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{\Huge\bf On RC 102-43-14}\\[6ex]
%
{\Large\bf Bernd Buldt}\\[3ex]
%
FB Philosophie, Universit\"at Konstanz\\
D-78457 Konstanz, Germany\\
{\em email}\/: bernd.buldt@uni-konstanz.de\\[3ex]
%
{\Large\bf November 2002}\\[6ex]
\end{center}
%
\section{Introduction}
%
When Carnap's star shone brighter on the philosophical heavens than it does
today, he was known for a lot of achievements; but even then his work in
infinitary logic was not regarded as one his merits. Rather on the
contrary, his work in this area is and always has been more or less
completely neglected---unduly neglected, I would like to add, since it is
here where he was more advanced than his most advanced contemporaries, and
where his general philosophical attitudes probably showed most clearly.
This was the main reason why I had chosen to address the meeting with a
survey on ``Carnap's Work in Infinitary Logic'' instead of a more
fashionable Carnapian topic.
However, due to length restrictions the present paper is not an elaboration
of the survey presented at the conference (which I hope to publish
elsewhere). Instead, it focuses on just one documentary evidence: a record
to be found among the Carnap Papers, which probably is the earliest
document proving his interest in and tackling with infinitary logic. Two
reasons led me to restrict the scope of the paper. On the one hand, this
peculiar note has deeply puzzled the relevant community since it first
became known some 10 years ago. On the other hand, it requires quite some
background to get elucidated sufficiently enough to make vanish the
general puzzlement it has caused. So, a separate treatment seemed
recommended.
The paper is organized as follows. The first section presents the
background I think necessary for an appropriate explanation of the note
which has proven so notoriously difficult to understand. This requires to
deal with the \orule, Hilbert, Bernays, and G\"odel, in this order. Thus
prepared, the second section explains first, why Carnap's note is so
puzzling. Then it attempts a sentence-for-sentence-interpretation of this
note, intended to do away with the mysteries surrounding it. A third and
final section offers a short summary of the methods employed, the
assumptions made, and the conclusions reached.
For those who are curious now as to what this note says---it is the one
with archive number ``RC 102-43-14''---here it is:
%
\bqt{\sf
Concerning \emc{Hilbert's new rule of inference}.\\
\emc{Me}: It seems to me that it does not yield more or less than the rule
of complete induction; therefore, merely a question of expediency.\\
\emc{G\"odel}: But Hilbert conceives of it differently, more broadly; the
condition is meant to be the following: ``If \ldots\ is provable with
metamathematical means whatsoever,'' and not: ``If \ldots\ is provable with
such and such means of formalized metamathematics.''\\
Therefore, \emc{complete induction [is] to be preferred} for my system.}%
\fn{``\slGE{Zu \emcG{Hilberts neuer Schlussregel}. \lb\ \emc{Ich}: Mir
scheint sie nicht mehr und nicht weniger zu leisten wie die Regel der
vollst\"andigen Induktion; daher blosse Zweckm\"assigkeitsfrage. \lb\
\emc{G\"odel}: Hilbert meint sie aber anders, umfassender; die Bedingung
ist so gemeint: \glq Wenn \ldots\ mit beliebigen metamathematischen Mitteln
beweisbar ist\grq, nicht so: \glq Wenn \ldots\ mit den Mitteln dieser und
dieser formu[!]lisierten Metamathematik beweisbar ist\grq. \lb\ Also
[ist] f\"ur mein System \emcG{vollst\"andige Induktion vorzuziehen}.}''
(RC 102-43-14; note, dated 12 July 1931; [K\"ohler 1991], p.~144 (=
[K\"ohler 2002a], p.~96)) -- {\em Note}\/: ``RC'' refers to the Carnap
Papers, housed at the University of Pittsburgh Library, with its mirror at
the \slGE{\em Philosophisches Archiv}, University of Konstanz; I extend my
gratitude to Dr.~Uhlemann, the curator of the {\em Archiv}, for her
constant helpfulness in all matters concerning `her' archive's collections.
``BP'' refers to the Bernays Papers, housed at the \slGE{\em
Wissenschaftshistorische Sammlung}, library of the \slGE{\em
Eidgen\"ossisch-Technische Hochschule} (ETH), Zurich; ``GP'' refers to the
G\"odel Papers, housed at the Firestone Library of Princeton University. I
thank all these institutions for their permission to quote. All translation
are mine, likewise the remarks in square brackets which suggest emendations,
point to omissions, etc. A dot ``$\cdot$'' indicates a line break in the
original text suppressed in the translation; underlining and other means of
emphasis in the original text are uniformly rendered as italics.}
\eqt
%
And for those not familiar with what infinitary logics are
all about, here a general orientation:
\bqt
{\sf An infinitary logic (IL) arises from ordinary first-order logic when
one or more of its finitary properties are allowed to become infinite:
\eg, by admitting infinitely long formulae or infinitely long or branched
proof figures. The need to extend first-order logic became pressing in the
late 1950's, when it was not only realized but also accepted, that this
logic is unable to express most of the fundamental notions of mathematics
and thus blocks their logical analysis. Because in many cases IL do not
suffer from these limitations, they are an essential tool in mathematical
logic since then.}%
\fn{[Buldt 1998], p.~769.}
\eqt
%%
%%
\section{Hilbert, Bernays, G\"odel, and the \orule}
%%
%%
Carnap's note is about ``Hilbert's new rule of inference,'' which was a
version of the \orule\ and caused, when Hilbert introduced it, at least
initially quite some frowning on part of those concerned with foundational
issues in logic and mathematics. So the first section shortly explains what
the
\orule\ is about, while the following three sections deal, respectively,
with what we either can safely guess about Hilbert's or do know about
Bernays' and G\"odel's views on this inference rule. This apparent detour
builds up, step by step, the background necessary for understanding
Carnap's note. For this note recorded a discussion Carnap had with G\"odel
on the \orule\ and only if we know what G\"odel knew at that time, we can
hope to shed some light on Carnap's minutes.
%%
%%
\subsection{The $\omega$-rule}
%%
%%
The \orule\ is an infinitary rule of inference that has been employed
within mathematical logic in various forms, depending on whether the
context is recursion theory, proof theory, or model theory. Thus, strictly
speaking, there is not `the' \orule, but a whole family thereof. The first
who came to think of and study a version of the \orule\ was Tarski in 1926;
but it was Hilbert who, in 1930, hit upon this rule as well and put it in
the limelight by his last two publications. In its simplest form it reads,
for all expressions $\varphi$ with one free variable:
\[
(\omega\mbox{-rule})\qquad
\forall\nN\bigm[\;\vdash\varphi(\num{n})\bigm]
\quad\Rightarrow\quad
\vdash\forall x\varphi(x).%
\]
A corollary to G\"odel's first incompleteness theorem shows all consistent
formal systems of arithmetic to suffer from $\omega$-incompleteness; \ie,
there is an expression $\psi$ (different for different formal
systems) with one free variable such that:
\[
(\omega\mbox{-incom.})\qquad\forall\nN\bigm[\;\vdash\psi(\num{n})\bigm]
\quad\&\quad
\nvdash\forall x\psi(x).
\]
This is why the \orule\ can be conceived of, though not necessarily so, as
a `natural' antidote to G\"odelian or $\omega$-incompleteness. For this
rule obviously removes exactly the kind of incompleteness G\"odel's first
theorem has unearthed.
According to the documentary evidence available (known to me),
Carnap learned about the \orule\ during the summer of 1931. Working at this
time on what was to become his \emc{Logical Syntax} and having been one of
the first to learn about G\"odel's first incompleteness theorem, the \orule\
suggested itself as a means to safeguard his logicist account of
mathematics from the threat of G\"odelian incompleteness. In fact, due to
his subsequent employment of the
\orule, it became even known for some time as ``Carnap's rule.''
Carnap's acquaintance with the \orule\ came through publications by
Hil\-bert, the then leading, though at the same time controversial,
preeminent figure in the field of logic and the foundations of mathematics,
whose axiomatic and metamathematical research programs exercised a
considerable influence on Carnap.
%
\bqt{\sf
We had a good deal of sympathy with the formalist method of Hilbert \ed and
learned much from this school \eds.}%
\fn{[Carnap 1963], p.~48.}
\eqt
%
Confronted with Hilbert's version of the \orule, Carnap asked G\"odel to
comment on it and found him well-prepared, for earlier the same year
G\"odel had had an exchange of letters on Hilbert's new move with Hilbert's
collaborator Bernays.
%%
%%
\subsection{Hilbert}
%%
%%
In the early 1930s the aging Hilbert was still the center of what was then
the world's leading mathematics department and his program for a secure
grounding of mathematics by metamathematical (proof-theoretical)
investigations made him a first authority also in foundational issues.
