }}$ is a canonical notation for an ordinal which is less than the Feferman–Schütte ordinal: it can be written using the Veblen functions as ${\displaystyle \varphi _{1}(\varphi _{\omega +1}(\varphi _{2}(0))+\varphi _{\omega }(0)^{\varphi _{3}(0)}42)^{\varphi _{1}(1729)\,\omega }}$.

Concerning the order, one might point out that ${\displaystyle \psi (\Omega ^{\Omega })}$ (the Feferman–Schütte ordinal) is much more than ${\displaystyle \psi (\Omega ^{\psi (\Omega )})=\varphi _{\varphi _{2}(0)}(0)}$ (because ${\displaystyle \Omega }$ is greater than ${\displaystyle \psi }$ of anything), and ${\displaystyle \psi (\Omega ^{\psi (\Omega )})=\varphi _{\varphi _{2}(0)}(0)}$ is itself much more than ${\displaystyle \psi (\Omega )^{\psi (\Omega )}=\varphi _{2}(0)^{\varphi _{2}(0)}}$ (because ${\displaystyle \Omega ^{\psi (\Omega )}}$ is greater than ${\displaystyle \Omega }$, so any sum-product-or-exponential expression involving ${\displaystyle \psi (\Omega )}$ and smaller value will remain less than ${\displaystyle \psi (\Omega ^{\Omega })}$). In fact, ${\displaystyle \psi (\Omega )^{\psi (\Omega )}}$ is already less than ${\displaystyle \psi (\Omega +1)}$.

To witness the fact that we have defined notations for ordinals below the Bachmann–Howard ordinal (which are all of countable cofinality), we might define standard sequences converging to any one of them (provided it is a limit ordinal, of course). Actually we will define canonical sequences for certain uncountable ordinals, too, namely the uncountable ordinals of countable cofinality (if we are to hope to define a sequence converging to them...) which are representable (that is, all of whose ${\displaystyle \Omega }$-pieces are less than the Bachmann–Howard ordinal).

The following rules are more or less obvious, except for the last:

