1 Introduction
As Kurt Gödel repeatedly stressed, his famous first incompleteness theorem is firmly based on the Liar paradox, which, in its simplest form, consists of the statement “This sentence is false,” in which he replaced “false” (for a specific formal theory T) by “unprovable in T.” Many of Gödel’s writings, though, reveal the conviction that the proof of the incompleteness theorem may as well be based on other antinomies, notably Richard’s antinomy. In his groundbreaking 1931 publication “Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme I,” Gödel writes: “The analogy between this result and Richard’s antinomy leaps to the eye; there is also a close relationship with the ‘liar’ antinomy,” and in a footnote he continues: “Every epistemological antinomy can likewise be used for a similar undecidability proof” [Reference Gödel7, p. 175]. Also, in one of his notebooksFootnote 1 preparing his publication, Gödel writes:
Man erkennt den nahen Zusammenhang dieses Beweises mit der Ant. Rich., und es lässt sich erwarten, dass auch die anderenFootnote 2 epist. Ant. ebenfalls sich zu analogen Beweisen umgestalten lassen, was auch tatsächlich zutrifft.
One recognises the close connection of this proof with the Richard antinomy, and it can be expected that also the otherFootnote 3 epistemological antinomies can be reshaped into analogous proofs, something that actually happens.
Gödel considered as one of the main ingredients of these “analogous proofs” the formal reproduction of the epistemological or semantical antinomies, focusing on their inner logical “shape” (Gestalt). In a typewritten manuscript of a lectureFootnote 4 given in Bad Elster in September 1931, Gödel writes:
Es wird nämlich auf diesem Wege möglich, die Antin. zweiter Art (Richard, in gewissem Sinn) formal nachzubilden, aber nat. so, dass sie ihres Charakters als Antin. entkleidet und zu korrekten Bew. für gewisse metamath. Sachverhalte werden.
Namely, it will become possible in this way to formally reconstruct the antinomies of the second kind (Richard, in a certain sense), but of course in a way that they, divested of their character as antinomies, turn into correct proofs of certain metamathematical facts.
It should be interesting to note that John von Neumann, in a reply to Gödel dated 29 November 1930, also independently stresses the importance of the antinomies. Here he states:Footnote 5
I believe I can reproduce your sequence of thoughts on the basis of your communication and can therefore tell you that I used a somewhat different method. You prove
$W \rightarrow A$
, I show independently the unprovability of W, though with a different kind of inference that likewise copies the antinomies.
We shall return to the question of the formal structure of the antinomies in Section 2.
Giving an actual example for the use of a paradox different from both the Liar and Richard paradoxes, Gregory Chaitin reports in Reference Chaitin[4] that Gödel, as an answer to his proposed use of the Berry paradox, told him in 1974 that “It doesn’t matter which paradox you use.” Details about Chaitin’s and other proofs of the incompleteness theorem based on the Berry paradox, also including George Boolos’ proof Reference Boolos[2], can be found in Reference Kikuchi, Kurahashi and Sakai[18].
It should be mentioned here that both Berry’s and Richard’s paradoxes have been as well used as a basis for the proofs of other metamathematical results. In Reference Chaitin[5], Chaitin uses Berry’s paradox to prove the impossibility of a proof that “any particular large program is elegant.” Here a computer program is defined to be elegant if no shorter program has the same output. And in his notebooks on quantum mechanics written in 1935, Gödel himself uses Richard’s paradox to show that the notion of “lawlike sequence” cannot be defined as “produced by a machine,” thus anticipating Turing’s famous results. This proof has been reconstructed in Reference Lethen[19].
Gödel’s general interest in the antinomies is most clearly expressed in notes preparing a course on elementary logic at the University of Notre Dame in the spring of 1939. These notes, published in Reference Adžić and Došen[1], read as follows:
The more detailed working out of logic
$\left \langle \text {...} \right \rangle $
has led to the discovery of the most interesting facts in modern logic. Namely to the fact that the evidences of natural thinking are not consistent with themselves, i.e., lead to contradictions which are called “logical paradoxes.”
This paper now further investigates Gödel’s view on these logical paradoxes, with a strong emphasis on the width of the spectrum of approaches which can be found in his writings. After presenting Gödel’s view upon the general logical shape or structure of the epistemological antinomies in Section 2, two versions of an antinomy called “antinomy of the intensional,” taken from his notebook series Resultate Grundlagen (Results on Foundations), are presented in Section 3. Section 4 then describes a new kind of antinomy closely related to Gödel’s ontological proof for the necessary existence of God. At the same time, type-theoretical solutions in connection with the existence of positive and evil properties are discussed. In Section 5, extensive notes on further theological paradoxes are reproduced. In Section 6, detailed philosophical remarks concerning the antinomies, taken from Gödel’s MaxPhil notebooks, are presented. As we shall see, these are closely connected to formal work in his mathematical workbooks (Arbeitshefte). Finally, in Section 7, Gödel’s early remarks on the connection between Leibniz’ Monadology and quantum mechanics are examined, emphasising a type-theoretic approach to the overall structure of the monadological worldviews.
The great bulk of the material that forms the basis for this paper is taken from unpublished notebooks found in Gödel’s Nachlass. Unless otherwise stated, these notes have been transcribed from the Gabelsberger shorthand system and translated by the present author. Gödel’s own footnotes are reproduced as such and are marked by the sign
$\vartriangleright $
; additions are given in
$\left \langle \text {angle brackets} \right \rangle $
. The material is used with permission of Institute for Advanced Study, unpublished Copyright Institute for Advanced Study. All rights reserved.
In addition to the unpublished material presented in this paper, two published sources on Gödel’s view on the antinomies may be of interest to the reader. The first one is the above-mentioned course on elementary logic given by Gödel in Notre Dame, published in [Reference Adžić and Došen1, pp. 106–115]. This course includes a whole chapter headed “Type theory and paradoxes.” The second source is Gödel’s famous paper “Russell’s mathematical logic” Reference Gödel and Schilpp[8], which comprises an extensive passage about the antinomies and Russell’s vicious circle principle. Also, Reference Wang[33] may serve as a useful supplement, especially Chapter 8.5, The Paradoxes and the Theory of Concepts, which emphasises the differences between the extensional, intensional, and epistemological paradoxes.
2 The logical structure of the antinomies
As we have already seen in the Introduction, Gödel considered what are known as epistemological antinomies as equivalent when it comes to the question if they are capable of forming a basis for certain metamathematical proofs. When examining this phenomenon, their logical “shape” (Gestalt) seems to be of central importance. In this section, we distinguish between what we call the internal and the external logical structure of an antinomy, thus closely following Gödel’s own investigations. Whenever new kinds of antinomies emerge, these two views may serve as possible classification tools.
2.1 The internal logical structure
The internal logical structure of an epistemological antinomy is represented by its own formal syntactic structure. In Reference Hilbert and Ackermann[11], which Gödel studied already in 1928 (cf. Reference von Plato[26]), two of the epistemological antinomies, namely the Liar’s and a form of Richard’s antinomy, are presented in logical notation.Footnote 6 A similar but slightly different approach is taken by Gödel in notes found in one of his notebooks on statistical mechanics and optics, written in 1935.Footnote 7 It should be interesting to observe that the chapter “Type Theory and Paradoxes” of his above-mentioned Notre Dame lectures ends, somewhat abruptly, with the words “The same paradox [i.e., the Liar’s] can be brought to a much more conclusive form as follows:” The notes in his book on mechanics and optics may now be regarded as the missing continuation. Here, Gödel represents the Liar paradox as follows:
$$\begin{align} P: (E\,p)\left[ \sim\! p\; \&\; A\, ass\, p \right],\end{align} $$
$$\begin{align} A\, ass\, P\; \&\; (p)\left[ (A\, ass\, p) \supset (p = P)\right]. \end{align} $$
Note how self-reference is achieved by the descriptive uniqueness of P asserted in proposition (2).
The second paradox formalised by Gödel is Grelling’s paradox, which he elsewhere describes as follows:Footnote 8
Ein epistemologisches Analogon zu 4 stammt von Grelling. Ein Eigenschaftswort soll autologisch bzw. heterologisch heissen, wenn die durch das Wort ausgedrückte Eigenschaft ihm zukommt bzw. nicht zukommt. Beispiele: “kurz” bzw. “lang”. “Heterologisch” ist dann weder autologisch noch heterologisch.
An epistemological analogue to 4 comes from Grelling. An adjective shall be called autological or heterological, if the property expressed by the adjective does or does not apply to itself, respectively. Examples: “short” and “long”. “Heterological” is then neither autological nor heterological.
