# FOM: A Foundation of Elementary Arithmetic

Thu Jan 6 16:20:46 EST 2000

```Introduction
As it is currently used, “foundations of arithmetic” can be a
misleading expression.  It is not always, as the name might indicate,
being used as a plural term meaning X = {x : x is a foundation of
arithmetic}.  Instead it has come to stand for a philosophico-logical
domain of knowledge, concerned with axiom systems, structures, and
analyses of arithmetic concepts.  It is a bit as if “rock” had come to
mean “geology.”  The conflation of subject matter and its study can be a
serious one, because in the end, one can lose sight of what one should
be doing in the first place.  Perhaps it is taking matters too
literally, but it seems that there is something to be said for taking
the term to represent X.  Doing so and accepting the term to have some
kind of significance, it is then natural to focus on the question of
what a foundation of arithmetic should be; and, if one exists, what one
is.  Whatever the case, that is what we shall do here.

Part I:  What should a foundation of arithmetic be?
Like most concepts, that of a foundation of arithmetic never
received some clear, initial definition before it developed into wide
currency.  Thus we lack a standard, and admittedly it is unlikely that
any exegesis of such a term can be perfect, much less universally
accepted.  Nonetheless, even without resorting to historical
verification, it is plausible that the notion of a foundation of a
mathematical discipline began as a metaphor, with the analogy of a
building.  Metaphors, like concepts, are rarely clear-cut, but again we
would probably not be too off the mark if we pointed to two
characteristics of a building's foundation as being crucial:  it
supports what is on top of it; and it is the most fundamental part of
the building, with nothing else supporting it.  Whatever the case, let
us assume these two qualities are the essential ones.
In mathematics what gives support are the axioms and rules of
deduction, from which one can prove the discipline’s theorems.  For they
are directly or indirectly responsible for every proof, and so to speak
are holding up the rest of the building.  Even the primitive concepts
and grammar of one’s language, which are obviously basic and important
to a theory, do not seem actually part of the foundation; they are more
likened to the glue, nails, and structure--clearly crucial, but not what
So a support for arithmetic must be a set of axioms (and rules)
sufficient to prove all its theorems.  But not all such sets are
supports.  For one thing the set should not be inconsistent nor should
it contain a falsehood.  Indeed, returning to the metaphor, we would say
that a set built on sand (as Weyl put it) does not support.  A
foundation must be firmly established on solid ground.  That is, for a
set to be a support, every of its axioms must have some justification,
some reason why we are able to assert it.
Now there is a certain amount of tension between the two essential
qualities we have just identified for a set of axioms to be a
foundation--a support must have a justification, and it must be
fundamental.  To be fundamental, an axiom’s justification cannot itself
be another assertion, since otherwise this new assertion would be an
axiom providing a deeper foundation.  That is, if I were to justify an
axiom A by saying proposition B implies it, then I should simply replace
A with B to have a deeper set.  Fortunately, there are other ways of
providing justifications.  For instance, to say that an axiom is
intuitively obvious provides it a justification but not a support, since
there is no proof intended or implied.
Before saying what holds arithmetic up, we obviously need to decide
what arithmetic is.  One could take a large view, and say that it
consists of every true sentence which can be written using arithmetic
terms (such as addition and multiplication), and that a foundation must
prove every such truth.  But in so doing, one of course runs up against
Godel's Incompleteness Theorem, with the resulting implication that
there can be no foundation for arithmetic in systems of a certain,
mechanical nature.  Since such a system is a fair way of ensuring that
all assumptions are clearly specified, we cannot avoid the conclusion.
Hence a large view of arithmetic, even if it may be the most natural, is
also a sterile one for a foundationalist.
However, it does seem a stretch to classify Godel’s theorem, even
when it is transcribed in arithmetic notation, as arithmetic.  It seems
more appropriate to limit the subject to--shall we say it?--simpler
propositions, such as 2 + 2 = 4, the Commutative Law of Multiplication,
and--even here it seems one is trespassing on a domain more properly
called “number theory”--the existence of an infinite number of prime
numbers.  In any case, to obviate sterile semantic arguments, let us
call this more limited (admittedly not well-defined) domain *elementary
arithmetic*.   It can be asked, without now receiving an immediately
negative reply, whether a foundation for elementary arithmetic exists
and, if so, what one is.
