Shaken Foundations or Groundbreaking Realignment? A Centennial Assessment of Kurt Gödels Impact on...

38
Shaken Foundations Shaken Foundations or or Groundbreaking Groundbreaking Realignment? Realignment? A Centennial Assessment of Kurt A Centennial Assessment of Kurt Gödel’s Impact on Logic, Gödel’s Impact on Logic, Mathematics Mathematics and and Computer Science Computer Science John W. Dawson John W. Dawson

Transcript of Shaken Foundations or Groundbreaking Realignment? A Centennial Assessment of Kurt Gödels Impact on...

Page 1: Shaken Foundations or Groundbreaking Realignment? A Centennial Assessment of Kurt Gödels Impact on Logic, Mathematics and and Computer Science Computer.

Shaken FoundationsShaken Foundations or or

Groundbreaking Groundbreaking Realignment?Realignment?

A Centennial Assessment of Kurt A Centennial Assessment of Kurt

Gödel’s Impact on Logic, MathematicsGödel’s Impact on Logic, Mathematics

andand

Computer ScienceComputer Science

John W. DawsonJohn W. Dawson

Page 2: Shaken Foundations or Groundbreaking Realignment? A Centennial Assessment of Kurt Gödels Impact on Logic, Mathematics and and Computer Science Computer.

““Die Arbeit über formal unentscheidbareDie Arbeit über formal unentscheidbare

Sätze würde wie ein Erdbeben empfunden.”Sätze würde wie ein Erdbeben empfunden.”

— — Sir Karl Popper, “Der wichtigste Beitrag seit Aristoteles”Sir Karl Popper, “Der wichtigste Beitrag seit Aristoteles”

““Gödel’s theorems produced a debacle.Gödel’s theorems produced a debacle.

… … Uncertainty and doubt concerning the Uncertainty and doubt concerning the

future of mathematics have replaced the future of mathematics have replaced the

certainties and complacencies of the past …certainties and complacencies of the past …

[so that] scientists must … be concerned [so that] scientists must … be concerned

about what mathematics can be confidently about what mathematics can be confidently

employed … .”employed … .”— — Morris Kline, Morris Kline, Mathematics: The Loss of CertaintyMathematics: The Loss of Certainty

Page 3: Shaken Foundations or Groundbreaking Realignment? A Centennial Assessment of Kurt Gödels Impact on Logic, Mathematics and and Computer Science Computer.

If we take the foregoing assess-If we take the foregoing assess-ments at face value, we might ments at face value, we might expect the mathematical land-expect the mathematical land-scape in the aftermath of Gödel’s scape in the aftermath of Gödel’s incompleteness theorems to have incompleteness theorems to have resembled that of San Francisco resembled that of San Francisco ten days before his birth:ten days before his birth:

Page 4: Shaken Foundations or Groundbreaking Realignment? A Centennial Assessment of Kurt Gödels Impact on Logic, Mathematics and and Computer Science Computer.
Page 5: Shaken Foundations or Groundbreaking Realignment? A Centennial Assessment of Kurt Gödels Impact on Logic, Mathematics and and Computer Science Computer.

But others, including Gödel himself,But others, including Gödel himself,

have taken a much more optimistichave taken a much more optimistic

view of those theorems:view of those theorems:

Page 6: Shaken Foundations or Groundbreaking Realignment? A Centennial Assessment of Kurt Gödels Impact on Logic, Mathematics and and Computer Science Computer.

“… “… what Gödel’s theorem actually tells us … can what Gödel’s theorem actually tells us … can

be viewed in a much more positive light, namely be viewed in a much more positive light, namely that the insights that are available to human that the insights that are available to human mathematicians … lie beyond anything that can mathematicians … lie beyond anything that can be formalized as a set of rules.”be formalized as a set of rules.”

— — Sir Roger Penrose, Sir Roger Penrose, Shadows of the MindShadows of the Mind

“ “I have come to cherish incompleteness for theI have come to cherish incompleteness for the support it lends to mechanism …, and to Turing’ssupport it lends to mechanism …, and to Turing’s thesis in particular.”thesis in particular.”

— — Judson Webb, Judson Webb, Mechanism, Mentalism, and Meta- Mechanism, Mentalism, and Meta- mathematicsmathematics

Page 7: Shaken Foundations or Groundbreaking Realignment? A Centennial Assessment of Kurt Gödels Impact on Logic, Mathematics and and Computer Science Computer.

So what impact So what impact has has Gödel’s Gödel’s work hadwork had

1.1. On mathematical logic?On mathematical logic?

2.2. On mathematics in general?On mathematics in general?

3.3. On computer science?On computer science?

Page 8: Shaken Foundations or Groundbreaking Realignment? A Centennial Assessment of Kurt Gödels Impact on Logic, Mathematics and and Computer Science Computer.