In this case, Gödel gives the following formalisation. Note the close correspondenceFootnote 9 to the Liar paradox as expressed in propositions (1) and (2).
$$\begin{align} Het(x) = (E\, f)\left[ \sim\! f(x)\; \&\; f\, Bed\, x \right], \end{align} $$
$$\begin{align} Het\, Bed\, \text{``}\mathit{Het}\text{''}\; \&\; (f)\left[ f\, Bed\, \text{``}\mathit{Het}\text{''} \supset f = Het \right]. \end{align} $$
In the case of Grelling, Gödel continues and writes down part of one direction of the proof of an apparent contradiction.
$$\begin{align} Het(\text{``}\mathit{Het}\text{''}) = (E\, f)\left[ \sim\! f(\text{``}\mathit{Het}\text{''})\; \&\; f\, Bed\, \text{``}\mathit{Het}\text{''} \right]\hspace{-1.5pt}, \end{align} $$
$$\begin{align} \sim Het(\text{``}\mathit{Het}\text{''}) \supset Het(\text{``}\mathit{Het}\text{''}). \end{align} $$
Equally, the other direction can now be concluded.
$$\begin{align} Het(\text{``}\mathit{Het}\text{''}) \supset\; \sim Het(\text{``}\mathit{Het}\text{''}). \end{align} $$
Gödel’s brief notes implement a structural unification of Hilbert and Ackermann’s formal version of the Liar paradox on the one hand, and of a formal version of Grelling’s paradox on the other hand, which he had obviously taken from Ramsey’s paper “The Foundations of Mathematics” Reference Ramsey[29], of which an excerpt can be found in one of his Excerptenhefte.Footnote 10 In order to achieve this syntactic unification, two steps had been necessary.
-
(a) Hilbert and Ackermann [Reference Hilbert and Ackermann11, Chapter 4] speak of a person
$\mathfrak {P}$
asserting the proposition
$\mathfrak {A}$
given as
$\forall X (Bh(X) \rightarrow \, ^\rightharpoondown \! X)$
. Here, Gödel changes the unary predicate
$Bh$
into his binary
$ass$
(for ‘asserts’). Note that Gödel calls the person A, whereas the proposition itself is called P. Also, the universal quantification is changed to an existential one. This is possible due to the uniqueness of the asserted proposition
$\mathfrak {A}$
, which Hilbert and Ackermann give as
$Bh(\mathfrak {A})$
and
$\forall X (Bh(X) \rightarrow \mathfrak {A}=X)$
. -
(b) Ramsey’s formalization of the predicate ‘heterological’, denoted by the name F, is given as
$Fx = :. (\exists \phi )\!: xR(\phi \hat {x})\,. \sim \! \phi x$
, using Principia Mathematica notation and taking R as the “relation of meaning,” which corresponds to Gödel’s
$Bed$
(Bedeutung).Footnote
11
This is supplemented only by the proposition
. Here, it has to be noted that Ramsey does not achieve uniqueness, and therefore his stipulations do not entail a contradiction. This must have been noted by Gödel, who immediately added the assertion
, which is indeed a direct translation of Hilbert and Ackermann’s variant of the Liar’s paradox.
Abstracting away from the details and taking
$\sim \! \Phi $
as an abbreviation for
$\Phi \supset \, \perp $
, both antinomies now lead to propositions of the (seemingly contradictory) “Gestalt”
$\Phi \supset (\Phi \supset \, \perp )$
and
$(\Phi \supset \, \perp ) \supset \, \Phi $
, which may be regarded as their internal logical structure.
At this point, it should be a worthwhile addition to have a look at another detailed rendering of the Liar paradox which can be found in Gödel’s notebook “Exceptenheft 7”Footnote 12 (p. 47) as item 7 of a list describing topics Gödel had discussed with Alfred Tarski in February 1942. Here he writes:
Ant. Lügner ohne Bezug auf empirische Tatsachen:
Wenn P ein Satz ist, so sei
$P'$
der Satz, in dem man die ersten 2 Worte von P durch “P” ersetzt. Und es sei U der Satz: Jede Aussage ergibt bei der Transf.
$'$
einen falschen Satz.
Dann ist
$U'$
weder wahr noch falsch, denn:
$U' \equiv $
(U ergibt bei der Tranformation
$'$
einen falschen Satz)
$\equiv U'$
ist falsch,
$U' =$
Die Aussage “Jede Aussage ist von sich falsch” ist von sich falsch
$.\! \equiv \!. \sim U'$
Antinomy Liar without any relation to empirical facts:
If P is a sentence, let
$P'$
be the sentence in which the first two words of P are replaced by “P”. And let U be the sentence: Every proposition yields, under the transformation
$'$
, a wrong sentence.
Then
$U'$
is neither true nor false, because:
$U' \equiv $
(U yields under the transformation
$'$
a wrong sentence)
$\equiv U'$
is wrong,
$U' =$
The proposition “Every proposition is wrong about itself” is wrong about itself
$.\! \equiv \!. \sim U'$
This notebook entry is accompanied by a loose sheet of paper headed “zu p. 47 oben,” containing Gödel’s thorough elaboration of the item just presented. For the interested reader, a detailed transcription is given in Appendix A.
2.2 The external logical structure
Assuming the two expressions
$\Phi \supset (\Phi \supset \, \perp )$
and
$(\Phi \supset \, \perp ) \supset \, \Phi $
now does seem to cause an uncomfortable feeling in our natural way of thinking, and indeed, they entail a contradiction in a classical setting. To see this, we give a formal proof,Footnote
13
at the heart of which one can find an instance of Peirce’s Law as well as the (slightly hidden) possibility of a multiple use of an assumption.
The appearance of Peirce’s Law naturally leads to the question if the internal structure of the Liar’s paradox, i.e., assumptions (1) and (2), would still lead to a contradiction in its absence. To be more precise, the internal structure of an antinomy obviously defines a borderline between those logics which do entail the absurdity
$\perp $
and those which do not. We call this partitioning of the landscape of formal logics the external structure of an antinomy.
Gödel himself investigated the phenomenon just described in his Arbeitsheft 7 Footnote 14 (p. 7) under the heading “Russell Antinomie ohne Negation mit alleiniger Verwendung des positiven Aussagenkalküls” (Russell Antinomy without negation, only using the positive propositional calculus). After some preliminary definitions that capture the internal structure of Russell’s paradox, and given as
$$ \begin{gather*} \sim\! p \equiv_{D\!f} p \supset F,\\ F \equiv_{D\!f} 1=2,\\ x \in a \equiv_{D\!f}\, \sim\! (x \in x),\\ a \in a \equiv\, \sim\! (a \in a), \end{gather*} $$
a short proof leads to the contradiction F.Footnote
15
Whereas the formula
$(p \supset (p \supset q)) \supset (p \supset q)$
is often considered an axiom (referred to as
$\mathsf {W}$
) in the positive implicational calculus, the formula
$((p \supset q) \supset p) \supset ((p \supset q) \supset q)$
, which we shall call
$\mathsf {W'}$
, remains unproved. A proof of
$\mathsf {W'}$
is given in Appendix B. Writing
$\perp $
for F and
$\Phi $
for
$a \in a$
, Gödel’s proof can now be summarised as follows:
R. L. Goodstein conjectured in Reference Goodstein[10] that “the rejection of the tertium-non-datur eliminates the reflexive paradox,” but Gödel’s proof now clearly shows that this cannot be the case. Instead, Gödel calls the expression
$\mathsf {W}$
the “entscheidende Formel” (crucial formula), and it can be shown that even weaker logics than the positive calculus, which include this very formula, are sufficient. As an example, we give a proof of the contradiction
$\perp $
in implicational
$\mathsf {B'W}$
-logic,Footnote
16
which is based solely on the following two axioms that represent transitivity and contraction:
As it turns out, it is even sufficient to use C. Meredith’s rule D of condensed detachment as the only rule of inference.Footnote
17
Roughly speaking, this rule combines the ordinary detachment modus ponens with ‘just enough substitution’ in the premisses. This in turn leads to a very simple and elegant proof notation as nothing else but a structure of ‘applications’ of axioms and intermediate theorems has to be indicated. Whereas the expression
$\mathsf {W'}$
, i.e.,
$((p \supset q) \supset p) \supset ((p \supset q) \supset q)$
, can now be proved by the term
$\mathsf {(B'B')W}$
, the complete proof of the contradiction
$\perp $
can be written as
$\mathsf {(((B'B')W)A')(WA)}$
where
$\mathsf {A}$
and
$\mathsf {A'}$
denote the assumptions
$\Phi \supset (\Phi \supset \, \perp )$
and
$(\Phi \supset \, \perp ) \supset \, \Phi $
, respectively.