It hardly seems likely that its existence can be held to be
intuitively obvious.  If it were, the same intuition would probably tell
us as well that there is a foundation for all of arithmetic, which again
by Godel is false.  It is not even sure that the answer is positive and
that a foundation exists.  The only way to answer the question in the
affirmative seems to be actually to construct and produce one.  At
least, I know of no other technique.
Remark on the indefinite article of our interrogation:  the
question is what *a*, and not *the*, foundation is.  Granted this goes
contrary to the metaphor, since a building has one and only one
foundation.  But given the manner in which the term has been
explained--support without support--, I don’t know of any argument for
uniqueness.  Certainly there are different ways of formulating what is
basically the same assertion, so at the least uniqueness holds only if
one has a notion of when assertions are “basically similar.”   In brief
the manner in which the query has been posed is intentional and appears
correct.  And it is the question of existence to which we will direct
our investigation here.
Admittedly “fundamental” is inherently a woolly and vague concept.
Even explaining that a set is fundamental if it cannot be provable from
a deeper set, helps only a little, since “deeper” is itself also
ambiguous.  Clearly a set with superfluous members is *not*
fundamental.  But simply the lack of superfluity does not guarantee
depth.  Perhaps the best way of putting it, as was already suggested
above, is whether one would justify axioms in the set by saying other
propositions imply them--if so, they are not fundamental.
As we have just put it, being fundamental is not related to being
more certain.  “2 + 2 = 4,” which presumably appears certain to almost
everyone, is nonetheless not fundamental.  At least, on reflection we
have the feeling that it is true because other propositions entail it.
The aim of a foundation is to provide understanding, not certainty.
Whatever the precise sense of the terms, it does seem that it is
not possible to be more fundamental than an axiom which just explains
what an arithmetic term means.  For instance, zero should be tied to the
situation where there are no objects, and addition to the putting
together of objects.  By explaining the essential idea, we are assured
it is reasonable to suppose that a foundation must be statable in such
terms, that somehow zero must be tied to the situation where there are
no objects and addition to the putting together of objects, because
otherwise the supposed foundation could presumably be proven from axioms
which did make the connection, and so which would be considered deeper.
Unhappily, it may turn out that some concepts--for example, that of
“natural number” itself--do not lend themselves to such essential
explanations.  One then states an axiom or axioms, which one takes to be
fundamental only because no alternatives appear more so.  So there can
never be confirmation that one’s axioms are in fact the deepest.
Refutation is always possible, as there is always the chance that a new,
deeper axiom will be proposed.  While less than ideal, such a situation
should not deter one from the attempt to state a foundation, since even
one step on a long journey is nonetheless an advance.

Part II:  What are *not* foundations?
Peano’s Axioms themselves, by their mere simplicity, suggest a
foundation.  Alas, they are not fundamental enough, since they do not
tie 0 to the situation of there being no objects, nor do they explain
addition by reference to the putting together of objects.  In brief the
level on which they are stated is higher than a foundationalist
desires.  While they support, they should be supported by something
deeper.
Nonetheless, there are still plenty of aspirants left, most notably
ZF set theory and Frege Arithmetic.  ZF defines 0 as the empty set, 1 as
the set of the empty set, addition as set union, and so on.  Since the
empty set represents no objects and set union is a way of putting things
together, ZF satisfies, for zero and addition at least, the demand to be
fundamental.
The technique of Frege Arithmetic, although older than ZF since it
originates (as its name suggests) with Frege, experienced a long period
of neglect, and has only recently been revived, by Crispin Wright in the
early 1980s.  Central to the theory is a proposition known (although
historically inaccurate, according to Michael Dummett) as Hume’s
Principle,  namely that
“The number of x = the number of y if and only if there is
a 1-1 correspondence between x and y.”