Because his program was partly designed to silence his critics by outdoing
them in their constructivism, his stress on finitary considerations and
finitary means in investigations into the foundations of logic and
mathematics was well-known.\fn{Secondary literature on Hilbert's program is
rich and divers; for the relevant aspects of his finitism \see [Buldt
2002], pp.~402--415, and the literature cited there.}
So it came as a surprise to the community when [Hilbert 1931a] introduced a
new ``finitary rule of inference'' which was nothing but a version of the
\orule:
%
\bqt{\sf
If it has been proved, that, every time $\frak z$ is a given numeral, the
formula ${\frak A}({\frak z})$ becomes a correct numerical
formula, then $\forall x\varphi(x)$ may be used as a first formula [in
a derivation, i.e., as an axiom].%
\fn{``\slGE{Falls nachgewiesen ist, da\ss\ die Formel $\lb\ {\frak
A}({\frak z})\ \lb$ allemal, wenn $\frak z$ eine vorgelegte Ziffer ist,
eine richtige numerische Formel wird, so darf die Formel $\lb\ \forall
x\varphi(x)\ \lb$ als Ausgangsformel angesetzt werden.}'' ([Hilbert 1931a],
p.~491 (= [Hilbert 1935], p.~194)) -- The paper is based on a lecture
Hilbert gave in Hamburg, December 1930, and was received by the journal 21
December 1930. Hilbert proposed the \orule\ (in a slightly different
formulation) also in [Hilbert 1931b], p.~121; but this latter paper was
read on 17 July 1931, 5 days after Carnap's meeting with G\"odel on July 12
(but \see footnote 22). This strongly suggests that Carnap's discussion
with G\"odel was triggered by [Hilbert 1931a].}
}\eqt
%
Let ``\hor'' be short for ``\orule\ according to Hilbert'' (in order to
distinguish it from other versions of the \orule); then we can restate it
more formally as:
\[
(\omega_H\mbox{-rule})\qquad\forall\nN\bigm[\varphi(\num{n})\mbox{ is
numerically correct}\bigm]
\quad\Rightarrow\quad
\vdash\forall x\varphi(x).
\]
%
Now what is important about this rule Hilbert showed already in his first
paper [1931a], namely:
%
\be
\item The \hor\ can consistently be added to a formal system
of arithmetic, say, \PA, the first-order system of
Peano-Arithmetic.\fn{\See [Hilbert 1931a], p.~491 (= [Hilbert 1935],
pp.~194\sq). -- Hilbert (and Bernays) usually worked with a formal system
called ``\Z'' (\see [Hilbert/Bernays 1934], p.~380 (= \S\,7.d.4); but since
\Z\ is, modulo one equality axiom, the same as the nowadays much more common
formalism \PA\ (and since all results carry over), I use \PA\ outside
quotations.}
%
\item It renders the resulting semi-formal system, say, \PAo,
$\Pi_1$-complete in the sense of Hilbert (abbreviated as ``H-complete for
$\Pi_1$''); in short:
\[
(\mbox{H-Com}_{\Pi_1})\qquad\forall
\varphi\in\Pi_1\bigm[\mbox{consistent}\,(\PAo\cup\{\varphi\})\
\Rightarrow\
\ \vdash_{\PAo}\varphi\bigm]\!.%
\fn{\See [Hilbert 1931a], p.~492 (= [Hilbert 1935], p.~195). There are
(annoyingly) many different notions of completeness; for a survey of their
definition and history, \see [Buldt 2001].}
\]
%
\ee
%
An immediate consequence is the syntactical
$\Pi_1$-completeness of
\PAo\ (consistency of \PAo\ assumed); in short:
\[
(\mbox{SynCom}_{\Pi_1})\qquad\forall\varphi\in\Pi_1\bigm[\;\vdash_{\PAo}\varphi\
\mbox{ or }
\vdash_{\PAo}\neg\varphi\bigm]\!.
\]
This completeness of the semi-formal system \PAo\ would have been enough to
escape the original formulation of G\"odel's incompleteness result.%
\fn{Contemporary readers may wonder why Hilbert rested contend at proving
$\Pi_1$-completeness and did not immediately show completeness in respect
to {\sf TA} (True Arithmetic, the set of all sentences of first-order
arithmetic true in the standard model of arithmetic). Though this proof is
a straightforward induction on the number of quantifiers once one assumes
closure under the \orule, it requires, if not the arithmetical hierarchy,
then at least the prenex normal form for arithmetical sentences; but the
latter was established only by [Kuratowski/Tarski 1931], while the first
dates back to [Kleene 1943] and [Mostowski 1947] respectively.}
Any application of the \hor\ requires, for all $n\in{\Bbb N}$, a numerical
evaluation of $\varphi(\num{n})$; that is why the \hor\ (like most versions
of the \orule) is considered an infinitary rule. But in his follow-up paper
Hilbert emphasized again that the \hor\ is a finitary rule of inference.
%
\bqt{\sf
Finally should be stressed the important and for our investigation crucial
fact, namely, that all axioms and inference schemes I called transfinite
[the \hor\ and the quantifier rules] do nevertheless have a strictly
finitary character: the instructions contained therein are performable
within what is finite.%
\fn{``\slGE{Endlich werde noch die wichtige und f\"ur unsere Untersuchung
entscheidene Tatsache hervorgehoben, die darin besteht, da\ss\ die
s\"amtlichen Axiome und Schlu\ss schemata \eds, die ich transfinit genannt
habe, doch ihrerseits streng finiten Charakter haben: die in ihnen
enthaltenen Vorschriften sind im Endlichen ausf\"uhrbar.}'' ([Hilbert
1931b], p.~121)}
}\eqt
%
We will turn to this startling claim in the following section.
%%
%%
\subsection{Bernays}
%%
%%
All this, and more, was known to G\"odel when Carnap started asking him
about the \hor\ during the summer of 1931. For already in January 1931
G\"odel was informed about the \hor\ and discussed it in an exchange of
letters with Bernays, then Hilbert's most important collaborator in
foundational issues at G\"ottingen.\fn{It is an open secret that the
lion's share of the work was done by Bernays; \see [Zach 1999; 2001] for a
beginning to do justice to Bernays' contributions to Hilbert's Program.}
Bernays communicated the \hor\ and the accompanying completeness result in
his second letter to G\"odel, dated 18 Januar 1931.\fn{The correspondence
between Bernays and G\"odel relevant here is partly reproduced in [Buldt
\etal~2002b], pp.~139--146, and will appear in its entirety in [G\"odel
200?].} Besides giving the \hor\ a slightly more general form, he more
importantly shed some light on how the numerical correctness check for the
$\varphi(\num{n})$'s in the antecedent of the \hor\ and on how the
consistency of $\PA\cup\{\varphi\}$ in H-Com$_{\Pi_1}$ are to be determined
according to Hilbert.
%
\bqt{\sf
If ${\frak A}(x_1,x_2,\ldots,x_n)$ is (according to your terminology)
a \emcG{recursive} formula, of which it can be shown, by finitary means,
that for arbitrarily given number values $x_1={\frak z}_1,\ x_2={\frak
z}_2\ \ldots\ x_n={\frak z}_n$ it results in a numerical identity, then the
formula $(x_1)(x_2)\ldots(x_n)\;{\frak A}(x_1\ldots x_n)$ may be used as a
first formula (\ie, as an axiom).
Now Hilbert proves by a simple argument that each formula
$(x_1)(x_2)$ $\ldots(x_n)\;{\frak A}(x_1\ldots x_n)$, where ${\frak
A}(x_1\ldots x_n)$ is a recursive formula shown (by a finitary
consideration) to be \emc{consistent} with the usual system of number
theory, is \emc{provable} in the system extended by the new rule.}%
%
\fn{``\slGE{Die Hilbertsche Erweiterung besteht nun in folgender Regel:
Wenn \lb ${\frak A}(x_1,x_2,\ldots,$ $x_n)$ \lb eine (nach ihrer
Bezeichnung) \emcG{rekursive} Formel ist, von der sich finit zeigen
l\"asst, dass sie f\"ur beliebig gegebene Zahlwerte $x_1={\frak z}_1,\
x_2={\frak z}_2\ \ldots\ x_n={\frak z}_n$ eine numerische Identit\"at
ergibt, so darf die Formel \lb $(x_1)(x_2)\ldots(x_n)\;{\frak A}(x_1\ldots
x_n)$ \lb als Ausgangsformel (d.\,h.~als Axiom) benutzt werden. \lb Hilbert
zeigt nun durch eine einfache \"Uberlegung, dass jede Formel \lb
$(x_1)(x_2)\ldots$ $(x_n)\;{\frak A}(x_1\ldots x_n)$, \lb bei welcher \lb
${\frak A}(x_1\ldots x_n)$ \lb eine rekursive Formel ist und welche
(durch eine finite \"Uberlegung) als \emcG{widerspruchsfrei} mit dem
gew\"ohnlichen System der Zahlentheorie \ed erwiesen ist, in dem durch die
neue Regel erweiterten System \ed \emcG{beweisbar} ist.}'' (GP 010015.45,
pp.~4--5; [Buldt \etal~2002b], pp.~139\sq) -- ``Recursive in G\"odel's
sense'' is what current usage knows as ``primitive recursive.''}
\eqt
%
Thus, required conditions for employing the \hor\ or for accepting
H-Com$_{\Pi_1}$ are finitary demonstrations of correctness and consistency.