The crucial role of axiom
$\mathsf {W}$
now seems to suggest the indispensability of the possibility to reuse assumptions, non-logical axioms, or—to put the phenomenon into a wider context—information in general. This conjecture seems to be supported by the fact that on the one hand, in a logical setting, the theorems of implicational
$\mathsf {BCK}$
and
$\mathsf {BCI}$
logics, which do not include
$\mathsf {W}$
, are exactly those which can be proved in natural deduction without multiple discharge of assumptions. On the other hand, in an algorithmic setting, they are exactly the types of those
$\lambda $
-expressions which do not make multiple use of the
$\lambda $
-bound parameter in their body. Details about these phenomena can be found, for example, in [Reference Hindley14, Chapter 6] and [Reference Sørensen and Urzyczyn32, Chapter 5]. It should be interesting to note that, concerning this matter, Gödel himself writes under his proof:
Die Formel
$\left [ p \supset (p \supset q)\right ] \supset (p \supset q)$
gilt in jedem System, in welchem
-
1. Impl. Regel,
-
2. wenn aus der Annahme
$A,B \quad C$
abgeleitet werden kann, so gilt
$A \supset (B \supset C)$
.
The formula
$\left [ p \supset (p \supset q)\right ] \supset (p \supset q)$
holds in every system in which
-
1. rule of implication,
-
2. if from the assumption
$A,B, \quad C$
can be deduced,
$A \supset (B \supset C)$
holds.
3 The antinomies of the intensional
When considering the internal logical structure of the Liar and Grelling paradoxes in Section 2.1, the repeated (implicit) use of definite descriptions meets the eye. For example, the stipulation
may as well be written as
,
$Het$
being the unique Bedeutung (meaning) of “Het,” denoted by Russell’s inverted iota operator. This proper description of the predicate
$Het$
may now be understood in both an extensional and an intensional manner, and it seems likely that these kinds of considerations led Gödel to what he called the “Antinomien des Intensionalen” (antinomies of the intensional). These antinomies can be found in book I (pp. 36–37) of an unpublished series of notebooks called Resultate Grundlagen (Results on Foundations).Footnote
18
$\underline {\text {Antinomien des Intensionalen}}$
Unter “Eigenschaft” soll verstanden werden eine in einer bestimmten Weise definierte Eigenschaft.
Sei
$\varphi $
eine Eigenschaft. Dann betrachte man die durch folgende Festsetzung definierte Eigenschaft
$\psi $
:
Dann:
-
1.
$\varphi \neq \psi $
, weil die Def. von
$\varphi $
und
$\psi $
verschieden sind. -
2.
$\varphi = \psi $
, nach Def. von
$\psi $
.
$\underline{\text{Antinomies of the intensional}}$
“Property” is to be understood as a property defined in a specific way.
Let
$\varphi $
be a property. Then consider the property
$\psi $
defined by the following stipulation:
Then:
-
1.
$\varphi \neq \psi $
, because the definitions of
$\varphi $
and
$\psi $
are different. -
2.
$\varphi = \psi $
, following the definition of
$\psi $
.
Gödel then continues with a second variant.
Weniger triviale Fassung: Sei F die folgende Funktion:
Wenn
$\mathfrak {r}$
eine Relation ist, so sei
$F(\mathfrak {r})$
die durch folgende Festsetzung definierte Eigenschaft
$\varphi $
:
$\varphi (x) = (\exists y)\, y\, \mathfrak {r}\, x$
. *
Dann gilt
$F(\mathfrak {r}) = F(\mathfrak {s}) \supset \mathfrak {r} = \mathfrak {s}$
.
Nun sei
$\varphi $
eine Eigenschaft, wo durch obige Äquivalenz * aus
$\mathfrak {r}$
definiert ist, und man definiert eine Relation
$\mathfrak {s}$
durch:
. Dann:
-
1.
$\mathfrak {s} \neq \mathfrak {r}$
, weil die Df. verschieden sind. -
2.
$\mathfrak {s} = \mathfrak {r}$
, weil nach Df. von
$\mathfrak {s} \quad F(\mathfrak {s}) = F(\mathfrak {r})$
.
Less trivial version: Let F be the following function:
If
$\mathfrak {r}$
is a relation, let
$F(\mathfrak {r})$
be the property
$\varphi $
defined by the following stipulation:
$\varphi (x) = (\exists y)\, y\, \mathfrak {r}\, x$
. *
Then
$F(\mathfrak {r}) = F(\mathfrak {s}) \supset \mathfrak {r} = \mathfrak {s}$
holds.
Now let
$\varphi $
be a property which is defined by the above equivalence * from
$\mathfrak {r}$
, and define a relation
$\mathfrak {s}$
by:
. Then:
-
1.
$\mathfrak {s} \neq \mathfrak {r}$
, because the definitions are different. -
2.
$\mathfrak {s} = \mathfrak {r}$
, because, following the definition of
$\mathfrak {s}, \quad F(\mathfrak {s}) = F(\mathfrak {r})$
.
Note that the implication
$F(\mathfrak {r}) = F(\mathfrak {s}) \supset \mathfrak {r} = \mathfrak {s}$
has to be interpreted intensionally, as a simple countermodel shows.Footnote
19
4 The Pseudo-God antinomy
We now turn to Gödel’s ontological proof of the necessary existence of God, which was first published in Reference Sobel[30] and subsequently included in Gödel’s Collected Works Reference Goldfarb[9]. Earlier variants of the proof, one of which was discovered in his Nachlass in 2019, are discussed in Reference Kanckos and Lethen[16].
As has already been argued in Reference Kanckos and Lethen[17], one of the key ingredients in all the known versions of the proof is based on Anselm of Canterbury’s definition or description of God through his positive properties. In Chapter XXIV of his Proslogion, Anselm writes:
Si enim singula bona delectabilia sunt, cogita intente quam delectabile sit illud bonum, quod continet iucunditatem omnium bonorum;
For, if individual goods are delectable, conceive in earnestness how delectable is that good which contains the pleasantness of all goods;
This is formalised by Gödel in the defining equivalence:
$$ \begin{align} G(x) \equiv (\varphi)\left[ P(\varphi) \supset \varphi(x) \right]. \end{align} $$
It expresses the idea that to be God-like is synonymous with having every positive property. Gödel follows, accordingly, a nominalistic approach that denotes the sum of all perfections as the metaphysical essence of God, whereas the subsisting being as an essence, that is so central to Anselm, only plays a subordinate role.
Most of the time when Gödel’s proof is discussed, the basis for the investigation is Dana Scott’s version of the proof, which has been published in [Reference Sobel30, Reference Sobel31].Footnote
20
Here, the impredicativity of the definition of the predicate G, which already surfaces in the Proslogion, is expressed by the additional axiom
$P(G)$
, denoting the positiveness of being God-like. Gödel himself avoids this direct labelling and takes an only slightly longer detour, the heart of which is the axiom
$P(\varphi ). P(\psi ) \supset P(\varphi. \psi )$
that expresses the compatibility of positiveness, about which Gödel writes in his notebook MaxPhil III
Footnote
21
(p. 154) under the simple heading “Axiome” (axioms):
Gutes widerspricht dem Guten nicht. (Die wahren Sätze sind ein widerspruchsfreies System, aber nicht die falschen.)
Good does not contradict what is good. (The true sentences are a system free of contradictions, but not the false ones.)
The axiom is accompanied by the footnote “and for any number of summands,” which is worth mentioning only if “any number” is taken to include the possibility to regard an infinite number of summands. Identification of the infinite conjunction of all positive properties with the universally quantified expression in the equivalence (8) then leads to the desired positiveness of the predicate G. Note that Scott comments his axiom
$P(G)$
in a very similar fashion: “Indeed,
$P(\phi )$
is a logical property and G is defined logically as an intersection of positive properties. Any such property ought also to be positive.”
In regard of definition (8), it should now become obvious that the infinite conjunction of positive properties contains itself as a (positive) conjunct. In order to avoid this phenomenon, we introduce the predicate
$G^*$
of being a Pseudo-God that is defined as follows:
$G^*(x)$
holds iff x has every positive property that does not contain itself as a conjunct. As one can now easily see,
$G^*$
includes itself as a property iff it does not include itself. We shall call this paradox, which is of course closely related to Russell’s antinomy, the Pseudo-God antinomy.
Many of Gödel’s widely scattered notes testify that he was fully aware of the situation just described, and the rest of this section will try to throw some light upon his attitude. In his notebook MaxPhil III he writes (pp. 53):
Bem.: Es ist klar, dass wir keinen adäquaten Begriff Gottes haben, sondern nur Approx. Aus der Annahme des Gottesbegriffs folgt wahrscheinlich die Existenz, aus anderen Sätzen mit Notwendigkeit, ähnlich wie beim Begriff der reellen Zahl.
It is clear that we do not have an adequate concept of God, but only an approximation. Existence probably follows from the assumption of the concept of God, from other sentences with necessity, similar to the concept of the real number.
