Kurt Gödel and the origins of computer science. John W. Dawson, Jr. Pennsylvania State University York, PA U.S.A. Kurt Gödel (1906 –1978). Considered the greatest mathematical logician of the twentieth century, he was one of the founders of recursion theory.
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John W. Dawson, Jr.
Pennsylvania State University
York, PA U.S.A.
Considered the greatest mathematical logician of the twentieth century, he was one of the founders of recursion theory.
A Princeton colleague of Alonzo Church and John von Neumann, his impact on computer science was seminal, but largely indirect.
*First published posthumously in Gödel’s Collected Works
The strong completeness theorem shows
that the rules of inference developed prior
to Gödel’s work are adequate for the pur-
pose of deriving all logical consequences of
a set of axioms. A computer incorporating
just those rules will thus be able to carry out
all such derivations.
The completeness theorem entails that the
statements provable in any first-order
axiomatization of number theory are those
that are true in all models of the axioms.
But for no recursive axiomatization will
those statements coincide with the state-
ments that are true of the natural numbers:
Any recursive axiomatization A of arithmetic
must either be inconsistent (and so prove
every statement φ ) or else fail to prove both
some variable-free statement φand its
negation. Hence if A is consistent, it must
fail to prove some statement that is true of
the natural numbers.
result. In the context of computer science,
as is well known, Alan Turing used the
correspondence between computer pro-
grams and formal systems to recast it as
The Halting Problem: No recursive pro-cedure can correctly determine, for an arbitrarily given computer running any specified program, whether or not the computer will eventually come to a halt.
For the development of computer science,
the proof Gödel gave of his incompleteness
results was more important than the
theorems themselves. Three aspects of
the proof were of particular significance:
overall structure of Gödel’s proof “looks very much
like a computer program” to anyone acquainted
with modern programming languages — unsur-
prisingly, since though “an actual … general-
purpose … programmable computer was still
decades in the future,” Gödel faced “many of the
same issues that those designing programming
languages and …writing programs in those
languages” face today.
As a further example of a numerical state-
ment that must be formally undecidable
within any consistent, recursive axioma-
tization of arithmetic, Gödel adduced a
numerical encoding of a statement asserting
the theory’s consistency.
Much later, in his Gibbs Lecture to the American
Mathematical Society (1951),Gödel would suggest
that the incompleteness theorems are relevant to
During the years 1932–33 Gödel turned his
attention to the decision problem — the
question whether there is an algorithm for
determining whether an arbitrary formula
of first-order logic is valid or not. He estab-
lished two results, concerning particular
prefixclasses of quantificational formulas:
Subsequently, on the basis of Church’s Thesis,
Church and Turing each demonstrated that the full
decision problem is algorithmically unsolvable.
Gödel spent the academic year 1933–34 in
Princeton at the Institute for Advanced Study.
That spring, in a course of lectures on exten-
sions of his incompleteness results, he defined
the notion of general recursive function, based on
an idea suggested to him by Jacques Herbrand.
Gödel’s was one of several definitions later shown
to characterize the same class of functions.
On 19 April 1935, in a talk before a meeting of the
American Mathematical Society, Church pro-
pounded his Thesis:
The class so characterized consists of exactly
those functions informally thought of as being
On 19 June 1935 Gödel made his last pre-
sentation to Karl Menger’s mathematical
colloquium. Titled “On the length of proofs”,
it appeared the following year as a two-page
article in the colloquium proceedings. It is
now regarded as the first announcement of
a “speed-up” theorem.
Gödel did not pursue the ideas in his length-
of-proof paper, which seems to have been
an afterthought to his incompleteness
Much later, in a letter to his terminally ill
friend von Neumann, Gödel made another
observation he did not follow up: The first
known formulation of the P = NP problem.
In his Gibbs Lecture, Gödel attempted
to draw implications from the incom-
pleteness theorems concerning three
problems in the philosophy of mind:
With regard to the first two questions, Gödel
argued that “Either … the human mind (even
within the realm of pure mathematics)
infinitely surpasses the powers of any finite
machine, or else there exist absolutely
unsolvable diophantine problems”. He
believed the first alternative was more likely.
Unlike commentators such as John Lucas
and Roger Penrose, Gödel did not assert
that the incompleteness theorems defini-
tively refute mechanism, nor (as Judson
Webb has asserted) that “the Church-Turing
thesis … is the principal bastion protecting
Gödel claimed that the existence of abso-
lutely unsolvable problems would seem
“to disprove the view that mathematics is …
our own creation; for [a] creator necessarily
knows all properties of his creatures”. He
admitted that “we build machines and still
cannot predict their behavior in every detail”.
But that objection, he said, is “very poor”:
nothing, but build them out of some
It seems he recognized that hardware
may behave unpredictably, but not
“The view that intelligence in machinery is merely a reflection of that of its creator is … similar to the view that … credit for the discoveries of a pupil should be given to his teacher. In such a case the teacher [sh]ould be pleased with the success of his methods of education, but [sh]ould not claim the results themselves unless he ha[s] actually communicated them to his pupil” ---Alan Turing, “Intelligent machinery”
that, unlike Turing or von Neumann, he was
never involved with the design or operation
of actual computers — even though it was at
the IAS that von Neumann built one of the
Consider also the statement that computer
pioneer Howard Aiken made five years later:
“If it should turn out that the basic logics of a machine designed for the numerical solution of differential equations coincide with the logics of a machine intended to make bills for a department store, I would re-gard this as the most amazing coin-cidence I have ever encountered.”
Despite his high regard for Turing’s work,
late in his life Gödel claimed that Turing had
erred in arguing that mental procedures
cannot go beyond mechanical procedures.
Specifically, he charged that Turing had
completely disregarded “that mind, in its
use, is not static but constantly developing.”
Contrary to Gödel’s charge, in his 1950
paper “Computing machinery and intelli-
gence”, Turing explicitly discussed learning
machines. There, too, he countered various
objections to the idea that machines could
simulate human thought, including the
“mathematical objection” based on the
Gödel and Turing never met, nor did they
exchange any correspondence of sub-
stance. Gödel delivered his Gibbs Lecture
one year after the appearance of Turing’s
paper quoted above, and his criticism of
Turing came long after Turing’s death.
One must therefore wonder:
Emil Post was an important contributor to
recursion theory, and helped to initiate the
field of automata theory. Twenty years
before Gödel, he also conjectured (and
nearly proved) the incompleteness of formal
number theory. His outlook, however, was
very different from Gödel’s.
Shortly afterward he wrote to Gödel to
expound further on his “near miss”. He
explained that he had set out to demonstrate
the existence of absolutely unsolvable prob-
lems, and thought he “saw a way of analyz-
ing ‘all finite processes of the human mind’ ”
that would establish incompleteness “in
general, … not just for Principia Mathema-
the development of computer science than
Gödel did. But as he himself conceded in
that same letter, “Nothing that I had done
could have replaced the splendid actuality of
your proof.” Sidetracked by his quest to
exhibit an absolutely unsolvable problem,
Post failed to give an example of one that
was undecidable in a particular formalism.
If Gödel were alive today, what would he think of the role of computers in mathe-matics?
Would he approve of their use in providing numerical evidence for conjectures?
What would he think of computer-assisted proofs?
Would he maintain his belief in the mind’s superiority?
Kurt Gödel, Collected Works (5 vols.), ed. Solomon Feferman et al. Oxford University Press, 1986–2003.
The Essential Turing, ed. B. Jack Copeland, Oxford University Press, 2004.
Solvability, Provability, Definability: The Collected Works of Emil L. Post, ed. Martin Davis. Birkhäuser, 1994.