I would like first to acknowledge Walter Felscher for his invaluable
information concerning Bettazzi proof – to which I will return on
another occasion. (His informations ignited many historical doubts in
my mind.) I would like now to contribute trying to clarify SOME issues
raised by Luigi Borzacchini and Dirk Schlimm related with first and
second order axiomatizations. In my opinion, the more elementary issue
concerning the categoricity of Peano axioms is pedagogically important
as a first step towards more general structures and may shed some light
over the potential tools available to Dedekind.
THE FIRST CATEGORICITY THEOREM
Call a structure <X, f, a> a Peano structure if X is a set, f is a
function X --> X, a is an element of X AND, in addition, the Peano
axioms are true in <X, f, a>, that is:
(1) f(X) is contained in X;
(2) f is injective;
(3) a does not belong to X;
(4) X is the intersection of all sets A contained in X such that f(A)
is contained in A, a belongs to A, and A contains X.
Note that only (4) is a second-order sentence in our basic alphabet,
since we have there a universal quantifier binding A. Dedekind theorem
132 in his Was sind und Was sollen die Zahlen (1887) states (in his
own words):
ALL SIMPLY INFINITE SYSTEMS ARE SIMILAR TO THE NUMBER-SERIES N AND
CONSEQUENTLY [by his theorem 33] TO ONE ANOTHER.
According to Dedekind, a set (a "system" in his terminology) X is SIMPLY
INFINITE if there exist a function f and an element a such that <X, f, a>
is a Peano structure by the above definition; and a set X is SIMILAR to
Y when (in present-day terminology) there exists a bijection X --> Y.
However, when we examine carefully Dedekind's proof, we see that he
proved a little bit more than what he stated in his own language. In
fact, his theorem 132 amounts to this:
If <X, f, a> and <Y, g, b> are Peano systems, then there exists an
ISOMORPHISM F between them.
That is to say, he showed how to construct a bijection F: X --> Y for
which F(a) = b and F(f(x)) = g(F(x)) for all x in X.
His basic tool for constructing F was his fundamental theorem 126
(theorem of the definition by induction) which states (again, in modern
terminology) that there exists an homomorphism F between any two Peano
structures. In the theorem 132 he proves by induction that such an
homomorphism is really a bijection.
PRE-HILBERTIAN?
Fifteen years after his Stetigkeit und irrationale Zahlen, Dedekind's
ATTITUDE (although not his wording) is entirely axiomatic and not
genetic in the sense of Hilbert. As far as rigor is concerned, there
is nothing "pre-Hilbertian" in his approach (except for his "proof" of
the existence of infinite sets and matters relating to more compact
formulations). Apparently, Dirk Schlimm used the adjective
"pre-Hilbertian" as synonymous with "not first order" but this use is
not entirely correct. It was Thoralf Skolem who emphasized the
"pathologies" involving first order axiom systems and paved the way for
the first-order axiomatic set theory that is common nowadays. Indeed,
he was the first to discover a non-standard model for the first-order
Peano axioms. Zermelo coined the term "Skolemism" for the reductionism
program of all mathematics to a first-order language.
ORDERED FIELDS AND BOLZANO
There is nothing in Dedekind work dealing with the categoricity of the
system of axioms for the real numbers; he did not address this problem,
although something of the spirit of the necessary tools can be found in
his work.
For us today (as for Hilbert essentially) THE real numbers are any
COMPLETE (or Dedekind-complete, as some prefer) ORDERED FIELD. The
condition of completeness can be formulated in many equivalent ways but,
in any case, it is a second-order condition – and that is why any two
complete ordered fields are isomorphic. Let us call an ordered field F
DEDEKIND-COMPLETE if every non-empty subset bounded from above has a
supremmum; CAUCHY-COMPLETE if is complete in the natural metric induced
by its order. Then we have:
(5)Dedekind-complete implies Archimedean (the converse is obviously
false);
(6)Dedekind-complete implies Cauchy-complete.
Here, again, the converse is false (difficult) but it IS true for
Archimedean fields. This last fact has a certain historical importance.
In 1817 Bolzano tried to PROVE that IR is Dedekind-complete! He stated
the Dedekind-completeness condition in terms of "proprieties" (instead
of "sets") in a somewhat tortuous way that corresponds to: for every
subset M of IR such that IR – M is not empty, if IR – M has an lower
bound, then IR – M has a greatest lower bound (an infimmum). His plan
was to prove Cauchy-completeness as a first step. It is easy for us
today to understand why he would never achieve his goal (without an
adequate axiomatization or the necessary logical instruments for a
genetic approach).
EXTREMAL AXIOMS
However, it is interesting to note that Hilbert original formulation of
the axiom of completeness did not mention upper bounds of sets as is
natural today and was for Bolzano. Instead, Hilbert stated it in many
occasions as something like
"the totality of real numbers contains, in the sense of a one-to-one
correspondence between elements, any other set whose elements satisfy
also the axioms that precede."
In other words, the real number system <R, +, ., 0, 1> is a MAXIMAL
Archimedean ordered field. Similarly, condition (4) above (that is,
the axiom of mathematical induction) can be reformulated as a MINIMAL
structure axiom. Here we have two classical examples of what Rudolf
Carnap called EXTREMAL AXIOMS, the addition of which to an
noncategorical axiom system OFTEN (but not always) yields a
categorical one. See Carnap’s Introduction to Symbolic Logic (Dover,
1958, p. 180) and references there.
REAL CLOSED FIELDS
Extremal axioms are in no way strange in abstract field theory. An
important example, connected with solutions of Hilbert’s 17th problem,
is provided by the REAL CLOSED FIELDS. A field F is real closed when
there exists an order relation < in F such that, equipped with <, F is
MAXIMAL algebraic ordered. This is a second order condition.
Interestingly, it can be proved that this is equivalent to
(i) every positive element is a square, and
(ii) every polynomial of odd degree has a root in F. (!!)
It is not difficult to see that conditions (i) and (ii) are equivalent
to a denumerable set of first-order sentences. Thus, the class of real
closed fields is first-order axiomatizable.
NON-AXIOMATIZABILITY
If I am not mistaken, Luigi raised the following question:
is the class of Dedekind-complete fields first-order axiomatizable?
Well, even the class of Archimedean ordered fields is not first-order
axiomatizable! This can be proved quickly by means of the famous
compactness theorem of first-order logic. It follows from what was said
above that the class of Dedekind-complete fields is not first-order
axiomatizable. Thus, Hilbert’s axiom of completeness is "intrinsically"
second-order and IR is not a first-order structure. An interesting
question is: can we characterize the set of all first-order proprieties
of IR? In a sense, the famous Tarski's principle provides an answer.
According to this result, the class of real closed fields shares with
IR all first-order proprieties. More precisely, any first-order propriety
of IR dealing with addition, multiplication and order holds in every
real closed field. (Conditions (i) and (ii) mentioned above are a
particular case of this.) This enables us to transfer results from IR
cheaply to an arbitrary real closed field.
Carlos Cesar