And regarding the circularity of the definition (8) of the predicate G he writes in MaxPhil IV Footnote 22 (p. 239):
$\underline {\text {Bem.}}$
(Phil.): Eine Zirkel-Df. (z.B. der philosophischen Grundbegriffe oder der logischen), d.h. Df. der Form
$\varphi = F(\varphi \, \psi \dots )$
, hat den Sinn eines Approx.-Verfahrens, d.h., wenn man für
$\varphi , \psi $
nur ganz angenäherte Begriffe (oder unbestimmte) einsetzt, so sind die so erhaltenen
$\varphi , \psi $
genauer, und das kann iteriert werden.
A circular definition (e.g., of the basic philosophical or logical concepts), i.e., definitions of the form
$\varphi = F(\varphi \, \psi \dots )$
, has the sense of an approximating procedure, i.e., if one uses only very approximate terms (or indefinite) for
$\varphi , \psi $
, the
$\varphi , \psi $
thus obtained are more precise, and this can be iterated.
Note that for Gödel, the notion of God, identified with the last principle, does indeed count as a fundamental metaphysical concept (Grundbegriff), as has already be pointed out in Reference Lethen[21].
Also, Gödel addressed the problem of circularity by considering type theories that involve both positive and negative (i.e., evil) properties, as can be observed, for instance, in a remark in his notebook MaxPhil X Footnote 23 (p. 70):
Es gibt aber verschiedene Arten des Guten (nicht nur verschiedene Int.), und es gibt Gutes, das eine Vielheit in sich hat. [Man ist gleichzeitig auf verschiedene Weisen glücklich.]
But there are different kinds of good (not just different intensities) and there is good that has a multitude inside. [One is happy in different ways at the same time.]
Whereas the different “intensities” allude to grades of positive properties already mentioned by Anselm of Canterbury, see Reference Kanckos and Lethen[17], the different “kinds” apparently constitute a hierarchy of types of positive properties, where—analogous to a set of elements of a certain type—a conjunction of properties of a certain type has itself a higher type than its conjuncts. We thereby encounter a third kind of type theory, one that constitutes a hierarchy of conjunctions of properties, next to those two kinds mentioned by Gödel in his Notre Dame lecture as the “hierarchy of extensions” and the “hierarchy of predicates.”Footnote 24 Note that Gödel states in the same lecture:
The lowest layer in the hierarchy of types described are the individuals or objects of the world; what these individuals are is an extralogical question which depends on the theory of the world which we assume.
This attitude can now of course simply be transferred to the “lowest layer” of positive properties. The circumstance that the nature of the individuals also has a logical component is briefly taken up again at the end of Section 6.
The close analogy between the different type theories leads us to Gödel’s views concerning the relation between a set and its elements on the one side, and between a being and its properties on the other. In MaxPhil V Footnote 25 (p. 327) he notes:
$x\, \varepsilon \, \mathfrak {a}$
hat irgendwie die Bedeutung:
$\mathfrak {a}$
ist ein Gesichtspunkt, von dem wir alle
$x\, \varepsilon \, \mathfrak {a}$
überblicken können.
$x\, \varepsilon \, \mathfrak {a}$
somehow has the meaning:
$\mathfrak {a}$
is a point of view from which we can survey all
$x\, \varepsilon \, \mathfrak {a}$
.
If this is taken to mean that
$\mathfrak {a}$
itself cannot be observed from this very point of view, a being (identified with the sum of its properties) will never be able to observe itself as a whole. And concerning exactly these restricted abilities of a (human) being, again mentioning and stressing an underlying type theory, Gödel writes in MaxPhil IV (p. 86):
$\underline {\text {Bem. Theol.}}$
: Vielleicht ist es der Vernunft möglich, alles unter ihr Liegende zu verstehen (Raum, Zahl, Materie, Tiere), aber nicht sich selbst und das über ihr Liegende (Engel und höhere Typen).
$\underline {\text {Remark theology}}$
: Perhaps reason is capable of understanding everything positioned under itself (space, number, matter, animals) but not itself and that which is positioned above itself (angels and higher types).
In summary, thus, Gödel not only confirms the Catholic dogma that the immediate vision of God transcends the natural power of cognition of the human soul, but also supports Ludwig Ott’s theory of the “mode of being,” which Ott, interpreting Thomas of Aquinas’ words “Cognitum est in cognoscente, secundum modum cognoscentis,” describes as follows: “As the nature is so is the cognition. When the mode of being of the object of cognition is higher than the mode of being of the subject of cognition, then the latter is from its nature incapable of immediately knowing the object of cognition in its essence” [Reference Ott23, p. 22].
Obviously alluding to John 8:12–14, this very circumstance is again expressed by Gödel in the same notebook (p. 265) as follows. Note that this remark hints at Gödel’s exceptional knowledge of the Bible.Footnote 26
$\underline {\text {Bem.}}$
(Theol.): “Wenn ich von mir selbst zeuge, spreche ich nicht die Wahrheit” bezieht sich vielleicht auf die theolog. Paradoxien?
$\underline {\text {Remark}}$
(theology): “When I testify of myself, I do not speak the truth” perhaps refers to the theological paradoxes?
Actually, the passage in question is given in King James Bible as:
Then spake Jesus again unto them, saying, I am the light of the world: he that followeth me shall not walk in darkness, but shall have the light of life. The Pharisees therefore said unto him, Thou bearest record of thyself; thy record is not true. Jesus answered and said unto them, Though I bear record of myself, yet my record is true: for I know whence I came, and whither I go; but ye cannot tell whence I come, and whither I go.
It is remarkable how closely Jesus’ answer to the Pharisees resembles both the idea of an approximation as well as an “understanding of the ‘progression,’ i.e., of the construction principle,” which Gödel, in connection with Russell’s antinomy, alludes to in philosophical remarks presented in Section 6, and which leads us back to the (approximating) definition of God through his properties.
Until now, we have mainly considered the conjunction of positive properties but Gödel’s notes also reveal that he was interested in types of negative or evil properties as well. Without any further explanations, he writes in his notebook MaxPhil IV (p. 287):
Antin. der Existenz des Bösen und der Willensfreiheit durch Typen theorie gelöst. [Welches ist die richtige Antwort in dieser und jener Welt?]
Antinomy of the existence of evil and free will resolved by type theory. [What is the correct answer in this and that world?]
Whereas Gödel’s view upon the dogma of free will and the apparent contradiction between free will and the existence of evil have already been discussed in Reference Lethen[21], his “solution” unfortunately remains unclear, and only a few fragments of the suggested approach could be traced. In a list of theological questions and remarks discovered in his Nachlass, item 46 combines the above mentioned idea of a “viewpoint” (Gesichtspunkt) and types of evil.Footnote 27 It reads as follows:
Die Güte Gottes gegen das Böse [d.h., Verzicht auf Rache = Zulassen eines Unrechts zu seinen Gunsten = Bereitschaft, ein Leid dafür auf sich auf sich zu nehmen] ist beschränkt. Das ist der Grund für die Existenz des Bösen in dieser Welt, aber für jedes Wesen größer, als dieses Wesen es fassen kann?
$\left \langle \text {...} \right \rangle $
Also: Der Teufel ist böser, als er es selbst fassen kann. D.h., er erkennt immer wieder eine neue größere Bosheit an sich. Jeder, der sich ein böseres Wesen vorstellen kann als er selbst, ist erlöst. Jeder, dessen Bosheit endlich ist [eine Grenze hat], ist erlöst, d.h., jeder, der irgendetwas Gutes an sich hat, aber nur bereit
$\left \langle \text {ist} \right \rangle $
, endlich viel Gutes einem endlich Bösen zu schenken. Erst indem dem Teufel unendlich viel Böses Mensch unendlich viel Gutes haben.
The goodness of God against evil (i.e., waiver of revenge = allowing an injustice in his favour = willingness to accept suffering for it) is limited. That is the reason for the existence of evil in this world, but larger for every being than this being can register?
$\left \langle \text {...} \right \rangle $
So: The devil is more evil than he can register. That is, he always recognizes a new greater malice in himself. Anyone who can imagine a more evil being than himself is saved. Everyone whose malice is finite [has a limit] is redeemed, that is, anyone who has something good about him, but is only willing to give a finite amount of good to a finite evil. Only by inflicting infinitely much evil to the devil man can have infinitely much good.
Returning to Gödel’s ontological proof, it remains an open question why, in the light of the above discussion, he told Oskar Morgenstern in 1970 that he was “satisfied” with his proof.Footnote 28 Of course, one possible answer might be that he simply aimed to formalise and imitate Anselm’s original argument without touching the underlying deeper questions hidden in the definition of God.