Following Charles Parsons, Wright observed the remarkable fact that the
inconsistent axiom in Frege’s system (Axiom V) is only needed to prove
HP.  Afterwards, it may be forgotten, since HP alone, with the rest of
second-order logic, suffices to prove Peano’s Axioms.  Frege Arithmetic
(FA) is just second-order logic plus HP, and George Boolos, who along
with Wright and Richard Heck led the research in this field, baptized as
“Frege’s Theorem” the fact that FA implies Peano’s Axioms.  The name is
well chosen, since Frege can in fact be considered to be the first to
have proven it.  Although to my knowledge Crispin Wright never advocated
the thesis which could be reinterpreted, using our terminology, as
saying that FA is a foundation for elementary arithmetic, it is still a
natural conjecture, and has many common points with Wright’s own
concerns.
So here are two strong candidates to be a foundation:  ZF on the
one hand, FA on the other.
As usually stated however, *neither* is in fact a foundation,
because despite appearances neither as usually stated is even about
arithmetic.  In order to have the look of arithmetic, both stipulate
that arithmetic symbols are to mean set-theoretic or logical concepts.
But one shouldn’t and can’t be deceived by appearances, and ZF is
nothing more than a type of set theory, and FA nothing more than a
variety of second-order logic.
Indeed, it is a philosophical truism (and a point of Paul
Benacerraf, among others) that you can’t *stipulate* what a concept
means, but only what a symbol or a word means.  A concept is what it
already is. It may be explicated, but cannot be dictated to mean
something (else).  While we can stipulate that the word ‘pig’ will mean
“creatures with wings,” we *can’t* stipulate that what the word ‘pig’
actually now means (the concept *pig*) is to be the same as “creatures
with wings.”  More baldly, while the word ‘pig’ can be stipulated to
mean “creature with wings,” *pigs* cannot be stipulated to fly.  If one
really wants the word ‘pig’ to mean “creature with wings,” then of
course one can insist that it be so, but then one must forget the usual
meaning of the word.  In particular, statements about these so-called
pigs are no longer about animals which go oink.  The stipulation has
changed the subject matter.
Stipulations should only be used when the term to be defined is
inconsequential.  A good test for this is if it can be replaced by a
nonsense word, such as ‘plok.’  For instance, instead of ‘prime number’,
we might define ‘plok’ to mean “natural number other than 1 divisible
only by 1 and itself”.  When one goes on to prove there are an infinite
number of ploks, no one will be seriously alarmed, because the
fundamental content of the assertion has not changed.  Having passed the
plok test, the definition of ‘prime number’ is a legitimate
stipulation.  However, ZF’s definition of the term ‘1’ depends crucially
on how the symbol is currently used.  It *needs* the symbol to be ‘1’ in
order to confuse its audience that the resulting assertions are actually
about the number *one* and are part of arithmetic.  Failing the plok
test, it is therefore not a stipulation, but can only, at best, be
considered an analysis of a concept.  The difference is important,
because while stipulations cannot be challenged--they are neither true
nor false--, analyses can be wrong, and misanalysis is possible.  ZF’s
definitions--analyses of mathematical concepts in terms of set
theory--can therefore be contested.  At the least, they should be stated
as additional axioms, so those who do want to discuss, question, or
disagree with them are allowed this option.
The same is true of FA.  While the status of Hume’s Principle is
the subject of philosophic dispute, Wright does sometimes speak of it as
if it were a stipulation.  Worse, when one defines what it means to be a
natural number (using the Fregean technique of the ancestral of the
successor relation), it is not stated as an assumption.  But again, it
can be contested that the analysis is correct, and so the definition
should be an axiom.
For the moment we are quibbling, because of course it is possible
simply to *add* axioms which have hitherto been held to be stipulative
definitions, and reclassify them, this time correctly, as analyses which
can be disputed.  Two questions then appear:  Are the analyses
*correct*, and can we *justify* them?  Answers to these will go a long
way to addressing our original concern, namely are these extensions of
traditional ZF and FA (call them ZF* and FA*), which contain axioms
rather than stipulative definitions, foundations of elementary
arithmetic.