While these conditions were not really made explicit in [Hilbert 1931a+b],
once spelled out, they help to explain Hilbert's startling claim that
the \hor\ is a finitary rule. For at that time Hilbert and Bernays were
still convinced that, first, Ackermann had established the consistency of
first-order arithmetic, and, second, that he had accomplished this by
purely finitary means. Consequently, Ackermann's consistency proof (which
was built around a numerical evaluation procedure, the so-called
$\epsilon$-elimination procedure) was held to furnish the \hor\ and
H-Com$_{\Pi_1}$ with what was needed, a finitary correctness and
consistency check. (In the light of G\"odel's second incompleteness theorem,
however, it was slowly realized that this consistency proof was
defective and thus it never was published.)\fn{\See [Hilbert/Bernays 1939],
\S\S\,1--3, for details on the $\epsilon$-calculus, Zach [2002] for a recent
assessment of Ackermann's original work, and [Ackermann 1940] as well as
[Hilbert/Bernays 1939], suppl.~V.B, for Ackermann's rectified proof, using
Gentzen's method of transfinite induction.} % That this is not merely a
historical speculation, but how things were seen at G\"ottingen early in
1931, is evidenced by Bernays, who wrote in the same letter:
%
\bqt{\sf
The consistency of the new rule follows from the method of
\slGE{A"ckermann} (or von Neumann) for demonstrating the consistency of
$\frak Z$.}%
\fn{``\slGE{Die Widerspruchsfreiheit der neuen Regel folgt aus der Methode
des Ackermannschen (oder auch des v.~Neumannschen) Nachweises f\"ur die
Widerspruchsfreiheit von $\frak Z$.}'' (GP 010015.45, p.~5; [Buldt
\etal~2002b],, p.~140) -- While the letter makes a claim only as to
the consistency of the \hor\ when added to $\frak Z$, I entertain the view
that Hilbert thought Ackermann's work using $\epsilon$-elimination would
also guarantee the finitary character of the \hor. This is a novel view and
might not find the enthusiastic approval of all Hilbert scholars. Hence, I
would like to stress that concerning my interpretation of Carnap's note
nothing in particular hinges on this reading of Hilbert.}
\eqt
%
Hence, Hilbert could emphasize exactly this, namely,
%
\bqt
the important fact that the \hor\ does have a strictly finitary character:
the instructions contained therein are performable within what is finite.%
\fn{\See footnote 9 for the exact wording.}
\eqt
%
It was as late as May 1931 that Bernays wrote to G\"odel that and where they
had erred in this respect:
%
\bqt{\sf
Also concerning Ackermann's proof for the consistency of number
theory, I believe I am about straightening things out now. It seems to me
that clearing up the facts consists in the following: Recursions of the
type \ed are, in general, \emc{not expressible within} the system $\frak
Z$.%
\fn{``\slGE{Auch betreffs des Ackermannschen Beweises f\"ur die
Widerspruchsfreiheit der Zahlentheorie glaube ich jetzt ins Klare zu
kommen. \lb Es scheint mir die Aufkl\"arung des Sachverhaltes darin zu
bestehen, dass Rekursionen vom Typ \ed im allgemeinen \emcG{nicht
innerhalb} des Systems \emcG{$\frak Z$ formulierbar} sind.}'' (GP
010015.47, pp.~1--2) -- But these observations took time to really sink in,
especially on Hilbert's side. Recall that his lecture [1931b], which
contains the explicit claim about the finitary character of the \hor\
quoted above, was delivered two months after Bernays wrote this letter.
Three observations may help to explain this discrepancy between Bernays'
letter and Hilbert's lecture. First, according to my understanding of the
Hilbert-Bernays relationship, this happened more often: Bernays was ahead
of Hilbert in accommodating to new facts, with Hilbert lagging stubbornly
behind (\see in this connection [Reid 1970], p. 172, describing Hilbert as
``slow to understand''). This might have very well increased through the
1930s, when Hilbert began to show visible signs of aging ({\em see}, \eg,
the anecdote reported \ibid, pp.~202\sq, and the whole ch.~24, \pass).
Second, we learn in a letter from Bernays to Heinrich Scholz, dated
1 December 1941 (preserved among the Scholz Papers, housed at the
\emcG{Institut f\"ur mathematische Logik}, University of M\"unster), that
he, Bernays, had not been engaged in polishing [Hilbert 1930] and
seeing it through the press. (Scholz was puzzled about the surprisingly
strong anti-Kantian undertones in [Hilbert 1930], which Bernays explained
by his non-participation.) That this was true also for [Hilbert 1931b] is
made credible by the fact that this last paper from Hilbert's pen featured
another terminology than that employed in the papers from 1918--1930, the
time of the active collaboration with Bernays---in fact, terminology-wise
[Hilbert 1931b] continues Hilbert's old terminology as employed in [Hilbert
1904]. Third, we know that the relationship between Bernays and Hilbert
saw, partly violent, disagreement over foundational issues (\see [Reid
1970], p.~173). Taking all this together, I'm inclined to think that a
certain alienation grew between Bernays and Hilbert, especially after
G\"odel's results became known: while Bernays advocated a more flexible
framework for finitism (\see [Bernays 1938]), Hilbert remained unconvinced
(\see his preface to [Hilbert/Bernays 1934]). The difference between
Bernays' letter and Hilbert's lecture was then, if not a sign of the
alienation that had arisen between the two, a sign of the different stance
the two took while trying to cope up with G\"odel's results. (In case of
the first alternative, the growing alienation, the fact, that Hilbert kept
Bernays as his assistent on private expenses until the situation in
Nazi-Germany became unbearable for Bernays in the spring of 1934 (when he
left for Zurich, \see [Reid 1970], pp.~205\sq), would then be solely due to
Hilbert's wish to see the two volumes of [Hilbert/Bernays] finally go to
the press.)} }\eqt
%
But Bernays' earlier letter contained more. It is interesting to see,
\eg, that he bothered to prove, using G\"odel's first incompleteness
theorem, that adding the \hor\ results in a non-conservative extension. We
can learn from it---and I consider this as highly important, for it warns
us not to read, anachronistically, modern knowledge into the historical
sources---that before G\"odel's first incompleteness theorem became known,
Hilbert and Bernays were not sure about the actual deductive strength of
the \hor. That is to say, it is by no means obvious, whether adding the
\hor\ was intended to attain a deductively more powerful formalism, or
whether it was to provide a mere point of attack for proof-theoretical
investigations.\fn{Certain weak versions of the \orule\ do indeed result in
conservative extensions and are hence of purely proof-theoretical
interest; \see [L\'opez-Escobar 1976] for an example.}
After the discussion of various such ramifications, Bernays added that it
would be desirable to have one inference rule instead of two (the \hor\ and
the induction axiom) and which would do the job of both of them. To this end
Bernays suggested a more general \orule, ``\bor'' for short, which came no
longer with any restriction on $\varphi$ and, reduced to the one-variable
case, reads:
\[
(\omega_B\mbox{-rule})\qquad\forall\nN\bigm[\;\vdash\varphi(\num{n})\bigm]
\quad\Rightarrow\quad
\vdash\forall x\varphi(x).%
\fn{``\slGE{Ist \lb ${\frak A}(x_1\ldots x_n)$ \lb eine (nicht notwendig
rekursive) Formel, in welcher als freie Indivi\-du\-en-Variablen nur
$x_1,\ldots,x_n$ auftreten und welche bei der Einsetzung von irgend
welchen Zahl-Werten anstelle von $x_1,\ldots,x_n$ in eine solche Formel
\"ubergeht, die aus den formalen Axiomen und den bereits abgeleiteten
Formeln durch die logischen Regeln ableitbar ist, so darf die Formel \lb
$(x_1)(x_2)\ldots(x_n)\;{\frak A}(x_1\ldots x_n)$ \lb zum Bereich der
abgeleiteten Formeln hinzugenommen werden.}'' (GP 010015.45, p.~11; [Buldt
\etal~2002b], p.~141)}
\]
Although his version of the \orule\ seemed to be much stronger, he had no
clue as to whether it guaranteed already closure under the new rule or
not. Hence, he posed this as a question to G\"odel.
%%
%%
\subsection{G\"odel}
%%
%%
It took even a G\"odel some time to digest the news. First of all he had of
course to safeguard his major discovery, the incompleteness results, from
the threat of completeness that came with the \orule(s). So he
responded only three months later; his letter is dated 2 April 1931. Two of
his insights reported in that letter are relevant for the present
context.
First, formal systems of arithmetic enlarged by either the \hor\ or the
\bor\ are not necessarily deductively closed; G\"odel's first
incompleteness theorem can be extended to cover also such semi-formal
systems of arithmetic.