5 Other theological antinomies
Unfortunately, Gödel does not explicitly tell which antinomies he is alluding to when he mentions the “theological paradoxes” in connection with Jesus’ speech. Nevertheless, a list headed “Beispiele scheinbar antin. Sachverhalte” (examples of apparently antinomical circumstances) can be found in one of his theological notebooks called Theologie 3, probably written in 1938.Footnote 29 On page 4 of this book, the following theological “antinomies” are recorded. Note the analysis of free will in items 6 and 6.1. (The transcriptions of the German originals can be found in Appendix C.)
$\underline {\text {Remark}}$
: Examples of seemingly antinomical circumstances
-
1. The sense of something consists in the fact that it has no sense.
-
2. Justice is served by the circumstance that someone suffers an injustice.
-
3. An omnipotent and omniscient spirit that changes his original intention (flood, mercy). (So his intention is to change his intention.)
-
4. How is it possible to sin against an almighty spirit? [Sin means to go against his commandment. The commandments are given to the wicked, for them not to be followed.]
-
5. If the priests destroy the original plan of salvation for a good man, then something better must take its place. Perhaps this is wrong, because he will demand accountability for the lost soul. This means that in order to do something good to a good person, one must do something evil to him.
-
5.1 How much good or evil another person has depends on other people [is given into their power]. But the dependence is not always direct, but sometimes wrong.
$\underline {\text {Is it also in the power of a person to decide}}$
$\underline {\text {whether someone will be blessed?}}$
Probably not in the power of an individual, but perhaps in the power of a totality [people’s voice, God’s voice]. With a totality, it can be the case that the power is only apparent in that they are created
$\underline {\text {in a way}}$
in which they do the will of God. (Also for an individual?) -
6. When an
omniscient spirit puts something “in someone’s power,” knowing in what sense he will use that power. So not in the power, then. -
6.1 ? Doesn’t it perhaps mean that the world plan is foreseen also for the case of the opposite and is also good?
? Or does “giving into the power” mean: I have given this into your power because I know that you will act in such and such a way, and nothing is foreseen for the opposite? In the second case, the consideration: What will happen if I do A, and what if I do
$\sim \! A$
? – is pointless, because nothing is foreseen in one of these cases, that is, there is no point in asking what will happen. But the world may be constructed in such a way
$\underline {\text {that something particular is to be expected because of its}}$
$\underline {\text {construction (its laws).}}$
Footnote
30
In any case, man does not know for what purpose (A,
$\sim \! A$
) something is put in his hand.
6 General philosophical remarks
This section presents a hitherto unpublished passage of Gödel’s notebook MaxPhil V, which, on its first page, bears the date “ca. 1. Mai
$\overline {42}\,-$
” and the comment “Das erste
$\frac {3}{4}$
dieses Heftes ohne Beziehung auf eine Lektüre [selbständig].” (The first
$\frac {3}{4}$
of this book without connection to any reading matter [independent].) Page 324, belonging to this “independent” work, is headed “Antinomien der Mengenlehre, p. 324–329” (antinomies of set theory). It should be interesting to note Gödel’s special emphasis on the constructive point of view, focusing on the idea that the elements of a set literally have to be constructed before the enclosing set comes into existence. In this very sense, we therefore encounter a causal relation between the elements and the enclosing set, which Gödel carefully discusses in his second remark, stressing the central role of the term “Seinsgrund” (reason for existence).Footnote
31
Now, the regarding of a set
$\mathfrak {a}$
as a point of view, from which its elements—and only its elements—can be observed (see Gödel’s fourth remark), clearly underlines a close analogy to a look back into the (branching) past. (The original German transcriptions are placed in Appendix D.)
$\underline {\text {Antinomies of set theory}}$
, p. 324–329
$\underline {\text {Remark}}\left \langle \text {1} \right \rangle $
(Foundations): Method for resolving the antinomies: To determine whether the concepts that occur are perceived or constructed, and from what they are constructed. The precise perception of the principles of construction must then reveal which propositions hold, i.e., where the error hides and what is correct in the proof.
$\underline {\text {Remark}}\left \langle \text {2} \right \rangle $
(Foundations):
-
1. The relation “B is defined with the help of (among others) A Footnote 32 ” or “B depends on A” or “A occurs among the reasons of being of B Footnote 33 ” means for sets the iterated
$\varepsilon $
-relationFootnote
34
[i.e.,
$\varepsilon ^*$
in Russell’s sense]. -
2. The relation of dependency defined in 1 is a partial order (and perhaps a partial well-order), therefore especially always completable to an order. This order is somehow the succession of construction or of being created.
-
3. It follows from 2 (with well-order) that the system of all sets is part of Zermelo’s systems constructed by the operations Power and
$\Sigma $
. (Or Zermelo’s system is a maximal system that adheres to 2.)Footnote
35
-
4. If one omits from 2 the condition of being well-ordered, it may be possible to realize other maximality conditions, namely: For every
$\varphi , m$
there is an
$\mathfrak {a}$
, such that:-
1.
$x\, \varepsilon \, \mathfrak {a} \equiv \varphi (x)$
for all x for which
$\sim \! \mathfrak {a}\, \varepsilon ^*\, x$
and -
2. for no
$x\, \varepsilon \, m$
holds
$\mathfrak {a}\, \varepsilon ^*\, x$
.Footnote
36
-
-
5. 2. can also be formulated as follows: There is a relation R:
$\underline {x \text { is a possible element of } y}$
, which does not contain cycles, and such that the above equivalence holds for all possible elements of
$\mathfrak {a}$
and all
$x\, \varepsilon \, m$
are possible elements of
$\mathfrak {a}$
.
$\underline {\text {Remark}}\left \langle \text {3} \right \rangle $
(Foundations): Mistake in Russell’s extensional antinomy: One can construct a set only
$\underline {\text {after}}$
all its elements have been constructed. [You can construct a multitude as a unit only after all of its members have been constructed.] For these are the basis [arguments] for the construction. Since one will never have constructed all things, one will never be able to construct the universal set. In a certain sense one “has” constructed the set of all numbers, namely by understanding the “and so on.” [By understanding the word “thing,” one would then have constructed all things? But no one understands this word, even if it is the concept that underlies all further constructions.] Or by understanding the “progression,” that is, the “construction principle.” Analogous then with ordinal numbers of the second class, inaccessible numbers, etc. This whole consideration formulated objectively would read?Footnote
37
$\underline {\text {Remark}}\left \langle \text {4} \right \rangle $
(Foundations): The type relation (A has a smaller type than B) plays such a fundamental role in all areas because the types are the “order classes” of the (transitivized) production relation [i.e., the relation of “reason of being”]. That there are different types for the same category of things means that the same things are created anew (and then better) (e.g., “justice”)? But without destroying the previously created ones?
Existence in time is something similar. The two conflicting propositions of Russell’s antinomy would be two beings, each committing suicide and placing the other on the throne. And this process continues to infinity. The universal set does not exist, because it would have to be created later than all things. (Or, it would appear among its own reasons of being.)
$x\, \varepsilon \, \mathfrak {a}$
somehow has the meaning:
$\mathfrak {a}$
is a point of view from which we can survey all
$x\, \varepsilon \, \mathfrak {a}$
[or a construction principle by which we can construct them]?
$\underline {\text {Remark}}\left \langle \text {5} \right \rangle $
(Foundations): The most plausible assumption about the reason for the antinomy is: The existence of an m, so that
$x\, \varepsilon \, m \equiv \varphi (x)$
for each x, is an extrapolation which is too far reaching. It does not hold
-
• for the nominalistic interpretation (? because there is no universally comprehensive language),
-
• for the idealistic one (Here we do not see at all what a “concept” in the objective sense is; if a mapping of facts to things, then the next argumentFootnote 38 ),
-
• in the extensional sense (because no extension can contain itself, and they are based one on the other),
-
• in the psychological sense (for no being, no matter how far extrapolating, has procedures that apply to all things).
The problem would be to see for each individual interpretation what is valid instead of this general statement.
$\underline {\text {Remark}}\left \langle \text {6} \right \rangle $
(Foundations): Russell’s antinomy and the consideration of the nature of extensionality [from which follows
] are an
example of recognition and real reason, namely of the non-existence of a class for each concept. With the concepts, I do not yet see the real reason for the non-existence of
.
$\underline {\text {Remark}}\left \langle \text {7} \right \rangle $
(Foundations): Consequence of the proposition that there exists a universally comprehensive language with an undecidable concept of the meaningful proposition?
$\underline {\text {Remark}}\left \langle \text {8} \right \rangle $
(Foundations): The resolution of the antinomies consists of the following points:
-
1. In the way of which inference of the chain of inferences is a fallacy.Footnote 39
-
2. Specification which weaker axiom is true, in place of the strong one used. [Because it is certainly not completely false.]