In the case of ZF*, there are good reasons to say the analysis
fails.  There is first of all, the dilemma noticed by Benacerraf.  ZF*
has different options on to how to define the number 2, for instance
either as {0,(0}} or as {{0}}.  How can one choose when both appear
equally good candidates?  Since the two options are provably different
(in ZF!), 2 cannot be both.  It seems, then, that 2 cannot be either.
This is not quite a knock-down argument, in the sense it is
conceivable that 2 just is (say) {0,(0}}, even if there is no way to
explain why it is so.  But since there is no explanation, there is no
justification, and so 2 = {0, {0}} cannot serve as part of a foundation.

Secondly, Occam’s razor is not kind to ZF* (and just plain old ZF),
whose ontological implications are, simply put, quite monstrous.  It
populates the universe with an unbelievable amount of entities, far more
than what a foundation of elementary arithmetic seems to warrant. If we
need all *that* to provide a justification for arithmetic, then it
hardly seems worth the bother.
Thirdly, we may ask what justifies the axioms of ZF.  There are
three major ones:
1) the “naive” idea of “set”, restricted to sets which are not too
“big”;
2) the iterative concept of “set”; and
3) what is probably best called,
ZF-works-and-that’s-all-that-matters.
Let’s take each in turn.
The first essentially accepts the “naive” concept of set but
locates the error which leads to the paradoxes in the fact that sets
which are too big are not really sets or perhaps are sets which cannot
belong to other sets, as Michael Hallett explains in his fine book on
the subject.  What bigness has to do with it, is not really explained,
except perhaps in Cantor’s theological meditations about the “absolutely
infinite”, which hardly seem to count.  To point out that otherwise
there is a contradiction, is of course just the case of the dog chasing
its tail.  What, after all, is it in the concept of “set” that excludes
the set of everything?   And why shouldn’t the set of everything belong
to itself?   The existence of a universal set is, of course, not
Separation, or some such, is added.  Although it takes us too far from
our course to argue here, philosophical arguments tend to show that the
paradoxes result from Separation --and impredicativity and
viciousness--and not the universal set.
The second theory advances the idea that sets are formed in stages;
at any particular stage, one puts together a set from sets already
formed in previous stages.  For instance, Dana Scott famously proved
that the axioms of ZF (minus Replacement) can be proved in a system
which essentially formalizes this idea of iteration, and some
philosophers advance this as a justification for ZF.  Note that if this
is so, then ZF cannot be a foundation for arithmetic.  For its axioms
would be justified by other propositions, so they are not fundamental,
and only Scott’s system could be a foundation.  Be that as it may, we
still would need to know what justifies the iterative theory.  While it
can perhaps be called “natural” (e.g. Boolos does), it hardly seems
evident or to be implicit in the concept “set”.  The *naive* concept is
evident, or would be evident if it were only consistent--that, after
all, is why we call it “naive”.
Another difficulty with the iterative concept is that it depends
for its intuition on the idea of “forming” sets at particular stages.
And, while finite stages seem to be permissible, there seems no
iron-clad argument to explain why a stage after all finite stages is.
So, even if the iterative concept is not *ad hoc* in the strictest
sense of the term, it does seem to be *ad hoc* in spirit, by not having
any sort of independent justification.  Indeed, since the iterative
concept effectively bans the universal set as well, the same criticisms
which were lodged against the first justification applies here as well.
Which brings us to the third type of justification--ZF works.  This
argument goes as follows:  ZF proves all or almost all of mathematics,
and since mathematics is used in the sciences, the success of science
serves as a justification of the axioms.  There may be some sense to
these observations in other contexts, but it does not have any in the
search for a foundation of arithmetic.  If arithmetic is the
justification for its own foundation, then we are caught in the vicious
circle of the house supporting the foundation, which is supposed to
support the house.  Secondly, if science is not supposed to hypothesize
unnecessary entities, neither it would seem should the mathematics which
is used in the sciences.  And as we have already noted, ZF’s ontological
needs are vast.