Gödel’s impact on logicGödel’s impact on logic

Simon Kochen has recalled that during his Simon Kochen has recalled that during his

oral examination for the doctorate, Steveoral examination for the doctorate, Steve

Kleene asked him: Kleene asked him:

““Name five theorems of Gödel.” Name five theorems of Gödel.”

The point of the question was to stress how The point of the question was to stress how

seminal Gödel’s work has been for theseminal Gödel’s work has been for the

development of logic. For example:development of logic. For example:

Page 9: Shaken Foundations or Groundbreaking Realignment? A Centennial Assessment of Kurt Gödels Impact on Logic, Mathematics and and Computer Science Computer.

1.1. In his doctoral dissertation, Gödel stated In his doctoral dissertation, Gödel stated and proved the and proved the Completeness Theorem Completeness Theorem for first-order logic (“Every consistent for first-order logic (“Every consistent first-order theory has a realization”); and first-order theory has a realization”); and in its published revision he also proved in its published revision he also proved the (countable) the (countable) Compactness Theorem Compactness Theorem

— — two cornerstones of two cornerstones of Model TheoryModel Theory..

Page 10: Shaken Foundations or Groundbreaking Realignment? A Centennial Assessment of Kurt Gödels Impact on Logic, Mathematics and and Computer Science Computer.

2.2. Gödel’s Gödel’s Incompleteness TheoremsIncompleteness Theorems restricted the scope of Hilbert’s restricted the scope of Hilbert’s Proof Proof TheoryTheory, and introduced notions (, and introduced notions (primitive primitive recursionrecursion) and techniques of proof ) and techniques of proof ((arithmetization of syntaxarithmetization of syntax) that became ) that became central to central to Recursion TheoryRecursion Theory..

Page 11: Shaken Foundations or Groundbreaking Realignment? A Centennial Assessment of Kurt Gödels Impact on Logic, Mathematics and and Computer Science Computer.

3.3. By showing that the Axiom of Choice and By showing that the Axiom of Choice and Cantor’s Continuum Hypothesis could be Cantor’s Continuum Hypothesis could be assumed without contradiction, Gödel’s assumed without contradiction, Gödel’s consistency results consistency results inin Set TheorySet Theory muted muted controversy over the former and (toge-controversy over the former and (toge-ther with Cohen’s work) led to a search ther with Cohen’s work) led to a search for axioms to decide the latter. Once for axioms to decide the latter. Once again, Gödel’s proofs introduced a new again, Gödel’s proofs introduced a new concept (that of concept (that of inner modelinner model).).

Page 12: Shaken Foundations or Groundbreaking Realignment? A Centennial Assessment of Kurt Gödels Impact on Logic, Mathematics and and Computer Science Computer.

4.4. Gödel’s Gödel’s Dialectica interpretationDialectica interpretation, based , based on the notion of on the notion of computable function of computable function of finite typefinite type, provided a new consistency , provided a new consistency proof for arithmetic and opened new proof for arithmetic and opened new directions in directions in Intuitionistic Proof TheoryIntuitionistic Proof Theory. .

(Earlier, his ‘(Earlier, his ‘negative translationnegative translation’ of classical’ of classical arithmetic into its intuitionistic counterpart arithmetic into its intuitionistic counterpart showed that if classical arithmetic is incon-showed that if classical arithmetic is incon- sistent, so is intuitionistic arithmetic.)sistent, so is intuitionistic arithmetic.)

Page 13: Shaken Foundations or Groundbreaking Realignment? A Centennial Assessment of Kurt Gödels Impact on Logic, Mathematics and and Computer Science Computer.

More generally, More generally, before Gödel’s work: before Gödel’s work:

The The conception of quantifiersconception of quantifiers was in flux. was in flux. (Some, e.g. Zermelo, regarded them as (Some, e.g. Zermelo, regarded them as infinitary conjunctions.)infinitary conjunctions.)

There was There was distrust of semantic notionsdistrust of semantic notions, , and of proofs involving and of proofs involving transfinite con-transfinite con-structionsstructions..

The importance of The importance of the distinction between the distinction between first-order and higher-order logicfirst-order and higher-order logic was not was not appreciated.appreciated.

Page 14: Shaken Foundations or Groundbreaking Realignment? A Centennial Assessment of Kurt Gödels Impact on Logic, Mathematics and and Computer Science Computer.

Within logic, Gödel’s results:Within logic, Gödel’s results:

Helped to establish first-order logic as a Helped to establish first-order logic as a tractable domain for metalogical study.tractable domain for metalogical study.

Demonstrated the connection between Demonstrated the connection between truth and provability in first-order theories, truth and provability in first-order theories, as well as the need for distinguishing as well as the need for distinguishing between them in the context of arithmetic.between them in the context of arithmetic.