%
\bqt{\sf
To start with, one can show that also the systems ${\frak Z}^*$,
${\frak Z}^{**}$ are not deductively closed.}%
\fn{``\slGE{Zun"achst kann man zeigen, da\ss\ auch die Systeme ${\frak
Z}^*$, ${\frak Z}^{**}$ nicht deduktiv abgeschlossen sind, \eds.}'' (BP Hs
975 1691a, p.~1) -- In the preceding letter, Bernays called ``${\frak
Z}^*$'' the system \Z\ as extended by the \hor\ and ${\frak Z}^{**}$ the
system extended by the \bor. The diligent reader may be confused here. For
we said above, that the \orule\ can be shown to guarantee {\sf
TA}-completeness, something that seems to be put into question now. The
explanation for this apparent conflict is, that the proof of {\sf
TA}-completeness assume closure under the \orule, which can be attained only
after a transfinite number of its application; \see footnote 33 for
details.}
\eqt
%
Second, G\"odel filed the complaint that,
%
\bqt{\sf
one cannot rest assured at the systems ${\frak Z}^{*}$, ${\frak
Z}^{**}$ as a satisfying grounding of number theory; first of all
because the very complicated and problematic notion of ``finitary proof''
is presupposed without closer mathematical specification.}%
\fn{``\slGE{"Ubrigens glaube ich, dass man sich \ed bei den Systemen
${\frak Z}^*$, ${\frak Z}^{**}$ als einer befriedigenden Begr"undung
der Zahlentheorie nicht beruhigen kann u.[nd] zw.[ar] vor allem deswegen,
weil in ihnen der sehr komplizierte und problematische Begriff `finiter
Beweis' ohne n"ahere mathem.[atische] Pr"azisierung vorausgesetzt wird
(bei Angabe der Axiomenregel).}'' (BP Hs 975 1691a, p.~7) -- The
reservation G\"odel uttered about the concept of ``finitary proof,'' was
the same the intuitionists of the time were challenged with, namely, to
make precise the notion of ``constructive proof.'' Interestingly enough,
G\"odel attempted in the same letter to specify a general condition any
finitary proof must satisfy.}
\eqt
%
Since Ackermann's consistency proof together with the accompanying
machinery of $\epsilon$-elimination was not yet published, no one outside
of Hilbert's G\"ottingen was able to see, as suggested above, why Hilbert
thought the application condition for the \hor---the check that
$\varphi(\num{n})$ is numerically correct for all $\nN$---can be fulfilled
with finitary means, and hence, why Hilbert could claim the finitary
character of the \hor.
%
Lacking this information, it was only natural for G\"odel to challenge
Hilbert instead with the much more general request to give a comprehensive
definition of the notion ``finitary proof.'' (What G\"odel requested is
more general, for the finitary check demanded by the application
condition for the \hor\ can be accomplished without defining in advance what
else might be finitary as well.)
Be all that as it may, important for the present paper is, that we
find G\"odel well-prepared to discuss the
\hor\ with Carnap.
%%
%%
\section{Carnap Meets the $\omega$-rule}
%%
Having gathered the necessary background information,
I will now proceed as follows. The first section describes the general
situation in which Carnap's first encounter with the
\orule\ took place and turns then to what I will call the `natural'
reading of note RC 102-43-14. Its goal is to show, why this
interpretation, though it forces itself onto the reader as `natural,' is
highly unsatisfactory; hence, this first section is entitled ``Problems.''
The second section attempts a more satisfying interpretation of RC
102-43-14 and the accompanying entry to Carnap's diary, dated 12 August
1931. It offers a sentence-for-sentence interpretation and is thus
divided into four subsections. Its goal is to promote an interpretation that
makes sense of the complete text of Carnap's notes and at the same time
avoids the problems of the `natural' reading. If successful, it would
present Carnap not as the fool the `natural' reading implies him to be,
but, on the contrary, as a top-notched foundational researcher of his
time. Sure enough, a much more favourable outcome; hence, this second
section is called ``Solutions.''
%%
%%
\subsection{Problems}
%%
%%
The summer of 1931 was the time when, after having abandoned the
first and ill-fated project \emcG{Untersuchungen zur allgemeinen
Axiomatik} (Investigations Into a General Axiomatics), Carnap started
writing
\emcG{Versuch einer Metalogik} (Essay on Metalogic), which would finally
become his
\emc{Logical Syntax of Language}.\fn{For the abandoned manuscript of the
\emc{Untersuchungen}, recently published as [Carnap 2001], \see [Coffa
1991], ch.~15, [K\"ohler 1991/2002a], \S\,3, and [Awodey/Carus 2001];
concerning the \emc{Metalogik} and its transformation into the \emc{Logical
Syntax}, \see [Carnap 1963], pp.~53--56.} The first documentary
evidence for Carnap's active interest in infinitary logic comes from this
context. It is one of the loose sheets that complement entries to his diary
and it carries the date 12 July 1931.%
\fn{Though the
sheet carries the date 12 July 1931, [K\"ohler 1991], p.~144 (= [K\"ohler
2002a], p.~96), suggests that the actual exchange took place on 30 August
1931 and that the record in question was added later to a previous one on
the same page. The reason K\"ohler adduces is an entry in Carnap's diary,
dated 30 August 1931, which reads: ``\slGE{Metalogik gearbeitet.
Nachmittags mit Feigl und G\"odel im Caf\'e.
\"Uber Hilberts neue Abhandlung; sehr bedenklich.''} If pressed for a
decision, I would, for various reasons, be more inclined to assume
several (at least two) discussions on the \hor. But since it does not make a
difference for the current purpose, I will not try to settle this issue
here.} %
Judging from its title, ``G\"odel Fragen'' (Questions to G\"odel),
and the notes he took, Carnap used this meeting (out of many) he had
with G\"odel for an inquiry about the views G\"odel held at that time on
certain technical and philosophical issues. Entries to his diary and other
accompanying sheets show, that this was a common practice among the two and
indeed some other sheets carry even the same title ``G\"odel Fragen'' (14
March and 9 June 1931).\fn{\See the material from the Carnap Papers
included to [K\"ohler 1991; 2002a+b].} But apparently they continued their
exchange beyond the questions Carnap had prepared in advance---as indicated
by a horizontal line in the manuscript dividing off the first section from a
second---in order to discuss [Hilbert 1931a], which had just arrived at the
library shelves.
It is astonishing, that Carnap's note seems to not reflect anything
of what was sketched in the preceding section; it simply reads:
%
\bqt{\sf
Concerning \emc{Hilbert's new rule of inference}.\\
\emc{Me}: It seems to me that it does not yield more or less than the rule
of complete induction; therefore, merely a question of expediency.\\
\emc{G\"odel}: But Hilbert conceives of it differently, more broadly; the
condition is meant to be the following: ``If \ldots\ is provable with
metamathematical means whatsoever,'' and not: ``If \ldots\ is provable with
such and such means of formalized metamathematics.''\\
Therefore, \emc{complete induction [is] to be preferred} for my system.%
\fn{RC 102-43-14; note, dated 12 July 1931; [K\"ohler 1991], p.~144 (=
[K\"ohler 2002a], p.~96; \see footnote 1 for the German text.}
}\eqt
%
Two problems hit one in the eye: Carnap's apparent glaring misunderstanding
and G\"odel's missing correction. For the first sentence
(``merely a question of expediency'') suggests that Carnap considered the
\hor\ and the rule of induction to be on par with each other, and hence,
that he---and also G\"odel, for there is no indication of G\"odel
setting things straight---apparently did not realize that the
\hor\ is much more powerful than the rule (or the axiom) of complete
induction. How could this possibly be? How could Carnap not realize, on
the spot, that the \orule\ must be deductively stronger than complete
induction? How could G\"odel leave Carnap at that, for we know that he knew
better at that time?
We are thus faced with a situation, where the `natural' reading of
the text and its wording---endorsed by everyone I have spoken with so
far---leads to a seemingly unacceptable conclusion. For this reading of
Carnap's note would force us to believe that not only Carnap but in
particular G\"odel would not have realized the difference between the two
rules.
But, in the light of what was mentioned above concerning G\"odel's
correspondence with Bernays, this `natural' reading is out of question and
we thus find us at the horns of a dilemma. For, alternatively, one could
hold onto the fact that G\"odel knew the difference between the two rules,
but that either G\"odel failed to let Carnap know as well (the one horn) or
that Carnap failed to understand what G\"odel might have said in this
respect and hence his notes too fail to reflect such knowledge (the other
horn).
This last alternative cannot be dismissed out of hand, since Carnap seems
to have been rather slow in grasping already the importance of G\"odel's
first incompleteness theorem. G\"odel told Carnap about his first theorem
as early as 26 August 1930; but there is no indication that Carnap, in
contrast to, say, von Neumann only a fortnight later, realized at that time
any of its foundational consequences.\fn{\See [Dawson 1985], p.~255,
[Dawson 1997], pp.~68--73, and [K\"ohler 1991; 2002a],\S\,4.1--3, for a
collection of the relevant material.} As unattractive as this alternative
is, as unlikely is, from what else we know, the first alternative. For
during the time Carnap wrote his \emc{Metalogik} G\"odel was always happy
to give advice or to help out with his technical expertise (\see footnote
23). Therefore, it makes absolutely no sense to assume G\"odel would have
withheld information from Carnap in this particular situation.
