I think your question is not as precise as you portray it.
First, let me point out that you have not actually defined
a function $z$, in the sense of giving a first order
definition of it in set theory, and you provably cannot do
so, because of Tarski's theorem on the non-definability of
truth. We simply have no way to express the relation x is
definable in the usual first-order language of set theory.
More specifically:
Theorem. If ZFC is consistent, then there are models
of ZFC in which the collection of definable natural numbers
is not a set or even a class.
Proof. If V is a model of ZFC, then let $M$ be an internal
ultrapower of $V$ by a nonprincipal ultrafilter on
$\omega$. Thus, the natural numbers of $M$ are nonstandard
relative to $V$. The definable elements of $M$ are all
contained within the range of the ultrapower map, which in
the natural numbers is a bounded part of the natural
numbers of $M$. Thus, $M$ cannot have this collection of
objects as a set or class, since it would reveal to $M$
that its natural numbers are ill-founded, contradicting
that $M$ satisfies ZFC. QED
In such a model, your definition of $z$ is not first order.
It could make sense to treat your function $z$, however, in
a model as an externally defined function, defined outside
the model (as via second-order logic). In this case, $z(n)$
only involves standard or meta-theoretic definitions, and
other problems arise.
Theorem. If ZFC is consistent, then there is a model
of ZFC in which $z(n)$ is bounded by a constant function.
Proof. If ZFC is consistent, then so is $ZFC+\neg
Con(ZFC)$. Let $V$ be any model of this theory, so that
there are no models of ZFC there, and the second part of
the definition of $z$ becomes vacuous, so it reduces to its
definable-in-$V$ first part. Let $M$ be an internal
ultrapower of $V$ by an ultrafilter on $\omega$. Thus, $M$
is nonstandard relative to $V$. But the function $z$,
defined externally, uses only standard definitions, and the
definable elements of $M$ all lie in the range of the
ultrapower map. If $N$ is any $V$-nonstandard element of
$M$, then every definable element of $M$ is below $N$, and
so $z(n)\lt N$ for every $n$ in $M$. QED
Theorem. If ZFC is consistent, then there is a model
of ZFC in which $f(n)\lt z(10000)$ for every natural number
n in the meta-theory.
Proof. If ZFC is consistent, then so is $ZFC+\neg
Con(ZFC)+GCH$. Let $V$ be a countable model of $ZFC+\neg
Con(ZFC)+GCH$. Since $V$ has no models of ZFC, again the
second part of your definition is vacuous, and it reduces
just to the definability-in-$V$ part. Let $M$ again be an
internal ultrapower of $V$ by an ultrafilter on $\omega$,
and let $N$ be a $V$-nonstandard natural number of $M$.
Every definable element of $M$ is in the range of the
ultrapower map, and therefore below $N$. In particular, for
every meta-theoretic natural number $n$, we have $f(n)\lt
N$ in $M$, since $f(n)$ is definable. Now, let $M[G]$ be a
forcing extension in which the continuum has size
$\aleph_N^M$. Thus, $N$ is definable in $M[G]$ by a
relatively short formula; let's say 10000 symbols (but I
didn't count). Since the forcing does not affect the
existence of ZFC models or Turing computations between $M$
and $M[G]$, it follows that $f(n)\lt z(10000)$ in $M[G]$
for any natural number of $V$. QED
Theorem. If ZFC is consistent, then there is a model
of ZFC with a natural number constant $c$ in which $z(n)\lt
f(c)$ for all meta-theoretic natural numbers $n$.
Proof. Use the model $M$ (or $M[G]$) as above. This time,
let $c$ be any $V$-nonstandard natural number of $M$. Since
the definable elements of $M$ all lie in the range of the
ultrapower map, it follows that every z(n), for
meta-theoretic $n$, is included in the $V$-standard
elements of $M$, which are all less than $c$. But $M$
easily has $c\leq f(c)$, and so $z(n)\lt f(c)$ for all
these $n$. QED
Martin Gardner's Annotated Alice: The Definitive Edition says only this:
Needless to say, all the Mock Turtle's subjects are puns (reading, writing, addition, subtraction, multiplication, division, history, geography, drawing, sketching, painting in oils, Latin, Greek). In fact, this chapter and the one to follow fairly swarm with puns. Children find puns very funny, but most contemporary authorities on what children are supposed to like believe that puns lower the literary quality of juvenile books.
Gardner then goes on to say that the "Drawling-master" is a reference to the art critic John Ruskin, and gives a couple of paragraphs of biographical information about Ruskin. Given Gardner's extensive knowledge of Carrolliana, it seems likely that Carroll never published any further comments about the "different branches of Arithmetic."
Best Answer
Recall that $NFU$ is the Quine-Jensen system of set theory with a universal set; it is based on weakening the extensionality axiom of Quine's $NF$ so as to allow urelements.
Let $NFU^-$ be $NFU$ plus "every set is finite". As shown by Jensen (1969), $NFU^-$ is consistent relative to $PA$ (Peano arithmetic). $NFU^-$ provides a radically different "picture" of finite sets and numbers, since there is a universal set and therefore a last finite cardinal number in this theory.
The following summarizes our current knowedge of $NFU^-$.
1. [Solovay, unpublished]. $NFU^-$and $EFA$ (exponential function arithmetic) are equiconsistent. Moreover, this equiconsistency can be vertified in $SEFA$ (superexponential function arithmetic), but $EFA$ cannot verify that Con($EFA$) implies Con($NFU^-$). It can verify the other half of the equiconsistency.
2. [Joint result of Solovay and myself]. $PA$ is equiconsistent with the strengthening of $NFU^-$ obtained by adding the statement that expresses "every Cantorian set is strongly Cantorian". Again, this equiconsistency can be verified in $SEFA$, but not in $EFA$.
3. [My result]. There is a "natural" extension of $NFU^-$ that is equiconistent with second order arithmetic $\sf Z_2$.
For more detail and references, you can consult the following paper:
A. Enayat. From Bounded Arithmetic to Second Order Arithmetic via Automorphisms, in Logic in Tehran, Lecture Notes in Logic, vol. 26, Association for Symbolic Logic, 2006.
A preprint can be found here.