FA* fares better than ZF* on the first objection raised above.  For
while it does permit multiple definitions of 0, they are different
without being inconsistent.  To see this, let “Nx : phi” be read as “the
number of x such that phi.”  Then FA* might state as an axiom either
(a) 0 = Nx : ~x = x; or
(b) 0 = Nx : ~ there exists y s.t. y = x.
Unlike the analogous situation with ZF*, (a) and (b) are not provably
inconsistent, and indeed in FA*, they are provably the same.  So with
FA*, we are not faced with an undecidable choice, with the consequence
that we are not forced to eliminate both options.
FA* uses traditional second-order logic, which, while never
explicitly advanced as such, can be seen as a solution to the
paradoxes.  After all, completely unrestricted, a logic with predicates
suffers from a version of Russell’s paradox, via the predicate which
does not satisfy itself.  Formally,
Q[P] <=> not P[P].
Second-order logic avoids contradiction by dichotomizing the universe
into thing and predicate.  Obviously, there is a grammatical difference
between a predicate (“is young”) and a referring concept (“Thomas”), but
(following Quine) second-order logic effectively eradicates this
dichotomy when it allows quantification over predicates.  “For all
predicates” sounds an awful lot like “for all things.”  Indeed, in order
to quantify over “P” in formula like “Px”, the latter must be read as
something like “x satisfies the predicate P;” so that, for instance,
“(x)(P)Px” can be read as, “for all x and all P, x satisfies P.”  That
is to say, the only grammatical predicate in second-order logic is
“satisfies,” and the predicates P are just a special type of thing.  So,
big letters are a subclass of small letters; and one should always be
able to substitute big letters for small.  For instance, “everything is
identical to itself” should normally be permitted to imply that “P is
identical to itself,” for any particular predicate P.  But this is not
just a minor fix, since then full-scale comprehension, which
is an incoherence in second-order logic:  between its need to quantify
over predicates and its dichotomy of the universe into thing and
predicate.  Resolving this in the most natural way leads back to the
rock and a hard place.
Finally, FA* does make a certain amount of ontological commitments,
obviously much less than ZF*, but still significant (for instance, it
says that there exists a number of the set of the natural numbers), and
still apparently more than arithmetic warrants.
However, there is a yet even more critical hurdle to FA*’s claim to
be a foundation:  HP does not appear to be even true, or at the least,
its truth is so questionable it is perverse to say that it can serve as
a justification for elementary arithmetic.  To be clear, HP is, as
stated above,
“the number of x is the number of y if and only if there
exists a 1-1 relation from x onto y” (HP).
One wants to react as Galileo did, by saying this *couldn’t* be true,
because a counterexample is x = the natural numbers and y = the even
numbers.  For on the one hand halving is a 1-1 onto relation between
them, and on the other the two sets cannot be equinumerous since the
even numbers are properly contained in the naturals.  Indeed, there is
what might be called a *standard* (I guess, to be fair, the Cantorians
would call it the “naive”!) view of number, that the only how-many
numbers are the natural numbers (and infinite sets do not have a
number).  While one can perhaps argue against this standard view, it is
hardly evidently false.  After all, people seemed to have had an
adequate idea of number before Cantor. And if it is not evidently false,
then HP, which implies immediately that the natural numbers have a
number, is not evidently true.
Even in the finite case, HP does not seem so sure.  What happens if
they’re aren’t any relationships, or any big relationships, or if
there’s a funny sort of relationship that we haven’t thought about?  Of
course, one can prove, in second-order logic, that relationships--and
the answers to these questions--are as one expects.  But it seems
unwarranted to assume facts which need to be proved, as part of the
justification for one’s axiom.  At the least, it makes it out to be less
than fundamental.
Given that it does not seem intuitively obvious, one should ask
where else may lay its justification.  There seem to be three
possibilities:  (1) it is a generalization from the finite case; (2) it
uses a principle of abstraction; or (3) it is a stipulation.  Let’s take
each in turn.
Of course generalizing from the finite to the infinite is fraught
with uncertainty, and there are many instances where it simply does not
hold.  So before one generalizes to get HP, one must somehow provide a
justification that it can be done.  That is, the request is simply
pushed back one step.  And, to my knowledge, there is no ready answer to
this second demand, which does not also beg the question.