Thus, the `natural' reading of the text leaves us gored on the one horn,
Carnap was slow off the mark---that's it. I regard this poor enough a
conclusion to encourage us looking for a different reading of the text and
thereby saving Carnap from the charge of being slow on the uptake. This
will be done (and, as I hope, be accomplished) in the next section.
%%
\subsection{Solutions}
%%
Trying my hands at a different reading of RC 102-43-14, I will proceed in
reverse order, \ie, I will start with the last sentence of this note and
work my way up to the first, providing a sentence-for-sentence
interpretation, finally turning to Carnap's related diary entry.
%%
%%
\subsubsection{``Therefore, complete induction is to be preferred for
my system.''}
%%
%%
If we assume what all available evidence supports, namely, that Carnap
referred here to what later became his \emc{Logical
Syntax} and bear in mind that Carnap made the rule of complete induction a
rule of his `Language I' but the \orule\ a rule of his `Language II,' then
part of the former bewilderment vanishes into thin air.%
\fn{\See [Carnap 1934; 1937], \S\S\,3--14, for `Language I,' and \ibid,
\S\S\,26--34, for `Language II,' as well as [Carnap 1935] or its later
incorporation to the English translation of the \emc{Logical Syntax},
[Carnap 1937], \S\,34a--i. -- A fact, often overlooked even in the scholarly
literature, is that the \orule\ is not restricted to `Language II' but
featured its first appearance already in the context of `Language I;' \see
\ibid, \S\,14, condition DC2. Likewise, the \orule\ is not a proper rule of
`Language II' but one of its metatheorems; \see \ibid, p.~120 (= Thm.
34f.10). Just to keep things simple, I will, for the moment being, skim
over these details and proceed as if the \orule\ belonged only to `Language
II.' The reason for doing so is that DC2, and hence the \orule, appeared in
the context of `Language I' only for expository purposes (a claim, that
would take too long to get established here).}
% For there is further evidence, that ``my system'' most probably refers
not to the whole work, but in particular to its `Language I.' But if this is
true, then the concluding sentence simply says, that the \orule\ is not
suited for a definite logic with a constructive spirit as `Language I' was
intended to be. Two observations support this reading of ``my system.''
First, `Language I' was the language of choice for Carnap:
%
\bqt{\sf
I had a strong inclination toward a constructivist conception. In my
book, \emc{Logical Syntax}, I constructed a language, called ``Language
I'', which fulfilled the essential requirements of constructivism.%
\fn{[Carnap 1963], p.~49.}
}\eqt
%
Second, the larger portion of the technical discussions with G\"odel during
1931 seems to stem from designing this language, \ie, it was `Language I'
with which Carnap was mostly concerned with at that time. A statement
typical for his notes during this time reads, \eg:
%
\bqt{\sf
I want make do \emc{without sentence variables, predicate variables} (and
variables for number functions).}%
\fn{``\slGE{Ich m\"ochte \emcG{ohne Satzvariable, Pr\"adikatsvariable} und
(Zahlfunktionsvariable) auskommen.}'' (RC \?; note, entitled ``G\"odel
Fragen,'' dated 9 June 1931; [K\"ohler 2002b], p.~112) -- His final
`Language I' met this demand, allowing only for individual (number)
variables, while `Language II' contained all sorts of variables; \see
[Carnap 1934; 1937], \S\,4, \S\,26 respectively. Details of the construction
of `Language II,' among them the important exchange on the notion of
analyticity, appear in the exchange with G\"odel only in 1932 and later;
\see the material reproduced in [K\"ohler 1991; 2002a+b].
}\eqt
%
Hence, if we have biographical reasons to believe that ``my system'' refers
to Carnap's `Language I,' then we can conclude that he (and G\"odel)
clearly understood the difference between the \orule\ and the rule of
complete induction. For then this clearly understood difference was the
very reason to include the induction rule to and to ban \orule s from
his `Language I.' Seen in this light, the conclusion, ``therefore, complete
induction for my system,'' makes perfectly sense.
%%
%%
\subsubsection{``\emc{G\"odel}: But Hilbert conceives of it differently,
more broadly; the condition is meant to be the following: `If \ldots\ is
provable with metamathematical means whatsoever,' and not: `If \ldots\ is
provable with such and such means of formalized metamathematics.'\,''}
%%
%%
Seen in the light of the preceding section, G\"odel's remarks fall into
place as well. For what Carnap preserved for later use was G\"odel's
distinction between ``provable with all metamathematical means whatsoever''
and ``provable with particular formalized metamathematical means,'' together
with the information that Hilbert endorsed the first interpretation.
But this answer of G\"odel presupposes that Carnap had asked about the
distinction contained therein. It suggests that Carnap had asked G\"odel
about meaningful conditions for applying the \orule---\ie, how Hilbert's
requirement for applying the \hor\ (``If it has been proved, that, every
time $\frak z$ is a given numeral \eds'') can be understood---, asked him,
what it could possibly mean, that Hilbert characterized the \hor\ as
a finitary rule. Carnap wanted to know, for, according to the last sentence
of his minutes, he was pondering the question whether he should better
include an \orule\ to his system or not. To get clear about this issue,
Carnap, always eager to give concepts a meaning as precise as possible,
probably even suggested ``provable by formalized means'' as an possible
explanandum for Hilbert's claim. And,
judging by G\"odel's response---``But Hilbert conceives of it
differently''---Carnap apparently favoured this reading of ``finitary'' as
``provable by formalized means.'' (More on this in the following section.)
In his reply G\"odel could rely on (part of) the information he had
gathered first hand from the correspondence with Bernays. So he could make a
authoritative claim---there is no sign that G\"odel had wavered between
several interpretations (``G\"odel thinks it more probable that
\ldots'')---as to what Hilbert's stance on this question actually was.
Though without knowing how tightly Hilbert's claims were connected to
Ackermann's consistency proof, the conditions for applying the \hor\
appeared G\"odel to be far too unspecific as to be mathematically useful;
this is evidenced by the above-quoted complaint he filed in his reply to
Bernays. We can even assume, that he did not withheld his opinion from
Carnap.
But all this was all grist again on Carnap's mills neither to follow Hilbert
nor, consequently, to include a version of the \orule\ to his own preferred
`Language I.' Following G\"odel's information that Hilbert did not see the
\hor\ the way he had initially thought, Carnap concluded that there was no
room for only vaguely specified ``metamathematical means whatsoever'' in his
definite `Language I;' otherwise, an \orule\ would have corrupted the
constructive purity of his `Language I.' In the light of the clarification
G\"odel was able to provide, the following ``therefore complete induction''
was hence a fully justified conclusion on Carnap's side.
%%
%%
\subsubsection{``\emc{Me}: It seems to me that it does not yield more or
less than the rule of complete induction; therefore, merely a question of
expediency.''}
%%
%%
Finally, we have to turn to the first sentence, according to which it is
simply ``a question of expediency'' whether `to go inductive' or `to go
$\omega$.' I will distinguish two possible scenarios; both have to (seek
to) answer the question that forces itself onto every reader of Carnap's
note, namely: Why is there no record of G\"odel correcting Carnap's
apparently outright mistaken opening statement?
%%
%%
\paragraph{The First Scenario.}
%%
%%
The first scenario assumes, that Carnap committed the embarrassing mistake
not to have realized, on the spot, that the \orule\ is more powerful than
the rule of induction. For, while induction does not decide the undecidable
G\"odel-sentence for \PA, the \orule\ does and is hence the stronger rule.
(This is the result of comparing the \orule\ with
$\omega$-incompleteness as given in \S\,0 above, which must have
hit also Carnap in the eye.) If this were true, \ie, if
Carnap did not see
$\Pi_1$-completeness to follow from the \hor, well, then Carnap was sure
enough set right by G\"odel; for G\"odel knew better, as we have seen
above. Moreover, this being the easy lesson we just assumed, we can
likewise assume that it did not require record afterwards---it simply
stuck. Thus, Carnap's note does not reflect the answer we expect from
G\"odel, because an oversight was corrected far too trivial to require a
written record for later perusal.
This reading of the text requires as an auxiliary hypothesis that Carnap
considered worth of being recorded in his minutes only what he expected to
be of later use for himself but which, if unrecorded, might get lost. This
seems to me both, a plausible and innocuous enough hypothesis to be
entertained. This scenario has, however, at least three weak spots. First,
it leaves unexplained why Carnap took down at all the first sentence of his
minutes. For, if it were the easy lesson that immediately stuck as this
scenario assumes, then why should have Carnap preserved his former error
at all? Second, it leaves, where we may expect a thread underlying the
recorded discussion, the first and second sentence completely unrelated to
each other; thus, this scenario does not provide a coherent meaning to the
whole note. Third, it forced us to believe that Carnap was unable to add
up two and two. For on the one hand Carnap knew about
the $\Pi_1$-incompleteness of \PA\ for approximately a year, while on the
other hand he read in Hilbert's paper a proof for the $\Pi_1$-completeness
of \PAo. Carnap was slow, perhaps, but definitely not blind.
