# The antinomy of Mr. Gödel

Chaïm Perelman, Jérôme Verstrynge

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Chaïm Perelman, Jérôme Verstrynge. The antinomy of Mr. Gödel. 2023. hal-04296690v2

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# THE ANTINOMY OF M. GÖDEL

TRANSLATION BY JÉRÔME VERSTRYNGE

ABSTRACT. In order to facilitate historical research in logic, we provide an English translation of a French article by Chaïm Perelman called 'L'antinomie de M. Gödel', published in the Bulletins de l'Académie royale de Belgique in 1936.

## The antinomy of M. Gödel,

by Ch. Perelman, F.N.R.S. aspirant

In a now famous article<fnref n="1" />, M. Gödel has proposed to demonstrate that inside an analytical system, like that of *Principia Mathematica*<fnref n="2" />, it is possible to construct indeterminable propositions.

By *indeterminable* proposition, we mean a proposition such that one cannot demonstrate either the truth of the affirmation or the truth of the negation of this proposition. To demonstrate that a proposition *p* is indeterminable, one must reduce the hypothesis that *p* is provable and that not-*p* is provable to a contradiction. To settle some terminology, let's say that a *provable* expression is true; by transposition, a false proposition is *unprovable*.

To construct an indeterminable proposition inside an analytical system, M. Gödel arithmetizes the formulas of logic, that is to say, he makes a determined natural number correspond to each logical sign; so that each formula can be expressed using a finite sequence of natural numbers, and each demonstration uses a finite sequence of finite sequences of natural numbers. Proving a proposition amounts to constructing a sequence of sequences of natural numbers such that the last sequence expresses the demonstrated proposition. Mr. Gödel shows that it is possible to construct, with the help of *Principia Mathematica* symbols, a formula $F(v)$ whose signification would be "v is a provable formula".

This being admitted, the construction of an indeterminable proposition turns out to be relatively easy.

Consider a sign corresponding to a propositional function; it will consist of a sequence of natural numbers containing a variable, that is, an empty

<footnote n="1">¹ Ueber formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme I. Aus den *Monatsheften für Mathematik und Physik*, XXXVIIIe, v., 1er cahier, Leipzig, 1931.</footnote>

<footnote n="2">² A. WHITEHEAD and B. RUSSELL. *Principia Mathematica*, 2nd ed., Cambridge, 1925.</footnote>
place, where one can put a random argument, which will be a natural num-
ber. Since all the formulas of this calculation consist of finite sequences of
numbers, the number of unique formulas one could construct cannot exceed
the order of what is countable, and it is possible to number these formu-
las, that is to associate each of them with an index which will be a natural
number.

If, amongst all the functions we have numbered, one considers as a value
for the variable the numerical sign corresponding to the index of each of
them, one will obtain sequences of natural numbers where some will be
provable and others will not. This fact makes it possible to distinguish two
kinds of indices: those linked to functions giving rise to provable expressions,
when one gives the variable their index as a value, and those which do not
have this property. *The latter indexes can be grouped into a set E.* Saying
"*n* is an element of *E*" will therefore be equivalent to "it is false that by
giving the variable of the function _nFx_ the value *n* we obtain a provable
expression".

Note that the propositional function "n is an element of E" also has
an index, say q; one can therefore designate it with the symbol $_{qFx}$. By
introducing this symbol into the last equivalence, we have "n satisfies ${}_qFx$
is equivalent to "it is false that n satisfies ${}_nFx$ is a provable expression", or
again: "$_qFn$ is equivalent to "${}_nFn$ is an unprovable expression".

From this last equivalence, it is easy to demonstrate that _qFq_ is an inde-
terminable expression, that is, the hypothesis of its provability, as much as
the provability of its negation, results in a contradiction.

Indeed, suppose that _qFq_ is provable; in this case, _qFq_ will be a true
expression, and it will be the same as its equivalent expression: "_qFq_ is an
unprovable expression". Hypothesizing the provability of _qFq_ implies thus
its negation, and must as a consequence be rejected.

Suppose that not-qFq is provable; it will therefore be true, like its equiv-
alent expression "it is false that qFq is an unprovable expression"; but this
is the same as claiming the demonstrability of_qFq_. Since hypothesizing the
provability of not-qFq also implies its negation, the qFq expression must be
considered as indeterminable.

Let's take a closer look at the reasoning of M. Gödel. A formal study of
the assumptions supporting it will convince us that the result of M. Gödel
is more modest than it seems at first glance. We will see, indeed, that the
only result of the reviewed article is the construction of a new antinomy, to
be added to those, now classic, of logic and set theory. The antinomy of M.
Gödel presents the same structure as the latter, and results, like them, from
a false equivalence laid down in premises<fnref n="3" />.

All of Mr. Gödel's reasoning revolves around a fundamental definition,
that of set E. An index is part of it if the expression, obtained by replacing

<footnote n="3">³ V. CH. PERELMAN, Les paradoxes de la logique, in *Mind*, april 1936, vol. XLV.
For an English translation, see: https://hal.science/hal-04294226</footnote>
the variable of the function by the index of the latter, is unprovable. This
allows us to lay down the equivalence defining set E:

$$
(n) \cdot n \in E \equiv \cdot \sim Dem_n Fn.
$$

(\sim \cdot Dem_n Fn is to be read: it is false that _n Fn is demonstrable.)