• First, get rid of the (iterated) base ${\displaystyle \delta }$ representations: to define a standard sequence converging to ${\displaystyle \alpha =\delta ^{\beta _{1}}\gamma _{1}+\cdots +\delta ^{\beta _{k}}\gamma _{k}}$, where ${\displaystyle \delta }$ is either ${\displaystyle \omega }$ or ${\displaystyle \psi (\cdots )}$ (or ${\displaystyle \Omega }$, but see below):
  • if ${\displaystyle k}$ is zero then ${\displaystyle \alpha =0}$ and there is nothing to be done;
  • if ${\displaystyle \beta _{k}}$ is zero and ${\displaystyle \gamma _{k}}$ is successor, then ${\displaystyle \alpha }$ is successor and there is nothing to be done;
  • if ${\displaystyle \gamma _{k}}$ is limit, take the standard sequence converging to ${\displaystyle \gamma _{k}}$ and replace ${\displaystyle \gamma _{k}}$ in the expression by the elements of that sequence;
  • if ${\displaystyle \gamma _{k}}$ is successor and ${\displaystyle \beta _{k}}$ is limit, rewrite the last term ${\displaystyle \delta ^{\beta _{k}}\gamma _{k}}$ as ${\displaystyle \delta ^{\beta _{k}}(\gamma _{k}-1)+\delta ^{\beta _{k}}}$ and replace the exponent ${\displaystyle \beta _{k}}$ in the last term by the elements of the fundamental sequence converging to it;
  • if ${\displaystyle \gamma _{k}}$ is successor and ${\displaystyle \beta _{k}}$ is also, rewrite the last term ${\displaystyle \delta ^{\beta _{k}}\gamma _{k}}$ as ${\displaystyle \delta ^{\beta _{k}}(\gamma _{k}-1)+\delta ^{\beta _{k}-1}\delta }$ and replace the last ${\displaystyle \delta }$ in this expression by the elements of the fundamental sequence converging to it.
• If ${\displaystyle \delta }$ is ${\displaystyle \omega }$, then take the obvious ${\displaystyle 0,1,2,3,\ldots }$ as the fundamental sequence for ${\displaystyle \delta }$.
• If ${\displaystyle \delta =\psi (0)}$ then take as fundamental sequence for ${\displaystyle \delta }$ the sequence ${\displaystyle \omega ,\omega ^{\omega },\omega ^{\omega ^{\omega }},\ldots }$
• If ${\displaystyle \delta =\psi (\alpha +1)}$ then take as fundamental sequence for ${\displaystyle \delta }$ the sequence ${\displaystyle \psi (\alpha ),\psi (\alpha )^{\psi (\alpha )},\psi (\alpha )^{\psi (\alpha )^{\psi (\alpha )}},\ldots }$
• If ${\displaystyle \delta =\psi (\alpha )}$ where ${\displaystyle \alpha }$ is a limit ordinal of countable cofinality, define the standard sequence for ${\displaystyle \delta }$ to be obtained by applying ${\displaystyle \psi }$ to the standard sequence for ${\displaystyle \alpha }$ (recall that ${\displaystyle \psi }$ is continuous and increasing, here).
• It remains to handle the case where ${\displaystyle \delta =\psi (\alpha )}$ with ${\displaystyle \alpha }$ an ordinal of uncountable cofinality (e.g., ${\displaystyle \Omega }$ itself). Obviously it doesn't make sense to define a sequence converging to ${\displaystyle \alpha }$ in this case; however, what we can define is a sequence converging to some ${\displaystyle \rho <\alpha }$ with countable cofinality and such that ${\displaystyle \psi }$ is constant between ${\displaystyle \rho }$ and ${\displaystyle \alpha }$. This ${\displaystyle \rho }$ will be the first fixed point of a certain (continuous and non-decreasing) function ${\displaystyle \xi \mapsto h(\psi (\xi ))}$. To find it, apply the same rules (from the base ${\displaystyle \Omega }$ representation of ${\displaystyle \alpha }$) as to find the canonical sequence of ${\displaystyle \alpha }$, except that whenever a sequence converging to ${\displaystyle \Omega }$ is called for (something which cannot exist), replace the ${\displaystyle \Omega }$ in question, in the expression of ${\displaystyle \alpha =h(\Omega )}$, by a ${\displaystyle \psi (\xi )}$ (where ${\displaystyle \xi }$ is a variable) and perform a repeated iteration (starting from ${\displaystyle 0}$, say) of the function ${\displaystyle \xi \mapsto h(\psi (\xi ))}$: this gives a sequence ${\displaystyle 0,h(\psi (0)),h(\psi (h(\psi (0)))),\ldots }$ tending to ${\displaystyle \rho }$, and the canonical sequence for ${\displaystyle \psi (\alpha )=\psi (\rho )}$ is ${\displaystyle \psi (0)}$, ${\displaystyle \psi (h(\psi (0)))}$, ${\displaystyle \psi (h(\psi (h(\psi (0)))))}$... If we let the ${\displaystyle n}$th element (starting at ${\displaystyle 0}$) of the fundamental sequence for ${\displaystyle \delta }$ be denoted as ${\displaystyle \delta [n]}$, then we can state this more clearly using recursion. Using this notation, we can see that ${\displaystyle \delta [0]=\psi (0)}$ quite easily. We can define the rest of the sequence using recursion: ${\displaystyle \delta [n]=\psi (h(\delta [n-1]))}$. (The examples below should make this clearer.)

Here are some examples for the last (and most interesting) case:

• The canonical sequence for ${\displaystyle \psi (\Omega )}$ is: ${\displaystyle \psi (0)}$, ${\displaystyle \psi (\psi (0))}$, ${\displaystyle \psi (\psi (\psi (0)))}$... This indeed converges to ${\displaystyle \rho =\psi (\Omega )=\zeta _{0}}$ after which ${\displaystyle \psi }$ is constant until ${\displaystyle \Omega }$.
• The canonical sequence for ${\displaystyle \psi (\Omega 2)}$ is: ${\displaystyle \psi (0)}$, ${\displaystyle \psi (\Omega +\psi (0))}$, ${\displaystyle \psi (\Omega +\psi (\Omega +\psi (0))),\ldots }$ This indeed converges to the value of ${\displaystyle \psi }$ at ${\displaystyle \rho =\Omega +\psi (\Omega 2)=\Omega +\zeta _{1}}$ after which ${\displaystyle \psi }$ is constant until ${\displaystyle \Omega 2}$.
• The canonical sequence for ${\displaystyle \psi (\Omega ^{2})}$ is: ${\displaystyle \psi (0),\psi (\Omega \psi (0)),\psi (\Omega \psi (\Omega \psi (0))),\ldots }$ This converges to the value of ${\displaystyle \psi }$ at ${\displaystyle \rho =\Omega \psi (\Omega ^{2})}$.
• The canonical sequence for ${\displaystyle \psi (\Omega ^{2}3+\Omega )}$ is ${\displaystyle \psi (0),\psi (\Omega ^{2}3+\psi (0)),\psi (\Omega ^{2}3+\psi (\Omega ^{2}3+\psi (0))),\ldots }$ This converges to the value of ${\displaystyle \psi }$ at ${\displaystyle \rho =\Omega ^{2}3+\psi (\Omega ^{2}3+\Omega )}$.
• The canonical sequence for ${\displaystyle \psi (\Omega ^{\Omega })}$ is: ${\displaystyle \psi (0),\psi (\Omega ^{\psi (0)}),\psi (\Omega ^{\psi (\Omega ^{\psi (0)})}),\ldots }$ This converges to the value of ${\displaystyle \psi }$ at ${\displaystyle \rho =\Omega ^{\psi (\Omega ^{\Omega })}}$.
• The canonical sequence for ${\displaystyle \psi (\Omega ^{\Omega }3)}$ is: ${\displaystyle \psi (0),\psi (\Omega ^{\Omega }2+\Omega ^{\psi (0)}),\psi (\Omega ^{\Omega }2+\Omega ^{\psi (\Omega ^{\Omega }2+\Omega ^{\psi (0)})}),\ldots }$ This converges to the value of ${\displaystyle \psi }$ at ${\displaystyle \rho =\Omega ^{\Omega }2+\Omega ^{\psi (\Omega ^{\Omega }3)}}$.
• The canonical sequence for ${\displaystyle \psi (\Omega ^{\Omega +1})}$ is: ${\displaystyle \psi (0),\psi (\Omega ^{\Omega }\psi (0)),\psi (\Omega ^{\Omega }\psi (\Omega ^{\Omega }\psi (0))),\ldots }$ This converges to the value of ${\displaystyle \psi }$ at ${\displaystyle \rho =\Omega ^{\Omega }\psi (\Omega ^{\Omega +1})}$.
• The canonical sequence for ${\displaystyle \psi (\Omega ^{\Omega ^{2}+\Omega 3})}$ is: ${\displaystyle \psi (0),\psi (\Omega ^{\Omega ^{2}+\Omega 2+\psi (0)}),\psi (\Omega ^{\Omega ^{2}+\Omega 2+\psi (\Omega ^{\Omega ^{2}+\Omega 2+\psi (0)})}),\ldots }$

Here are some examples of the other cases:

• The canonical sequence for ${\displaystyle \omega ^{2}}$ is: ${\displaystyle 0}$, ${\displaystyle \omega }$, ${\displaystyle \omega 2}$, ${\displaystyle \omega 3}$...
• The canonical sequence for ${\displaystyle \psi (\omega ^{\omega })}$ is: ${\displaystyle \psi (1)}$, ${\displaystyle \psi (\omega )}$, ${\displaystyle \psi (\omega ^{2})}$, ${\displaystyle \psi (\omega ^{3})}$...
• The canonical sequence for ${\displaystyle \psi (\Omega )^{\omega }}$ is: ${\displaystyle 1}$, ${\displaystyle \psi (\Omega )}$, ${\displaystyle \psi (\Omega )^{2}}$, ${\displaystyle \psi (\Omega )^{3}}$...
• The canonical sequence for ${\displaystyle \psi (\Omega +1)}$ is: ${\displaystyle \psi (\Omega )}$, ${\displaystyle \psi (\Omega )^{\psi (\Omega )}}$, ${\displaystyle \psi (\Omega )^{\psi (\Omega )^{\psi (\Omega )}}}$...
• The canonical sequence for ${\displaystyle \psi (\Omega +\omega )}$ is: ${\displaystyle \psi (\Omega )}$, ${\displaystyle \psi (\Omega +1)}$, ${\displaystyle \psi (\Omega +2)}$, ${\displaystyle \psi (\Omega +3)}$...
• The canonical sequence for ${\displaystyle \psi (\Omega \omega )}$ is: ${\displaystyle \psi (0)}$, ${\displaystyle \psi (\Omega )}$, ${\displaystyle \psi (\Omega 2)}$, ${\displaystyle \psi (\Omega 3)}$...
• The canonical sequence for ${\displaystyle \psi (\Omega ^{\omega })}$ is: ${\displaystyle \psi (1)}$, ${\displaystyle \psi (\Omega )}$, ${\displaystyle \psi (\Omega ^{2})}$, ${\displaystyle \psi (\Omega ^{3})}$...
• The canonical sequence for ${\displaystyle \psi (\Omega ^{\psi (0)})}$ is: ${\displaystyle \psi (\Omega ^{\omega })}$, ${\displaystyle \psi (\Omega ^{\omega ^{\omega }})}$, ${\displaystyle \psi (\Omega ^{\omega ^{\omega ^{\omega }}})}$... (this is derived from the fundamental sequence for ${\displaystyle \psi (0)}$).
• The canonical sequence for ${\displaystyle \psi (\Omega ^{\psi (\Omega )})}$ is: ${\displaystyle \psi (\Omega ^{\psi (0)})}$, ${\displaystyle \psi (\Omega ^{\psi (\psi (0))})}$, ${\displaystyle \psi (\Omega ^{\psi (\psi (\psi (0)))})}$... (this is derived from the fundamental sequence for ${\displaystyle \psi (\Omega )}$, which was given above).

Even though the Bachmann–Howard ordinal ${\displaystyle \psi (\varepsilon _{\Omega +1})}$ itself has no canonical notation, it is also useful to define a canonical sequence for it: this is ${\displaystyle \psi (\Omega )}$, ${\displaystyle \psi (\Omega ^{\Omega })}$, ${\displaystyle \psi (\Omega ^{\Omega ^{\Omega }})}$...

Start with any ordinal less than or equal to the Bachmann–Howard ordinal, and repeat the following process so long as it is not zero:

• if the ordinal is a successor, subtract one (that is, replace it with its predecessor),
• if it is a limit, replace it by some element of the canonical sequence defined for it.

Then it is true that this process always terminates (as any decreasing sequence of ordinals is finite); however, like (but even more so than for) the hydra game:

1. it can take a very long time to terminate,
2. the proof of termination may be out of reach of certain weak systems of arithmetic.

To give some flavor of what the process feels like, here are some steps of it: starting from ${\displaystyle \psi (\Omega ^{\Omega ^{\omega }})}$ (the small Veblen ordinal), we might go down to ${\displaystyle \psi (\Omega ^{\Omega ^{3}})}$, from there down to ${\displaystyle \psi (\Omega ^{\Omega ^{2}\psi (0)})}$, then ${\displaystyle \psi (\Omega ^{\Omega ^{2}\omega ^{\omega }})}$ then ${\displaystyle \psi (\Omega ^{\Omega ^{2}\omega ^{3}})}$ then ${\displaystyle \psi (\Omega ^{\Omega ^{2}\omega ^{2}3})}$ then ${\displaystyle \psi (\Omega ^{\Omega ^{2}(\omega ^{2}2+\omega )})}$ then ${\displaystyle \psi (\Omega ^{\Omega ^{2}(\omega ^{2}2+1)})}$ then ${\displaystyle \psi (\Omega ^{\Omega ^{2}\omega ^{2}2+\Omega \psi (\Omega ^{\Omega ^{2}\omega ^{2}2+\Omega \psi (0)})})}$ then ${\displaystyle \psi (\Omega ^{\Omega ^{2}\omega ^{2}2+\Omega \psi (\Omega ^{\Omega ^{2}\omega ^{2}2+\Omega \omega ^{\omega ^{\omega }}})})}$ and so on. It appears as though the expressions are getting more and more complicated whereas, in fact, the ordinals always decrease.