%%
%%
\paragraph{The Second Scenario.}
%%
%%
The second scenario avoids these weak spots and does so by a reading of the
opening sentence in the light of G\"odel's subsequent answer. It is brought
about by entertaining the hypothesis of the first scenario about what
Carnap did not think necessary to include to his notes; it supplies some
of the context Carnap had no reason to include to his short and personal
minutes.
G\"odel's answer drew on the distinction between ``provable with
all metamathematical means'' and ``provable with particular formalized
means.'' That is why I assumed above, first, that Carnap requested from
G\"odel some clarification of what Hilbert could possibly had in mind, when
he stated that the correctness of $\varphi(\num{n})$ can finitarily be
proven for all $\nN$; and second, that Carnap at first preferred the reading
``provable with formalized means.''
Further, we know that, by 1934---and there is no reason to assume otherwise
for the summer of 1931---in order to make more precise Hilbert's notion of
``provable with finitary means'' Carnap's best guess was to equate it with
``provable with definite means,'' which in turn he specified as ``provable
within `Language I'.'' In addition, Carnap not only knew very well G\"odel's
arithmetization technique, but even granted it a modest further development
for his own purposes.%
\fn{\See [Carnap 1937] pp.~129, 173, for equating ``finitary'' with
``definite;'' [Carnap 1934; 1937], \S\,15 for the characterization of
`Language I' as definite, and \ibid, \S\S\,18--24, for his knowledge of the
arithmetization technique.} %
Thus, taking these considerations together, Carnap's initial understanding
of the \hor\ must have been something like:
\[
(\omega_{C}\mbox{-rule})\qquad
\vdash_{{\cal L}1}\forall x\bigm[{\sf Pr}_{\sf
L1}(\gn{\varphi(\dot{x})})\bigm]
\quad\Rightarrow\quad
\vdash_{{\cal L}1}\forall x\varphi(x)
\]
(the index ``${\cal L}1$'' refers to the formal system of `Language I'). I
do not propose, of course, that during the discussions with G\"odel Carnap
wrote down on the coffee house table the
\cor\ exactly as given above. Rather, the peculiar formulation of the
\cor\ is intended to make explicit what Carnap's best guess
could have been, while striving for making precise Hilbert's claim that the
\hor\ is finitary.%
\fn{The conclusions I draw do not depend on the exact wording of the
\cor\ (which goes under the name ``formalized'' or ``arithmetized \orule'');
it suffices that it reflects the `spirit' of Carnap's assessment of the
\hor. For the arguments to follow, I do not even need to assume that he
actually tried his hands at formalizing the \hor. The reason to state the
\cor\ is solely due to the intention to give the following discussion a
firmer basis by providing a specific example. I freely admit therefore,
that Carnap's best attempt to formalize the \hor\ would, most probably,
have been:\\[0.5ex]
%
\centerline{$
(\omega_{C^*}\mbox{-rule})\qquad\forall\nN\bigm[\vdash_{LI}{\sf Pr}_{\sf
LI}(\gn{\varphi(\num{n})})\bigm]
\quad\Rightarrow\quad
\vdash_{LI}\forall x\varphi(x)
$.}\\[0.5ex]
%
For the trick of working with the functional expression
``$\gn{\varphi(\dot{x})}$'' was introduced by Bernays only eight years later
(``${\frak B}(\{x\})$'' in his notation); \see [Hilbert/Bernays 1939],
pp.~322--326. But the \cor\ as given in the main text, with the universal
quantification performed within the formal system, better reflects what
Carnap would have aimed at; he preferred strictly formal
procedures that remain completely within the formalism (\see [Carnap 1934;
1937], \S\,22).}
%
What I do assume, however, is that Carnap thought of formalizing the
antecedent of the \orule; for otherwise, as indicated above, the reference
as to ``provable with formalized means'' would make no sense. (One may
recall here that any unformalized version of the \hor\ would not have
been even worth consideration---because of
its infinitary character---for inclusion to his `Language I.')
Now we are prepared to give specific meaning to Carnap's conjecture, that
the \hor\ did not ``yield more or less than the rule of complete
induction.'' In order to compare the relative strength of both rules, we
need, first, a basic formal system without induction or an \orule; second,
this basic formalism should be finitary. Both requirements are fulfilled by
`Language I' without induction. Let ``{\sf IND}'' denote the rule of
complete induction and ``${\cal L}^{\mbox{-}}1$'' ${\cal L}1$ without {\sf
IND}. Then we can restate Carnap's Conjecture (``CC'' for short) that,
%
\bqt{\sf
It seems to me that it [\ie, the \hor\ made precise in the form of the \cor]
does not yield more or less than the rule of complete induction},
\eqt
%
as:
\[
(CC)\qquad\{\varphi:{\sf IND}\vdash_{{\cal L}^{\mbox{-}}1}\varphi\}\
\stackrel{?}{\approx}\
\{\varphi:\omega_c\mbox{-rule}\vdash_{{\cal L}^{\mbox{-}}1}\varphi\}.
\]
But stating CC is not the embarrassing mistake the first impression of
Carnap's opening sentence made us believe it were. Carnap does not put into
question the $\Pi_1$-completeness as proved by Hilbert, but ponders the
question how useful a formalized (`constructivized') \orule, like
the \cor, might be for his own purposes. Four remarks along this line.
First, at a time when completeness issues that come with \orule s had
not been settled, Carnap's conjecture was a serious one: How much gain in
completeness can we expect from adopting a (formalized) \orule? Recall
that both, Bernays and G\"odel, bothered to prove results even for
non-formalized \orule s most logicians would consider as trivial
today: $\PA\subsetneq\PA^{\omega_H}$ (Bernays) and
$\PA^{\omega_B}\subsetneq{\sf TA}$ (G\"odel). Logicians were able to study
formalized \orule s---after a first step taken by [Rosser 1937]---only by
the late 1950s.\fn{[Shoenfield 1959] and [Feferman 1962] are the
milestone papers in case.} Hence, I take it, an answer to CC was by no
means obvious in 1931, and, consequently, Carnap was not the fool the
`natural' reading of RC 102-43-14 or the first scenario suggested he was.
Second, Carnap was right with his conjecture insofar formalized
\orule s yield only a modest strengthening of the underlying formal system.
To see this, recall that Carnap was explicit about the requirement, that
only a finite number of applications of an infinitary rule are allowed.
%
\bqt{\sf
We must do this [the evaluation in `Language II'] in such a way that this
process of successive reference comes to an end in a finite number of
steps.}%
\fn{[Carnap 1937], p.~106 (= [Carnap 1935], p.~173).}
\eqt
%
Let ``${\cal F}^{\omega_*}_\alpha$,'' with $\alpha$ an infinite limit
ordinal, denote a formal system of arithmetic \F, in which less than
$\alpha$ applications of the $\omega_*$-rule are allowed, with
$*$ either $H$, $B$, or $C$. Then, according to the finiteness condition
just quoted and his intention to use a formalized \orule, Carnap
was interested only in one of the smallest and weakest of these systems,
\ie, ${\cal L}^{\mbox{-}}1^{{\omega}_c}_\omega$. Due to the lack of
induction, ${\cal L}^{\mbox{-}}1^{{\omega}_c}_\omega$ is at most as strong
as ${\cal L}1$:
\[
{\cal L}^{\mbox{-}}1^{{\omega}_c}_\omega\stackrel{?}{\approx}{\cal L}1.
\]
To see this, one can argue as follows (modulo much handwaving): ${\cal L}1$
allows (in the limit) for at most $\omega$-many applications of the rule of
complete induction; substitute each application of induction in ${\cal L}1$
with an application of the \cor\ in ${\cal L}^{\mbox{-}}1^{{\omega}_c}$;
then the deductive strength of ${\cal L}^{\mbox{-}}1^{{\omega}_c}_\omega$
amounts (at the very most) to that of ${\cal L}1$. This way of estimating
the deductive power of ${\cal L}^{\mbox{-}}1^{{\omega}_c}_\omega$ does not
presuppose anything that were not accessible to Carnap. (And using other
\orule s would not change the general picture.) In the absence of full
proofs settling CC (available only much later), a rough estimation of the
deductive power to be expected from ${\cal
L}^{\mbox{-}}1^{{\omega}_c}_\omega$ does thus not give cause for any hopes.