Justifying HP by a principle of abstraction is, in itself,
attractive.  However, doing so implies that HP is provable from another
proposition, which being more general would also be deeper.  So FA*
cannot be a foundation, only FA + {some principle of abstraction}.  The
question then becomes, is this latter a foundation?
Now as Dummett has pointed out, abstraction is also the basis of
Frege’s inconsistent Axiom V, which indeed is the fully general
version.  So not all abstractions evidently work, and there must be some
finetuning as to which principle is exactly being contemplated, and
why.  This Wright has gamely tried to provide.  First, the general
principle of abstraction is given as,
(F)(G) (Sx : Fx) = (Sx : Gx) iff F ~ G,
where ~ is some equivalence relation.
Wright explains--and this is an exact transcription, except for the
change in notation--that one is not allowed to make this assertion
precisely when an S object falling under some concept, F, of which it is
Sx : Fx entails that it falls under every F of which it is Sx : Fx.  If
the reader comprehends this in one go, he is evidently more capable than
myself.  Since the rule is difficult to understand and so able only with
difficulty to claim the status of an intuitively obvious assertion, what
can justify our accepting it and thus Wright’s (restricted) abstraction
principle?   That is to say, Wright should not merely construct a rule
which separates the “good” from the “bad.”  The technical accomplishment
is only half the work, and largely the easiest part.  By all
rights--even if he is not interested in the foundation question--he
should provide an explanation why his rule should be adopted, other than
Since Wright entitles his paper “The Harmless Impredicativity of N=,” he
apparently would not explain his rule by appealing in the direction of
impredicativity, and so it is mysterious just how he can pull it out of
his philosophical hat.
But suppose even that Wright can correctly explain his rule.
Nonetheless, it is inconsistent!  As Dummett points out, it lets through
Frege’s own fix of Axiom V, which is true only in singleton domains.
And Boolos has other examples of applications of the rule which force
the universe to be finite. Since the abstraction principle also implies
HP, which in turn implies PA and that the universe is infinite, the
restricted principle is inconsistent as it stands.
But even grant Wright his abstraction principle.  He *still* cannot
justify that HP, deduced from an abstraction principle, is about natural
numbers, because he has no control over what the abstracted entity is.
That is to say, the principle allows one to abstract from the
equivalence relation “x is equinumerous to y”, to an object “the plok of
x.”  But there is no guarantee that the plok of x will be the natural
number of x!  *This* is an additional assertion, which somehow must be
stated as an axiom in any system which wants to be a foundation for
elementary arithmetic or (as Wright merely wants) prove the Peano
Axioms.  Again, since it can hardly be asserted to be intuitively
obvious, we can again ask what justifies our saying it is so.  Indeed,
given that it goes against the standard view of number, it seems that it
is *not* even so.
The third possible justification, that HP can be stipulated,
certainly takes the principle out of harm’s way, because, as we have
already remarked, stipulations cannot be challenged.  However, as we
have argued, the concept “number” cannot be stipulated, only the word.
And, if the word is involved, the definition does not change what the
*concept* number is, or what *arithmetic* is.  The definiendum might as
well have been “plok” as “number”.  And had it been “plok”, we would all
probably agree that the stipulation, of itself, has nothing to do with
arithmetic.
Now, as Frege formulates it and Dummett stresses, it is possible to
break up HP into two steps:
“the number of x is the number of y if and only if x and y
are equinumerous”--
call this HPa--and
“x and y are equinumerous if and only if there exists a 1-1
relation from x onto y”--
call this HPb.  HPa appears relatively harmless.  HPb, on the other
hand, seems to have a certain amount of meat to it, and so again we are
entitled to ask what justifies it.  Not surprisingly, we are likely to
get back the same answers as we did for HP.  Generalization from the
finite to the infinite, no more works here than it did before, for HP
itself.  Moreover, as this is not a case of abstraction, there can be no
appeal to such a principle.