-
3. To provide the real reason why the strong one does not hold and the weak one does. [The antinomy is the basis for the knowledge, because of the correctness of all the other inferences, of the non-validity of the stronger one.]Footnote 40
-
4. To give the real basis for the evidence of the antinomy, i.e., to analyze how the evidence for the false axiom arises, on the basis of tacit construction principles and perceptions. [This is a deepening of 1.](4. is something similar to the analysis of how the perceptions arise from the sensations. The entire matter is probably easier in the case of the extensional antinomy and should therefore be done there initially. Is it at all possible?)
-
2’. To make it plausible that with the weaker axiom, the antinomy does not occur.
The crucial point would be 3.
Gödel’s second remark is paralleled by formal work which can be found in his Arbeitsheft 14.Footnote
41
Following four pages (pp. 44–47) headed “Fregesche Auflösung der Antinomien,” Gödel investigates an approach which he calls “Antinom.: Lösung ohne Typenth. (aufgrund einer Art Zirkelfreiheit)” (Antinomy: solution without type theory (based on a kind of freedom from circularity)). As a basic principle, he postulates the existence of a set
$\mathfrak {a}$
of those entities x with a given property
$\varphi $
which are, in a certain way, independent of the set
$\mathfrak {a}$
, thus closely following his own Remark 5. Defining the independence relation
$x\, U \mathfrak {a}$
as
$\sim x\, \varepsilon ^* \mathfrak {a}$
, where
$\varepsilon ^*$
denotes iterated set inclusion, the adjusted and restricted stipulation is then given as
$(\exists \mathfrak {a})(x)\left [x\, \varepsilon \, \mathfrak {a} \equiv \varphi (x)\,. \,x\, U \mathfrak {a}\right ]$
.
Gödel then proceeds to prove the existence of the set m of individuals with an arbitrary given property
$\varphi $
, written as
$x\, \varepsilon \, m \equiv \varphi (x) .\, \mathcal {I}(x)$
, where the notion of being an individual, in a second try,Footnote
42
is defined as
$\mathcal {I}(x) \equiv (u)(u\, \varepsilon \, x \equiv u = x)$
, paired with an axiom that states the existence of at least two different individuals. Unfortunately, Gödel finally gets stuck in an attempted proof of the existence of a set
$\mathfrak {a}$
for every property
$\varphi $
which fulfils the equivalence
$x\, \varepsilon \, \mathfrak {a} \equiv \varphi (x)\,. \,x \varepsilon M$
, M denoting the powerset of the set I of individuals. Note that it might have been exactly this failure, combined with an unusual notion of individuality, which led Gödel to his subsequent comment “schlecht” (bad) in Remark 2, item 4.
7 Monads and types
Yet another of Gödel’s perspectives on the antinomies was first briefly mentioned by Jan von Plato in Reference von Plato[25]. In 1935, Gödel wrote two notebooks on the foundations of quantum mechanics, together comprising one single list of about 340 items.Footnote 43 Only one year later, he wrote a third book entitled Aflenz 1936, which partly copies and partly elaborates the items listed in the earlier two books.Footnote 44 Also, some of the items were completely dropped, possibly because Gödel found an answer to a question posed or simply because he was no longer interested in a certain remark.
Russell’s antinomy reappears in Gödel’s books on quantum mechanics, mentioning the universal set as an analogon to an objective thing-in-itself. Gödel thus transfers the antinomiesFootnote 45 into a physical context, presenting the analogy in the following table (Aflenz, item 250):
This close analogy now immediately focuses Gödel’s attention upon the phenomenon of an approximation of a theory of an “objective world,” which he formulates, in the corresponding item in QM2, as an explicit task to be tackled.
Hauptproblem: Für die physikalische Theorie dasselbe leisten, was die Typentheorie für die Logik geleistet hat. D.h., welches ist die
$\underline {\text {Struktur}}$
derjenigenFootnote
46
Folge von Theorien, welche die klassische Theorie einer “objektiven Welt” zu ersetzen und zu approxim. hat? Ein wesentlicher Bestandteil dabei muss offenbar die “atomistische” (monadologische) Struktur der Welt
$\left \langle \text {sein} \right \rangle $
.
Main problem: To achieve the same for the physical theory, as type theory has achieved for logic. I.e., what is the
$\underline {\text {structure}}$
of theFootnote
47
sequence of theories, which has to replace and to approximate the classical theory of an “objective world”? Obviously, the “atomistic” (monadologic) structure of the world has to be a crucial component.
As already foreshadowed in this remark, Gödel’s books on quantum mechanics clearly reveal that he had studied Leibniz’ Monadology in depth as early as 1935 and regarded it as a suitable frame for the new kind of physics that was arising at the time. In item 266 he writes:
Die metaphysischen Systeme sind nichts anderes als verschiedene Rahmen für physikalische Theorien. [Wie sieht das Platonische aus?] Bisher wurde immer nur das Demokrit’sche System als Rahmen in der wirklichen Physik verwendet. In der modernen Physik wird es anders werden. Vielleicht das Leibniz’sche an seine Stelle?
The metaphysical systems are nothing else but different frameworks for physical theories. [What does the Platonic one look like?] So far only the Democritean system has been used as a framework in real physics. It will be different in modern physics. Maybe the Leibnizian one in its place?
As we shall see, the Leibnizian system offered to Gödel a neat possibility to approximate an objective physical reality, closely linking the monads with the idea of different worldviews. The connection between monads and worldviews is described in item 250 of QM2.
Vielleicht ist
$\left \langle \text {die} \right \rangle $
Leibniz’sche Monadolgie eine der Zwischenstufen zwischen Solips. und objektiver Theorie. Charakter: Man hat nicht “ein” “wahres” Weltbild, sondern ebenso viele verschiedene Weltbilder, als es Monaden gibt.
Perhaps the Leibnizian monadology is one of the intermediate stages between solipsism and objective theory. Character: One does not have “one” “true” worldview, but as many different worldviews as there are monads.
This close connection is of course no stranger to the Monadology. Here Leibniz states (Section 57):Footnote 48 “And as the same town, looked at from various sides, appears quite different and becomes as it were numerous in aspects [perspectivement]; even so, as a result of the infinite number of simple substances, it is as if there were so many different universes, which, nevertheless are nothing but aspects [perspectives] of a single universe, according to the special point of view of each Monad.”
Accordingly, a first step which precedes Gödel’s general concept of a monadologic type theory is what he calls the “Anerkennung des Du” (acceptance of the you) in item 254 of the Aflenz book.Footnote 49 Thus, an infinite set of solipsistic views constitutes a first layer of still disconnected monads, that represent the individuals of Gödel’s theory. This disconnectedness now corresponds to an absence of what Leibniz calls a “universal harmony” in Section 59 of the Monadology. A first level of harmony in a monadologic structure is given by Gödel again in item 250 of QM2:
$\left \langle \text {Die} \right \rangle $
Leibniz’sche Monadologie ist als Zwischending zwischen Solips. und objektiver Welt nicht willkürlich, denn:
$\left \langle \text {Der} \right \rangle $
Grund, weswegen man bei Solips. nicht stehen bleiben kann, ist das Du. D.h., es bestehen gleichberechtigte “Selbste”, zwischen denen dann offenbar Abhängigkeiten bestehen müssen. [Das ist das Schema (Struktur) der Monadentheorie.] (Die Art dieser Abhängigkeiten kann am besten beschrieben werden durch Spiegelung, aber nicht Spiegelung des Universums, sondern bloß Spiegelung seiner selbst durch tausende von Spiegeln.)
Leibniz’s monadology as an in-between between solipsism and the objective world is not arbitrary, because: The reason why one cannot stop at solipsism is the you. I.e., there are equal “selves” between whom there must obviously be dependences. [This is the schema (structure) of the monad theory.] (The nature of these dependences can best be described as mirroring, but not mirroring of the universe, but merely mirroring of itself through thousands of mirrors.)
Here, Gödel follows Leibniz’ description of a universal harmony amongst the infinite set of monads or simple substances exactly, at least as far as the mirroring relation between the monads is concerned. In Section 56 of the Monadology, Leibniz writes: “Now this connexion or adaptation of all created things to each and of each to all, means that each simple substance has relations which express all the others, and, consequently, that it is a perpetual mirror of the universe.” Approximating an objective physical theory by means of a kind of monadologic type theory, Gödel certainly has to reject Leibniz’ “consequence,” a mirroring of the entire universe.Footnote 50 Also, for him the actual mirroring is an iterated process and takes place in distinct layers of monadologic theories, thus paralleling the layers of set-theoretical types. In item 250b of the Aflenz book, Gödel describes this process of iteration as follows:
Zum Prinzip, welches von T zu
$T'$
führt:
$T_0$
, es sind nur die Monaden da (ohne Inhalt, d.h. Vorstellungen).
$T_1$
, jede Monade enthält das Bild der übrigen (leeren) Monaden.
$T_2$
, jede Monade enthält das Bild der übrigen Monaden mit den Vorstellungen aus
$T_1$
, usw.