Concerning the first statement, one could introduce, for any ordinal ${\displaystyle \alpha }$ less or equal to the Bachmann–Howard ordinal ${\displaystyle \psi (\varepsilon _{\Omega +1})}$, the integer function ${\displaystyle f_{\alpha }(n)}$ which counts the number of steps of the process before termination if one always selects the ${\displaystyle n}$'th element from the canonical sequence (this function satisfies the identity ${\displaystyle f_{\alpha }(n)=f_{\alpha [n]}(n)+1}$). Then ${\displaystyle f_{\alpha }}$ can be a very fast growing function: already ${\displaystyle f_{\omega ^{\omega }}(n)}$ is essentially ${\displaystyle n^{n}}$, the function ${\displaystyle f_{\psi (\Omega ^{\omega })}(n)}$ is comparable with the Ackermann function ${\displaystyle A(n,n)}$, and ${\displaystyle f_{\psi (\varepsilon _{\Omega +1})}(n)}$ is comparable with the Goodstein function. If we instead make a function that satisfies the identity ${\displaystyle g_{\alpha }(n)=g_{\alpha [n]}(n+1)+1}$, so the index of the function increases it is applied, then we create a much faster growing function: ${\displaystyle g_{\psi (0)}(n)}$ is already comparable to the Goodstein function, and ${\displaystyle g_{\psi (\Omega ^{\Omega ^{\omega }\omega })}(n)}$ is comparable to the TREE function.

Concerning the second statement, a precise version is given by ordinal analysis: for example, Kripke–Platek set theory can prove[4] that the process terminates for any given ${\displaystyle \alpha }$ less than the Bachmann–Howard ordinal, but it cannot do this uniformly, i.e., it cannot prove the termination starting from the Bachmann–Howard ordinal. Some theories like Peano arithmetic are limited by much smaller ordinals (${\displaystyle \varepsilon _{0}}$ in the case of Peano arithmetic).

It is instructive (although not exactly useful) to make ${\displaystyle \psi }$ less powerful.

If we alter the definition of ${\displaystyle \psi }$ above to omit exponentiation from the repertoire from which ${\displaystyle C(\alpha )}$ is constructed, then we get ${\displaystyle \psi (0)=\omega ^{\omega }}$ (as this is the smallest ordinal which cannot be constructed from ${\displaystyle 0}$, ${\displaystyle 1}$ and ${\displaystyle \omega }$ using addition and multiplication only), then ${\displaystyle \psi (1)=\omega ^{\omega ^{2}}}$ and similarly ${\displaystyle \psi (\omega )=\omega ^{\omega ^{\omega }}}$, ${\displaystyle \psi (\psi (0))=\omega ^{\omega ^{\omega ^{\omega }}}}$ until we come to a fixed point which is then our ${\displaystyle \psi (\Omega )=\varepsilon _{0}}$. We then have ${\displaystyle \psi (\Omega +1)={\varepsilon _{0}}^{\omega }}$ and so on until ${\displaystyle \psi (\Omega 2)=\varepsilon _{1}}$. Since multiplication of ${\displaystyle \Omega }$'s is permitted, we can still form ${\displaystyle \psi (\Omega ^{2})=\varphi _{2}(0)}$ and ${\displaystyle \psi (\Omega ^{3})=\varphi _{3}(0)}$ and so on, but our construction ends there as there is no way to get at or beyond ${\displaystyle \Omega ^{\omega }}$: so the range of this weakened system of notation is ${\displaystyle \psi (\Omega ^{\omega })=\phi _{\omega }(0)}$ (the value of ${\displaystyle \psi (\Omega ^{\omega })}$ is the same in our weaker system as in our original system, except that now we cannot go beyond it). This does not even go as far as the Feferman–Schütte ordinal.