By process of elimination, that leaves the third possibility.  And
at first glance it seems possible that HPb is just a stipulation.  After
all, since HP has been broken up into two steps, HPb can be a
stipulation without implying that number itself has been stipulated,
because HPa intervenes.  This is all to the good, since again, we
*cannot* stipulate what number means, unless we want to give up on
talking about arithmetic.  However, if HBb is just a stipulation of what
the the word “equinumerous” means, then its appearance in any
proposition can simply be removed in favor of what it has been fixed to
mean.  So HPa then simply becomes HP!  And so breaking HP into two,
serves no purpose, because it hasn’t really been divided at all.

Part III:  What is a Foundation?
Negativism is all well and good, but let us try to be constructive
and sketch--with no claim of originality--what appears to be a
foundation of elementary arithmetic.  First, as was suggested above, the
solution to the paradoxes is in some version of impredicativity or
viciousness.  An analysis of the situation (which, I’m afraid, will not
be given here, but which makes no great claims of profundity) leads to a
statement of comprehension--call it Pred--similar to Weyl’s formulation
and as it is found in systems like ACA. (Solomon Feferman deserves a
great deal of credit for his study of such predicative systems.)  First,
introduce terms {x,y,z,... : phi} to stand for the predicate of those
x,y,z,... which satisfy phi.  That is, {x : phi} is a term, {x,y : phi}
is a term, and so forth.  Then
(x)(y)(z)... ( {x,y,z,... : phi}x,y,z,.. <=> phi )
provided phi does not contain either any bound variable or
x,y,z,... in the predicate position.
Remark for philosophical reasons (there is no gain on the mathematical
side) that Pred is slightly less restrictive than the usual
impredicative formulation, since it allows bound variables in phi in the
argument place.
For those who are worried about ontological commitments, the
introduction of the terms {x,y,z,... :  phi} makes none.  It essentially
asserts there is a predicate in the case when there is already a
predicate (phi), so there are no “new” entities required.  Pred, which
makes no ontological assertion, is itself essentially about when a
predicate can be assured to be well-behaved.
Now a common objection to a schema like Pred is that it is too weak
to allow a definition of the natural numbers N.  For instance, Pred
cannot be used if phi corresponds to the ancestral of the successor
relation, as in
Nx iff (P) ( P0 & (y)(Py => P(y+1)) => Px )
(Induct)
because there exists a bounded variable, P, in the predicate place.  And
of course it is precisely Induct which FA uses to define the natural
numbers.  However, it should be clear to the reader who has accepted the
argument in Part II, that this is not a real drawback.  True, the
natural numbers N cannot be defined in this way, but N *cannot* and
*should not* be defined in any way, shape, or form.  It has to be taken
as a given, because if it is not, the resulting system is not even about
arithmetic. N cannot be defined, and axioms about N cannot be avoided,
in *any* system hoping to be a foundation of elementary arithmetic.
Another objection is that the system is not sufficiently strong to
“do” real analysis.  Now granted that the aim of those who study ACA is
to see what mathematics can be developed from it, so evidently such a
remark may be germane.  But it would not make sense in the present
context, unless one assumes that somehow real analysis *does* or
*should* have the same justification as elementary arithmetic.  Now this
seems implausible at best, since anyway on a superficial glance, the
status of real analysis looks different than the status of elementary
arithmetic.  Besides, without entities which are real numbers (which,
according to the philosophy expressed here, should be assumed rather
than “constructed”), it is hard to see how a foundation of arithmetic
*could* be one of real analysis as well.  Of course, if real analysis
were to fall out of the same wash as elementary arithmetic for free,
then I as much as the next person would be perfectly willing to accept
the gift.  But it should not be a condition of a foundation of
arithmetic that it also be a foundation for real analysis.  In brief, I
know of no argument supporting the great Catholic imperative--that all
of mathematics must or should emanate from the same single universal
source.  And until that time (if ever) it is firmly established as
necessary, it cannot be fairly used as a bludgeon.
In spite of seeing things very nearly aright on the paradoxes, ACA
assumes the Peano axioms.  Thus it never ties 0 to the situation where
there are no objects, and so on.  This is easily rectified, so let’s do
it!