Regarding the principle that leads from T to
$T'$
:
$T_0$
, only the monads are there (without content, i.e., ideas).
$T_1$
, each monad contains the image of the other (empty) monads.
$T_2$
, each monad contains the image of the other monads with the ideas from
$T_1$
, etc.
It should be worth mentioning here that Gödel’s description of the “empty” Monad without any contents does not necessarily contradict Leibniz’ idea that the simple substances do comprise an internal activity, as described in Section 18: “All simple substances
$\left \langle \text {...} \right \rangle $
have a certain self-sufficiency which make them the sources of their internal activities and, so to speak, incorporeal automata.” Being “empty” rather underlines an independence of the other monads of the same type-theoretical layer
$T_0$
. In the light of Gödel’s definition of being an individual in the preceding section,
$\mathcal {I}(x) \equiv (u)(u\, \varepsilon \, x \equiv u = x)$
, one could conclude that a Monad on this layer mirrors only itself, thus allowing for a kind of reflexive introspection.
Finally, Gödel’s concept of a typed monadology is nicely summed up in the following remark, to be found in item 248 of his Aflenz book. Note the emphasis on the close relation between the “old kind of physics” and solipsism on the one hand, and between quantum mechanics and monadology on the other.
Bei konsequenter Durchführung der pos. Quantenmechanik muss irgendwo das intersubjektive Moment hineinkommen (Loskommen vom Solipsismus). D.h., die unendlich vielen Gesichtspunkte (subjektive Weltbilder) sind gesetzmäßig verknüpft, ohne dass diese gesetzmäßige Verknüpftheit auf ein gemeinsames “Ding-an-sich” zurückgeführt wirdFootnote 51 (= Unterschied gegenüber alter Physik). Hier transfinites Moment, da die Abbildung selbst wieder abgebildet wird.
When consistently carrying out pos. quantum mechanics, the intersubjective element has to enter somewhere (getting rid of solipsism). I.e., the infinitely many points of view (subjective worldviews) are lawfully interwoven, without this lawful interwovenness being traced to a common “thing-in-itself”Footnote 52 (= difference from old physics). Here a transfinite element, because the mapping itself is mapped again.
8 Conclusion
As this paper has been able to point out, Gödel looked upon the phenomenon of the antinomies and a solution by means of type theories from surprisingly many angles, including logical, physical, philosophical, theological, and even biblical perspectives. And yet, the material collected and transcribed from various of his notebooks clearly shows that these perspectives are not to be understood as separate isolated views. Rather, the emphasis apparently lies on their close interconnectedness. This in turn seems to imply that further studies of Gödel’s scientific investigations will have to take the whole spectrum of his writings into consideration, many of which are still slumbering undiscovered in his Nachlass.
Acknowledgements
The research for this article is part of the Godeliana project led by Jan von Plato in Helsinki, Finland, which has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No. 787758) and from the Academy of Finland (Decision No. 318066). I would like to thank Jan von Plato, without whose vast knowledge of Gödel’s Nachlass this paper would not have come into existence. Also, I would like to thank Maria Hämeen-Anttila and Annika Kanckos for fruitful comments on a first draft of this paper, as well as Marcia Tucker at the IAS for her generous support. Finally, my sincere thanks for many comments, which helped to improve this paper, go to the anonymous referee.
9. Appendix. “Lügner ohne Bezug auf empirische Tatsachen”
The following elaboration can be found on a single sheet of paperFootnote 53 inserted into Gödel’s notebook “Excerptenheft 7.”
$\fbox {1}$
$\underline {\text {zu p. 47 oben}}$
Eine All-Aussage
$(x)F(x)$
ist von sich falsch, wenn
$\sim \! F$
(“
$(x)F(x)$
”).
Die Aussage:
ist äquivalent mit ihrer Negation, denn die Aussage in “” bedeutet
$ \underline {\text {von sich}}$
: “Jede All-Aussage ist von sich falsch” ist von sich falsch, also
$\left \langle \text {ist} \right \rangle $
das Ganze seine eigene Negation.
Genauer:
$p =$
Zeichenreihe,
$x =$
Variable
$(p) =$
für alle Zeichenreihen
$An\!f(p) =$
unter Anführungszeichen setzen
$An\!f(p) =$
“p”
$Sb(x,y) =$
Resultat der Subst. von x für alle Variablen in y
$F =$
ist falsch
$Bed(An\!f) = an\!f$
,
$Bed(An\!f) =$
“”
$Bed(f) = F$
$Bed(sb) = Sb$
Footnote
54
$Bed\; An\!f(p) = p$
$An\!f\; Bed\; p = p$
$Bed(e) = \varepsilon $
$Bed(xy) = Bed(x) \cdot Bed(y)$
“z” ein Name von z Footnote 55
$Bed$
“z” = z
“
$Bed\; p$
” = p
$\fbox {2}$
$\underline {\text {zu p. 47 oben}}$
Man bildet den Ausdruck
$\underbrace {\Pi x\; sb\; an\!f\; x,x\;\, e\; f}_{U}$
Nun bildet man den Satz
$Sb(An\!f\; U, U)\; \varepsilon \; F$
Es gilt dann
$Sb(An\!f\; U,U) = sb\lfloor an\!f\; An\!f\; U, An\!f\; U\rfloor \; e\; f$
Footnote
56
Daher:
$Bed[Sb(An\!f\; U, U)] = Sb[An\!f\; U, U]\; \varepsilon \; F$
Aber:
Besser direkt:
$A\, \varepsilon \, W\; .\equiv .\; Bed(A)$
10. Appendix. Proof of
$\mathsf {W'}$
in the positive implicational calculus
In [Reference Hilbert and Bernays12, Chapter 3], three different axiomatizations of the positive implicational calculus are given, and two of them include the axiom
$\mathsf {W}$
. Our proof is based on the following system:
By the use of modus ponens and substitution as rules of inference, the proof of
$\mathsf {W'}$
can then be presented as follows. Note that axiom
$\mathsf {K}$
remains unused.
With condensed detachment as the only rule of inference, the above proof can be simply represented as
$\mathsf {((BW)(CB))}$
.
11. Appendix. German transcriptions for Section 5
$\\ \underline {\text {Bem.:}}$
Beispiele scheinbar antinom. Sachverhalte
-
1. Der Sinn von etwas besteht darin, dass es keinen Sinn hat.
-
2. Der Gerechtigkeit wird dadurch Genüge getan, dass jemandem ein Unrecht geschieht.
-
3. Ein allmächtiger und allwissender Geist, der seine ursprüngliche Absicht ändert (Sintflut, Erbarmungen). (Seine Absicht besteht also darin, seine Absicht zu ändern.)
-
4. Wie ist es möglich, gegen einen allmächtigen Geist zu sündigen? [Sünde bedeutet, zu widerhandeln gegen sein Gebot. Die Gebote werden den Bösen gegeben, damit sie nicht gehalten werden.]
-
5. Wenn die Priester einem Guten gegenüber den ursprünglichen Heilsplan zerstören, so muss etwas Besseres an die Stelle. Vielleicht ist das auch falsch, denn er wird Rechenschaft fordern für die verlorene Seele. D.h., um einem Guten etwas Gutes zu tun, muss man ihm etwas Böses tun.
-
5.1 Wie viel Gutes oder Böses ein anderer Mensch hat, hängt von anderen Menschen ab [ist in ihre Macht gegeben]. Aber die Abhängigkeit ist nicht immer direkt, sondern manchmal verkehrt.
$\underline {\text {Ist es auch in die Macht eines anderen Menschen gegeben, ob jemand}}$
$\underline {\text {seelig wird oder nicht?}}$
Wahrscheinlich nicht in die Macht eines Einzelnen, aber vielleicht einer Gesamtheit [Volkes Stimme, Gottes Stimme]. Bei einer Gesamtheit trifft vielleicht zu, dass die Macht eben insofern nur scheinbar, als sie eben
$\underline {\text {so}}$
erschaffen sind, dass sie Gottes Willen tun. (Auch bei einem Einzelnen?) -
6. Wenn ein
allwissender Geist etwas “in die Macht” jemands legt, wo er doch weiß, in welchem Sinn er diese Macht verwenden wird. Also doch nicht in der Macht. -
6.1 ? Heißt es nicht vielleicht, der Weltplan ist auch für den Fall des Gegenteils vorgesehen und auch gut?
? Oder heißt “in die Macht geben”: Ich habe das in deine Macht gegeben, weil ich weiß, dass du so und so handeln wirst, und für das Gegenteil ist nichts vorgesehen? Im zweiten Fall ist die Überlegung: Was wird geschehen, wenn ich A tue, und was, wenn ich
$\sim \! A$
tue? – sinnlos, denn für einen Fall ist gar nichts vorgesehen, d.h., es hat keinen Sinn zu fragen, was geschehen wird. Aber die Welt kann so konstruiert sein,
$\underline {\text {dass aufgrund ihrer Konstruktion (ihrer Gesetze) etwas}}$
$\underline {\text {Bestimmtes zu erwarten ist.}}$
Footnote
57
Jedenfalls weiß der Mensch nicht, zu welchem Zweck (A,
$\sim \! A$
) etwas in seine Hand gegeben ist.