The conclusion as to
whether the \cor\ (or any other \orule) is to preferred over the rule of
induction---because a rough estimate gives
${\cal L}^{\mbox{-}}1^{{\omega}_c}_\omega\approx{\cal L}1$ (with
`${\cal L}1={\cal L}^{\mbox{-}}1+{\sf IND}$'!)--- is, ``therefore, merely a
question of expediency.''%
%
\fn{Consequently, if completeness results can be expected at all for
${\cal L}^{\mbox{-}}1^{{\omega}_c}_\alpha$, then only for ordinals
$\alpha$ much bigger than $\omega$. To see this, first consider the \bor\
and assume that a definitional extension of ${\cal L}1$, denoted by ``${\cal
L}{\bf1}$,'' equals \PA. Then it follows from [Rosser 1937] that
${\cal L}{\bf1}^{{\omega}_B}_{\omega+n}$ is $\Pi_{2n}$-complete, for all
$\nN_0$, and from [Goldfarb 1975], that ${\cal
L}{\bf1}^{{\omega}_B}_{\omega^2}$ is {\sf TA}-complete. Taking into
account, that the weaker system ${\cal L}^{\mbox{-}}1^{{\omega}_B}_\alpha$
has to catch up on induction, one will arrive at completeness results only
for $\alpha>\omega+n$, $\alpha>\omega^2$ respectively. Now turn to the
\cor. We know from [Feferman 1962] (and [Kreisel 1965], p.~255 (remark
2(i)), who pointed out that, instead of the reflection principle
$\forall x[{\sf Pr}(\gn{\varphi(\dot{x})})]\rightarrow\forall x\varphi(x)$,
employed by Feferman, the corresponding rule, \ie, the \cor, will do as
well), that ${\cal L}{\bf1}^{{\omega}_c}_\alpha$ can be {\sf TA}-complete;
but only if, in order to define the $\alpha$'s, a suitable path through
$\cal O$, the class of all recursive ordinals, will be chosen. This shows
how much weaker the \cor\ is compared to the \bor\ and thus how unlikely
completeness results are for the even weaker system
${\cal L}^{\mbox{-}}1^{{\omega}_c}_\alpha$. These results show further,
that, even if Carnap's constructive scruples should not have prevented him
from using stronger \orule s, like the \hor\ or the \bor, he would have had
to allow for a transfinite number of applications of these \orule s in
order to arrive at a considerable gain in completeness. In fact, this
was what happened to his `Language II;' \see [Carnap 1938].}
Third, Carnap was not the hard core logicist he usually is portrayed as.
His indebtedness to Frege (and other logicists) notwithstanding, he
entertained a non-foundationalist, pragmatic attitude towards (the
foundations of) mathematics, oriented at its applicability.
%
\bqt{\sf
Since \ed I came to philosophy from physics, [I] looked at mathematics
always from the the point of view of its application in empirical science.}%
\fn{[Carnap 1963], p.~48.}
\eqt
%
Accordingly, all his logic books do not only stand out by featuring
practical examples of how logic can be applied to the empirical sciences,
but he was even willing to settle foundational problems in terms of
applicability.
%
\bqt{\sf
According to my principle of tolerance, I emphasized that [\ldots\ if
there are] methods which, though less safe because we do not have a proof
of their consistency, appear to be practically indispensable for physics
[\ldots\ then] there seems to be no good reason for prohibiting these
procedures so long as no contradictions have been found.}%
\fn{\ibid, p.~49.}
\eqt
%
Hence I take it that another question Carnap presumably had was, what
increase in applicability do I get from adopting an \orule, possibly
formalized? By the lights of the preceding paragraph, his judgement must
have been devastating. There is no reason to sacrifice a form of reasoning
so well-entrenched as induction is in favour of a highly artificial rule,
designed for proof-theoretical purposes, without any apparent gain in
completeness. Questions of expediency strongly suggested to stick to
induction.
Fourth, G\"odel's commentary finally answers a question. `No, Carnap, you
cannot restate the \hor\ as narrowly as the \cor, because ``Hilbert
conceives of it differently, more broadly.''\,'
%%
%%
\subsubsection{``Hilbert's new paper; highly questionable.''}
%%
%%
Imagine Carnap, amidst of moulding his `Language I,' reading
[Hil\-bert 1931a] and asking himself, at what price more completeness? Sure,
adopting the \hor\ (and perhaps even adopting a formalized version of it)
would be a gain in completeness, but exactly how much completeness? And
would the gain in completeness be only a virtual, merely `logical' one, or
also an increase in the applicability of formalized number theory? But the
most pressing question for Carnap, I presume, must have been whether some
gain in completeness is worth sacrificing the definiteness of his `Language
I'---for G\"odel had informed him that the \hor\ should be conceived of
``more broadly'' than he initially was prepared to do. Weighing up a
probably small increase in completeness (of doubtful value) with loosing the
definiteness of his preferred `Language I,' does not the loss outstrip the
benefit, such that the net gain is at most zero (if not negative)?
We thus arrive at another conjecture of Carnap, namely, that employing an
\orule\ has, in terms of its philosophical net gain, in its wake a real
disadvantage:
\[
\{\mbox{losses of employing an \orule}\}
\ \stackrel{?}{\mbox{outstrip}}\
\{\mbox{benefits of employing an \orule}\}.
\]
Having all this in mind, he confided the sceptical entry to his diary,
``Hilbert's new paper; highly questionable.''
%%
%%
\section{Conclusion}
%%
%%
Sure enough, my interpretation
(like any other) of Carnap's difficult to understand note does not offer
more than guarded speculations. But this is the way historical
studies more or less are. RC 102-43-14 is (like a fragment of a
pre-Socratic), taken by itself, evidence too poor to allow for historical
reconstruction. Hence, I added two assumptions. (To labour the obvious, one
always needs some insights to get new ones.) The first assumption was to
read RC 102-43-14 as Carnap's personal and hence elliptical minutes to
which we have to add what Carnap had in and on his mind
these days. The second assumption was about what Carnap had on his mind
during the summer of 1931. What all evidence seems to suggest is that
Carnap was busy working on what later became his \emph{Logical Syntax} and
was focussing especially on his `Language I' during the relevant time in
question. In addition, we could draw on the
\emph{Logical Syntax} for information as to how Carnap's developed views
looked like. I regard both assumptions as highly plausible. The more scanty
the facts, the more important becomes coherence for historical truth. The
two assumptions made, enabled us to give RC 102-34-14 a coherent reading
that does justice to all other documentary evidence, while the
``\,`natural' reading'' and the ``first scenario'' do not. Thus, I'm
inclined to think the present paper is justified from a methodological
point of view.
So what is the bottom line? The hard facts are as follows. During the summer
of 1931 Carnap got to know about the \orule\ from Hilbert's then most
recent publication; he learned in particular that it can consistently be
added to an arithmetic formalism and that it renders this formalism
$\Pi_1$-complete. The conjectured facts are as follows. Contrary to the
first impression his note conveys, Carnap (and of course G\"odel)
understood very well the differences between the rule of induction and the
\orule. In fact, it was precisely this comprehension that made Carnap
shrink back from building the \orule\ into this `Language I.' Further,
according to the discussion as reconstructed from his notes, there was
also no hope for gaining more completeness through an \orule\
so formalized as to make it fit into his `Language I.' A highly
doubtful gain in completeness was not worth the sacrifice in
constructivity; he rather stuck to induction. The twist that enabled this
interpretation was, essentially, to read Carnap's note on Hilbert's new
rule as not referring to the \hor\ in the first place, but to a
formalized version of it, like the \cor, for such a rule was
of prime interest for Carnap while designing his `Language I.'
Carnap's work in infinitary logic did not stop here. On the contrary, it was
this acquaintance with the \hor\ that actually got him started to do
serious work in infinitary logic. From now on he will be concerned, for a
period of more than 10 years, with developing a satisfying account of
infinitary logic, which finally culminated in his theory of junctives (for
which \see [Carnap 1943], \S\S\,19--24).%
\fn{If there is
historical truth---and I firmly believe there is---than it is not one but
many. In order to arrive at historical truth(s) it is sufficient to tell a
story that complies with the facts and the evidence available; but there
are---and I'm likewise convinced of this ---always many stories satisfying
this requirement. I told my story on RC 102-43-14; if others were
encouraged to come up with better stories, I would be gratified.}
%%
%%
\vspace{2cm}
%%
%%
\begin{center}
{\Large\bf Bibliography}\\[3ex]
\end{center}
%%
%%
\setlongtables
\begin{longtable}{crp{9.86cm}}
%%
\name{Awodey, Steve \& Carus, Andr\'e}
%
& [2001] & ``Carnap, Completeness, and Categoricity: The
\emc{Gabelbarkeitssatz} of 1928,'' in: \emc{Erkenntnis} {\bf 45},
pp.~145--172.\ze
%
\name{Bernays, Paul}
%
& [1941] & ``\slFE{Sur les questions m\'ethodologiques actuelles de la
the\'eorie Hilbertienne de la d\'emonstration},'' in: \emcF{Les entretiens
de Zurich sur les fondements et la m\'ethode des sciences math\'ematiques,
6--9 d\'ecembre 1938}, ed.~by Ferdinand Gonseth, Zurich: Leemann,
pp.~144--152.\ze
%
& --- & \see [Hilbert/Bernays 1934; 39].\ze
%
\name{Buldt, Bernd}
%
& [1998] & ``Infinitary Logics,'' in: \emc{Routledge Encyclopedia of
Philosophy}, ed.~by Edward Craig, London \& New York: Routledge,
vol.~4, pp.~769--772.\ze
%
& [2001] & ``Vollst\"andigkeit/Unvollst\"andigkeit,'' in:
\emcG{Historisches W\"or\-ter\-buch der Philosophie}, ed.~by
Gottfried Gabriel \etal, Basel: Schwa\-be, vol.~11, pp.~1136--1141.\ze
& [2002] & \slGE{``Philosophische Implikationen der G\"odelschen S\"atze?