Big letters, small letters:  big letters can always be substituted for
universally quantified small letters.  One can suppose the reverse--that
small letters can always be substituted for universally quantified big
letters (i.e. there is no difference between small and big, and the use
of the one or the other is just a stylistic choice)--but it is not
necessary and probably better not to for philosophic reasons.

Mn,P means “n numbers the things satisfying P”
(x) means “for all x”
[x] means “there exists x”
{x : phi} means “the predicate of those x satisfying phi” (and is to be
considered a “big letter”)
sn,m means “m is the successor of n”
Nn means “n is a natural number”
phi(i\j) substitutes the term i in place of all free occurrences of j.
(We presuppose that phi is such that i, upon substitution, does not
become bound.)

COMPREHENSION
(x)(y)(z)... ( {x,y,z,... : phi}x,y,z,.. <=> phi )
provided phi does not contain either any bound variable or x,y,z,... in
the predicate position.

NUMBER OF
(n)(m)(P) ( Mn,P & Mm,P => n = m)
(n)(P)(Q) ( (x)(Px <=> Qx) & Mn,P => Mn,Q )
(n)(P)(a)(b) ( ~Pa & ~Pb & Mn,{x : Px v x = a}
=> Mn,{x : Px v x = b} )
(n)(P)(a) ( Mn,P => [m] Mm,{x : Px v x = a} )

ZERO
(P) ( M0,P <=> (x) ~Px )

ONE
(P) ( [a](x)(Px <=> x = a) => M1,P )

TWO, THREE, FOUR (up to 4 so that “2 + 2 = 4” is provable)
(P) ( [a][b](~ a = b & (x)(Px <=> x = a v x = b)) => M2,P )
etc.

NATURAL NUMBER
(n)(P) ( Mn,P => Nn )
(n)(m) ( sn,m => Nn )

SUCCESSOR
(n)(m)(P)(a) ( Mn,P & ~Pa & Mm,{x : Px v x = a} => sn,m )
(n)(m)(P)(a) ( sn,m & ~Pa =>
(Mn,P <=> Mm,{x : Px v x = a}) )

INDUCTION
phi(0\n)
(n)(m) ( sn,m & phi  ? phi (m\n)  )
----------------------------------------
(n) ( Nn => phi )

(n)(m)(k)(A)(B) ( Mn,A & Mm,B & ~[x](Ax & Bx) =>
(+n,m,k <=> Mk,{x : Ax v Bx}) )

SUBTRACTION (-)
(n)(m)(k)(A)(B) ( Mn,A & Mm,B & (x)(Bx => Ax) =>
(- n,m,k <=> Mk,{x : Ax & ~Bx}) )

MULTIPLICATION (*)
(n)(m)(k)(A)(B) ( Mn,A & Nm & (x) (Ax => Mm,{y : Bx,y}) &
(x)(y)(z)(Bx,y & Bz,y => x = z)
=> ( *n,m,k <=> Mk,{y : [z](Bz,y & Az)} ) )

Remark that the Subtraction axiom does not say what happens when n
< m.  That is, -1,3,5 (aka 1 - 3 = 5) does not result in a
contradiction.  Indeed, we can neither prove -1,3,x nor prove ~-1,3,x
for any x.  Of course, we could state another axiom which would clarify
this, but none of the propositions which this axiom would let us prove
is important.
A crucial proof (as it is in Frege Arithmetic) is the one that
there are an infinite number of things and its cousin, that for any n,
there exists a P such that Mn,P.  It follows the line of Dedekind’s
argument, that the empty predicate is one, the empty predicate union the
predicate x = the empty predicate are two, and so on.  That is, the
theorem essentially follows from the infinitary presuppositions of the
underlying language. Once it is proven that there are n objects for
every n, the hypotheses in the axioms for Successor, Addition,
Subtraction, and Multiplication become satisfiable.
It is possible to state and prove the following two theorems (which
usually are definitions), which seem by right to be the Fundamental
Theorems of Arithmetic:  that addition is just repetition of the
successor relationship; and that multiplication is just repetition of