12. Appendix. German Transcriptions for Section 6
$\\ \underline {\text {Antinomien der Mengenlehre}}$
, p. 324–329
$\underline {\text {Bem.}}\left \langle \text {1} \right \rangle $
(Gr.): Methode zur Auflösung der Antinomien: Von den vorkommenden Begriffen feststellen, ob sie wahrgenommen oder konstruiert sind, und woraus konstruiert. Die genaue Wahrnehmung der Konstruktionsprinzipien muss dann ergeben, welche Sätze gelten, d.h., wo der Fehler steckt und was an der Beweisführung richtig ist.
$\underline {\text {Bem.}}\left \langle \text {2} \right \rangle $
(Gr.):
-
1. Die Relation “B ist definiert mit Hilfe (unter anderem) von A Footnote 58 ” oder “B hängt ab von A” oder “A kommt unter den Seinsgründen von B vorFootnote 59 ” bedeutet für Mengen die iterierte
$\varepsilon $
-RelationFootnote
60
[d.h.
$\varepsilon ^*$
im Russell’schen Sinne]. -
2. Die in 1. definierte Relation der Abhängigkeit ist eine partielle Ordnung (und vielleicht eine partielle Wohlordung), daher insbesondere immer zu einer Ordnung ergänzbar. Diese Ordnung ist irgendwie die Reihenfolge der Konstruktion oder des Erschaffen-Werdens.
-
3. Aus 2. (mit Wohlordnung) folgt, dass das System aller Mengen ein Teil der durch die Operationen Pot. und
$\Sigma $
konstruierten Zermelo’schen Systeme ist. (Oder das Zermelo’sche System ein Max.-System unter Festhaltung von 2.)Footnote
61
-
4. Lässt man aus 2. die Wohlordnungsbedingung weg, so ist es vielleicht möglich, andere Max. Bed. zu realisieren, nämlich: Für jedes
$\varphi , m$
gibt es ein
$\mathfrak {a}$
, so dass:-
1.
$x\, \varepsilon \, \mathfrak {a} \equiv \varphi (x)$
für alle x, für welche
$\sim \! \mathfrak {a}\, \varepsilon ^*\, x$
und -
2. für kein
$x\, \varepsilon \, m$
gilt
$\mathfrak {a}\, \varepsilon ^*\, x$
Footnote
62
-
-
5. 2. kann man auch so formulieren: Es gibt eine Relation R:
$\underline {x \text { ist m}\ddot{\rm o}\text{gliches Element von } y}$
, welche keine Zykel enthält, und sodass die obige Äquivalenz für alle möglichen Elemente von
$\mathfrak {a}$
gilt und alle
$x\, \varepsilon \, m$
mögliche Elemente von
$\mathfrak {a}$
sind.
$\underline {\text {Bem.}}\left \langle \text {3} \right \rangle $
(Gr.): Fehler in der ext. Ant. Russell: Man kann eine Menge erst konstruieren,
$\underline {\text {nachdem}}$
alle ihre Elemente konstruiert sind. [Man kann die Vielheit als Einzelding erst konstruieren, nachdem die einzelnen Elemente konstruiert sind.] Denn diese sind die Basis [Argumente] für die Konstruktion. Da man niemals alle Dinge konstruiert hat, kann man niemals die Allmenge konstruieren. In einem gewissen Sinne “hat” man die Menge aller Zahlen konstruiert, nämlich durch Verstehen des “und so weiter”. [Durch Verstehen des Wortes “Ding” hätte man dann alle Dinge konstruiert? Aber niemand versteht dieses Wort, obwohl es doch der Begriff ist, welcher allen weiteren Konstruktionen zugrunde liegt.] Oder durch Verstehen des “Fortschreitens”, d.h. des “Konstruktionsprinzips”. Analog dann bei Ordinalzahlen der zweiten Klasse, unerreichbaren Zahlen, etc. Diese ganze Überlegung objektiv formuliert würde lauten?Footnote
63
$\underline {\text {Bem.}}\left \langle \text {4} \right \rangle $
(Gr.): Die Typenrelation (A hat kleineren Typus als B) spielt deswegen in allen Gebieten eine so fundamentale Rolle, weil die Typen die “Ordnungsklassen” der (transitivierten) Erzeugungsrelation [d.h. Relation von “Seinsgrund”] sind. Dass es verschiedene Typen für dieselbe Kateg. von Dingen gibt, bedeutet, dass dieselben Dinge nochmals (und dann besser) erschaffen werden (z.B. “Gerechtigkeit”)? Aber ohne Zerstörung des vorher Erschaffenen?
Existenz in der Zeit ist etwas ähnliches. Die beiden widersprechenden Sätze der Russell’schen Antinomie wären zwei Wesen, deren jedes Selbstmord begeht und das andere auf den Thron setzt. Und dieser Prozess geht ins Unendliche fort. Die Allmenge existiert nicht, denn sie müsste später erschaffen sein als alle Dinge. (Oder: Sie würde unter ihren eigenen Seinsgründen vorkommen.)
$x\, \varepsilon \, \mathfrak {a}$
hat irgendwie die Bedeutung:
$\mathfrak {a}$
ist ein Gesichtspunkt, von dem wir alle
$x\, \varepsilon \, \mathfrak {a}$
überblicken können [oder ein Konstruktionsprinzip, durch das wir sie konstruieren können]?
$\underline {\text {Bem.}}\left \langle \text {5} \right \rangle $
(Gr.): Die plausibelste Annahme über den Grund der Antinomie ist: Die Existenz eines m, sodass
$x\, \varepsilon \, m \equiv \varphi (x)$
für jedes x, ist eine zu weit gehende Extrapol. Sie gilt
-
• nicht für die nom. Interpr. (? weil es keine allumfassende Sprache gibt),
-
• nicht für die idealistische (Hier sehen wir gar nicht, was ein “Begriff” im objektiven Sinn ist; wenn eine Zuordnung von Sachverhalten zu Dingen, dann das nächste ArgumentFootnote 64 ),
-
• nicht im ext. Sinn (denn keine Ext. kann sich selbst enthalten, und sie basieren eine auf der anderen),
-
• nicht im psychol. Sinn (denn kein noch so weit extrapol. Wesen hat Verfahren, die sich auf alle Dinge beziehen).
Das Problem wäre, für jede einzelne der Interpret. zu sehen, was statt dieser allgemeinen Aussage gilt.
$\underline {\text {Bem.}}\left \langle \text {6} \right \rangle $
(Gr.): Die Antin. Russell und die Betrachtung des Wesens der Ext. [woraus folgt
] sind ein Beispiel für Erkenntnis und Realgrund, nämlich für die Nicht-Existenz einer Klasse für jeden Begriff. Bei den Begriffen sehe ich noch nicht den Realgrund für die Nicht-Existenz von
.
$\underline {\text {Bem.}}\left \langle \text {7} \right \rangle $
(Gr.): Folgerung aus dem Satz, dass es eine allumfassende Sprache mit unentscheidbarem Begriff des sinnvollen Satzes gibt?
$\underline {\text {Bem.}}\left \langle \text {8} \right \rangle $
(Gr.): Die Auflösung der Antin. besteht in folgenden Punkten:
-
1. Auf die Weise, welcher Schluss der Schlusskette ein Fehlschluss ist.Footnote 65
-
2. Angabe, welches schwächere Ax. anstelle des verwendeten starken wahr ist. [Denn es ist ja sicher nicht vollkommen falsch.]
-
3. Den Realgrund dafür angeben, warum das starke nicht gilt und das schwache gilt. [Für die Nicht-Geltung des stärkeren ist aufgrund der Richtigkeit aller anderen Schlüsse die Antinomie ein Erkenntnisgrund.]Footnote 66
-
4. Den Realgrund für die Evidenz der Antinomie angeben, d.h., das Zustandekommen der Evidenz des falschen Axioms aufgrund unausgesprochener Konstruktionsprinzipien und Wahrnehmungen analysieren. [Das ist eine Tieferlegung von 1.]
(4. ist etwas ähnliches wie die Analyse des Zustandekommens der Wahrnehmungen aus den Empfindungen. Das Ganze ist wahrscheinlich leichter im Falle der ext. Antinomie und sollte daher dort zunächst gemacht werden. Ist das Ganze überhaupt möglich?)
-
2’. Plausibel machen, dass mit dem schwächeren Axiom die Antinomie nicht auftritt.
Der wesentliche Punkt wäre 3.