Ein Bericht,''} in: [Buldt \etal~2002b], pp.~395--438.\ze
%%
\nameed{\phantom{Buldt, Bernd} et al.}{eds}
%
& [2002a] & \eG{Kurt G\"odel, Wahrheit und Beweisbarkeit. Vol.~1:
Dokumente und historische Analysen}{, Wien:
H\"older-Pichler-Tempsky.}
%
& [2002b] & \eG{Kurt G\"odel, Wahrheit und Beweisbarkeit. Vol.~2:
Kompendium zum Werk}{, Wien: H\"older-Pichler-Tempsky.}
%
\name{Carnap, Rudolf}
%
& [1934] & \eG{Logische Syntax der Sprache}{ (Schriften zur
wissenschaftlichen Weltauffassung; 8), Wien: Springer, $^2$1968.}
%
& [1935] & ``\slGE{Ein G\"ultigkeitskriterium f\"ur die S\"atze der
klassischen Mathematik},'' in: \emcG{Monatshefte f\"ur Mathematik und
Physik} {\bf 42} (1935), pp.~163--190.\ze
%
& [1937] & \emc{Logical Syntax of Language}, London: Kegan.\ze
%
& [1938] & ``Review of [Rosser 1937],'' in: \emc{Journal of Symbolic
Logic} {\bf 3}, p.~50.\ze
%
& [1943] & \emc{Formalization of Logic} (= Studies in Semantics; 2),
Cambridge/MA: Harvard UP, $^2$1959.\ze
%
& [1963] & ``Intellectual Autobiography,'' in: \emc{The Philosophy of
Rudolf Carnap}, ed.~by Paul Arthur Schilpp, La Salle/IL: Open Court,
pp.~3--84.\ze
%
& [2001] & \emcG{Untersuchungen zur allgemeinen Axiomatik}, postum ed.~by
Th. Bonk \& J.~Mosterin, Hamburg: Meiner.\ze
%
\name{Carus, Andr\'e}
%
& --- & \see [Awodey/Carus 2001].\ze
%
\name{Coffa, J.~Alberto}
%
& [1991] & \emc{The Semantic Tradition from Kant to Carnap. To the
Vienna Station}, Cambridge: Cambridge UP.\ze
%
\name{Dawson, John W.~Jr.}
%
& [1985] & ``The reception of G\"odel's incompleteness theorems,'' in:
\emc{PSA 1984: Proceedings of the 1984 Biennial Meeting of the
Philosophy of Science Association}, Michigan: PSA, vol.~2, pp.~253--271.\ze
%
& [1997] & \emc{Logical Dilemmas. The Life and Work of Kurt G\"odel},
Wellesley/MA: Peters.\ze
%
\nameed{DePauli-Schimanovich, Werner}{ed.}
%
& --- & \see [Buldt \emc{et al.} 2002a,b].\ze
%
\name{Feferman, Solomon}
%
& [1962] & ``Transfinite recursive progressions of axiomatic theories,'' in:
\emc{Journal of Symbolic Logic} {\bf 27} (1962), pp.~259--316.\ze
%
\name{G\"odel, Kurt}
%
& [200?] & \emc{Collected Works, Vols 4--5: Correspondence}, ed.~by Solomon
Feferman \etal, New York: Oxford UP.\ze
%
\name{Hilbert, David}
%
& [1905] & ``\"Uber die Grundlagen der der Logik und Arithmetik,'' in:
\emcG{Verhandlungen des Dritten Internationalen mathematiker-Kongresses
in Heidelberg vom 8.--13.~August 1904}, Leipzig: Teubner, pp.~174--185.\ze
& [1930] & ``Naturerkennen und Logik,'' in: \emcG{Naturwissenschaften}
{\bf 18}, pp.~959--963.\ze
%
& [1931a] & ``Die Grundlegung der elementaren Zahlenlehre,'' in:
\emcG{Mathematische Annalen} {\bf 104}, pp.~485--494 (partly
repr.~in: [Hilbert 1935], pp.~192--195).\ze
%
& [1931b] & ``Beweis des Tertium non datur,'' in: \emcG{Nachrichten von
der Gesellschaft der Wissenschaften zu G\"ottingen aus dem Jahre 1931,
Mathematisch-Physikalische Klasse}, 120--125.\ze
%
& [1935] & \emcG{Gesammelte Abhandlungen, Bd.~3}, Berlin: Springer.\ze
%
\nameph{Hilbert, David}{Bernays, Paul}
%
& [1934] & \eG{Grundlagen der Mathematik I}{, Berlin: Springer, $^2$1968.}
%
& [1939] & \eG{Grundlagen der Mathematik II}{, Berlin: Springer $^2$1970.}
%
\name{Kleene, Stephen Cole}
%
& [1943] & ``Recursive predicates and quantifiers,'' in:
\emc{Transactions of the American Mathematical Society} {\bf 53} (1943),
pp.~41--47.\ze
%
\nameed{Klein, Carsten}{ed.}
%
& --- & \see [Buldt \etal~2002a,b].\ze
%
\name{K\"ohler, Eckehardt}
%
& [1991] & ``G\"odel und der Wiener Kreis,'' in: \emcG{Jour fixe
der Vernunft. Der Wiener Kreis uind die Folgen}, ed.~by P.~Kruntorad,
Wien: H\"older-Pichler-Tempsky, 1991, pp.~127--158 (\see [K\"ohler
2002a]).\ze
%
& [2002a] & ``G\"odel und der Wiener Kreis,'' in: [Buldt \etal~2002a],
pp.~83--108 (rev.~version of [K\"ohler 1991]).\ze
%
& [2002b] & ``Rudolf Carnap -- Kurt G\"odel. Gespr\"ache und Briefe
1928--1940,'' in: [Buldt \etal~2002a], pp.~109--128.\ze
%
& --- & \see [Buldt \etal~2002a,b].\ze
%
\name{Kreisel, Georg}
%
& [1965] & ``Review of [Feferman 1962],'' in: \emcG{Zentralblatt f"ur
Mathematik und ihre Grenzgebiete} {\bf 117} (1965), pp.~254--256.\ze
%
\name{Kuratowski, Kazimierz \& Tarski, Alfred}
%
& [1931] & ``Les op\'erations logiques et les ensembles projectifs,'' in:
\emcG{Funda\-men\-ta Mathematicae} {\bf 17} (1931), pp.~240--247.\ze
%%
\name{L\'opez-Escobar, E. G. K.}
%
& [1976] & ``On an extremely restricted $\omega$-rule,''
in: \emph{Fundamenta Mathematicae} {\bf 20}, pp.~159-172.\ze
%
%
\name{Mostowski, Andrzej}
%
& [1947] & ``On definable sets of positive integers,'' in: \emcG{Fundamenta
Mathematicae} {\bf 34}, pp.~81--112.\ze
%
\name{Reid, Constance}
%
& [1970] & \emc{David Hilbert}, New York: Springer.\ze
%
\name{Rosser, James Barkley}
%
& [1937] & ``G\"odel theorems for non-constructive logics,'' in:
\emc{Journal of Sym\-bolic Logic} {\bf 2}, pp.~129--137.\ze
%
\name{Shoenfield, Joseph R.}
%
& [1959] & ``On a restricted $\omega$-rule,'' in: \emcF{Bulletin de
l'Acad\'emie Polonaise des Sciences, S\'erie des sciences math\'ematique,
astronomique et physique} {\bf 7}\,(1), pp.~ 405-407.\ze
%
\nameed{St\"olzner, Michael}{ed.}
%
& --- & \see [Buldt \etal~2002a,b].\ze
%
\name{Tarski, Alfred}
%
& --- & \see [Kuratowski/Tarksi 1931].\ze
%
\nameed{Weibel, Peter}{ed.}
%
& --- & \see [Buldt \etal~2002a,b].\ze
%
\name{Zach, Richard}
%
& [1999] & ``Completeness before Post: Bernays, Hilbert, and the
development of propositional logic,'' in: \emph{Bulletin of Symbolic Logic}
{\bf 5}, pp.~331-366.\ze
%
& [2001] & \emph{Hilbert's Finitism: Historical, Philosophical, and
Metamathematical Persepctives}, Ph.D., Berkeley; available at
\flq http://www.ucalgary.ca/\~{}rzach\frq.\ze
%
& [2002] & ``The Practice of Finitism: Epsilon Calculus and Consistency
Proofs in Hilbert's Program,'' to appear in: \emph{Synthese}.\ze
\end{longtable}
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\end{document}