Since the $n \in E$ expression designates a function whose index is $q$, we
obtain, with a simple replacement in the previous equivalence,

$$
(n) \cdot {}_qFn \equiv \cdot \sim \cdot Dem_n Fn.
$$

This equivalence being affirmed for all values of *n*, it must also be true if
one replaces *n* with one of its values, namely *q*:

$_qFq \equiv \cdot \sim \cdot Dem_qFq;$

however, this last formula affirms the equivalence of an expression with
the affirmation of its unprovability. One can notice the analogy of this re-
sult with the liar paradox, where one affirms, in an expression, the falsity
of it. Nonetheless, Mr. Gödel seems to bring off, not a paradox, but a
valuable result. Indeed, while in paradoxes one arrives at the equivalence of
a proposition and its negation, one has managed here to demonstrate the
equivalence of a proposition, not with its negation, but with the affirmation
of its unprovability. Now, this result does not seem, a priori, contradic-
tory; instead of classifying it among the paradoxes, it was considered as a
mathematical discovery of the highest importance.

In the following pages, we will demonstrate that from the fundamental
equivalence from which Mr. Gödel starts it is possible to deduce paradoxes
laying down the equivalence of a proposition and its negation, that Mr.
Gödel's reasoning is based on false premises, just like the other antinomies,
and that one should therefore not be surprised to see Mr. Gödel achieve an
extraordinary result, whereas it is easy to come, from these same premises,
to a contradictory result.

Consider the equivalence:

$$
(n) \cdot n \in E \equiv \cdot \sim \cdot Dem_n Fn. \quad (1)
$$

Let's apply the transposition principle to it:

$$
(n) \cdot \sim n \in E \equiv \cdot Dem_n Fn. \quad (2)
$$

In what follows, we will transform these first equivalences to make their
second member equivalent, not to the truth of the first, but to its provability.
We will start by transforming equivalence (2).

Let's note that set E has been defined as the one of all *n*, such that
~ *Dem*<sub>*n*</sub>*Fn*. Its complementary will be the set of all *n*, that *Dem*<sub>*n*</sub>*Fn*.
It follows that if, for a given *n*, *<sub>n</sub>Fn*, is provable, the expression ~ *n* ∈ *E* is
not only true but also provable; it will be enough, in fact, to complete the
demonstration of *<sub>n</sub>Fn* by the definition of set E to demonstrate ~ *n* ∈ *E*.

How set E was introduced allows us to lay down:
$$ (n) \cdot Dem_n Fn \supset Dem \sim n \in E. \quad (3) $$

On the other hand, since every provable expression is true, we have

$$ (n) \cdot Dem \sim n \in E \supset \sim n \in E. \quad (4) $$

From equivalence (2), we can get:

$$ (n) \cdot \sim n \in E \supset Dem_n Fn. \quad (5) $$

Since from (4) and (5), one can get, by syllogism,

$$ (n) \cdot Dem \sim n \in E \supset Dem_n Fn, \quad (6) $$

one can deduce from (3) and (6)

$$ (n) \cdot Dem \sim n \in E \cdot \equiv \cdot Dem_n Fn. \quad (7) $$

Since $n \in E$ can be written $qFn$, proposition (7) is equivalent to

$$ (n) \cdot Dem \sim qFn \cdot \equiv \cdot Dem_n Fn. \quad (8) $$

By applying the same reasoning to proposition (1) which resulted in proposition (8), using equivalence (2), we obtain the equivalence

$$ (n) \cdot Dem_q Fn \cdot \equiv \cdot \sim \cdot Dem_n Fn. \quad (9) $$

For argument $q$, equivalences (8) and (9) become

$$ Dem \sim qFq \cdot \equiv \cdot Dem_q Fq \quad (10) $$

and

$$ Dem_q Fq \cdot \equiv \cdot \sim \cdot Dem_q Fq. \quad (11) $$

Considering propositions (10) and (11), which one could easily deduce from the definition of set E, laid down by Mr. Gödel, one will not be surprised by the result obtained by him, which exploits only to a small extent the possibilities it offered. It makes it possible, in fact, to demonstrate, in addition to the existence of an indeterminable proposition, a proposition whose truth and falsity can be demonstrated simultaneously, and a proposition whose provability is equivalent to unprovability. Here we are, right in the middle of paradoxes. This harvest of results, however, is richer than that obtained in classical paradoxes where one is satisfied enough to demonstrate that the truth of a proposition is equivalent to its falsity; in fact, the notion of provability allows us to obtain four possibilities for a given proposition: it can be provable or unprovable, and the same is true of its negation.

How was it possible to demonstrate propositions (10) and (11), whose contradictory character is nevertheless explicit? They were deduced from propositions (8) and (9), which constitute false equivalences. The first states the equivalence of the provability of a function to that of its negation, and this for all the values of the variable; in the second proposition, we affirm the formal equivalence of the provability of a function to its unprovability. The fact that within the first member of these equivalences is located, in place of the variable, one of the arguments of the function makes it possible to limit
the contradiction; but it explodes when we introduce the same argument
into the second member of the equivalence.

As these false equivalences are deduced, with the greatest ease, from the
definition of set E, it is this which must be called into question; by taking
a close look, in fact, one realizes that it consists of a false equivalence, that
the principle of contradiction forces us to reject.

By saying, in his article, that his demonstration was related to paradoxes,
Mr. Gödel said too little. It is, in fact, a new antinomy, having identically
the same structure as known paradoxes, which he had just built. And, just
like classic paradoxes, the one that Mr. Gödel constructed in his famous
article results from a contradiction lying in the premises.