Showed the fruitfulness of transfinite Showed the fruitfulness of transfinite methods in set theory, and focused the methods in set theory, and focused the attention of set theorists on the rank attention of set theorists on the rank hierarchy.hierarchy.

Page 15: Shaken Foundations or Groundbreaking Realignment? A Centennial Assessment of Kurt Gödels Impact on Logic, Mathematics and and Computer Science Computer.

Philosophically, Gödel’s results:Philosophically, Gödel’s results:

Showed the untenability of the logistic Showed the untenability of the logistic thesis that all of mathematics is subsumed thesis that all of mathematics is subsumed within one all-embracing system of logic.within one all-embracing system of logic.

Caused a redirection of formalist efforts in Caused a redirection of formalist efforts in proof theory away from finitary methods.proof theory away from finitary methods.

Clarified what sorts of consistency proofs Clarified what sorts of consistency proofs may be meaningfully sought; andmay be meaningfully sought; and

Led him to champion mathematical Led him to champion mathematical Platonism as a foundational philosophy.Platonism as a foundational philosophy.

Page 16: Shaken Foundations or Groundbreaking Realignment? A Centennial Assessment of Kurt Gödels Impact on Logic, Mathematics and and Computer Science Computer.

Reactions to the incompleteness Reactions to the incompleteness theoremstheorems

Initially, there was widespread Initially, there was widespread conster-conster-nation nation concerning their implications, andconcerning their implications, and

Many mathematicians exhibited Many mathematicians exhibited lack of lack of understandingunderstanding of details of the proofs. of details of the proofs.

But:But: AcceptanceAcceptance followed quickly, and followed quickly, and In their wake,In their wake, controversy controversy over competing over competing

foundational philosophies foundational philosophies largely died largely died downdown, apart from Intuitionism., apart from Intuitionism.

Page 17: Shaken Foundations or Groundbreaking Realignment? A Centennial Assessment of Kurt Gödels Impact on Logic, Mathematics and and Computer Science Computer.

Outside of logicOutside of logic

Gödel’s theorems attracted little notice. Gödel’s theorems attracted little notice. Then as now, most mathematicians Then as now, most mathematicians regarded foundational issues as the regarded foundational issues as the province of logicians province of logicians — questions — questions largely largely irrelevant to their own research.irrelevant to their own research.

Gödel’s results have had hardly any effect Gödel’s results have had hardly any effect on mathematical on mathematical practicepractice. In particular,. In particular,

Page 18: Shaken Foundations or Groundbreaking Realignment? A Centennial Assessment of Kurt Gödels Impact on Logic, Mathematics and and Computer Science Computer.

The quest for The quest for natural natural examples of unde-examples of unde-cidable arithmetical statements (those cidable arithmetical statements (those arising outside a logical context) goes on. arising outside a logical context) goes on. Though progress has been made toward Though progress has been made toward that goal (by Harvey Friedman and that goal (by Harvey Friedman and others), the results remain debatable.others), the results remain debatable.

Many mathematicians, however, do seem toMany mathematicians, however, do seem to

have adopted Platonism as their “working have adopted Platonism as their “working

philosophy”.philosophy”.

Page 19: Shaken Foundations or Groundbreaking Realignment? A Centennial Assessment of Kurt Gödels Impact on Logic, Mathematics and and Computer Science Computer.

Works by Gödel of special Works by Gödel of special relevance to computer sciencerelevance to computer science

1.1. Completeness of first-order logic (1930)Completeness of first-order logic (1930)

2.2. Incompleteness of formal number theory Incompleteness of formal number theory (1931)(1931)

3.3. 1934 IAS lectures on formally undecid-1934 IAS lectures on formally undecid-able statements (first published 1965)able statements (first published 1965)

4.4. ““On the lengths of proofs” (1936)On the lengths of proofs” (1936)

Page 20: Shaken Foundations or Groundbreaking Realignment? A Centennial Assessment of Kurt Gödels Impact on Logic, Mathematics and and Computer Science Computer.

5.5. ““Some basic theorems on the founda-Some basic theorems on the founda-tions of mathematics, and their philo-tions of mathematics, and their philo-sophical implications” (Gibbs Lecture, sophical implications” (Gibbs Lecture, *1951)*1951)

6.6. Letter to von Neumann (*1956)Letter to von Neumann (*1956)

*First published posthumously*First published posthumously

in Gödel’s in Gödel’s Collected WorksCollected Works

Page 21: Shaken Foundations or Groundbreaking Realignment? A Centennial Assessment of Kurt Gödels Impact on Logic, Mathematics and and Computer Science Computer.

Significance for computer scienceSignificance for computer science

1.1. The The completeness theoremcompleteness theorem established established that the rules of inference developed that the rules of inference developed prior to Gödel’s work were adequate for prior to Gödel’s work were adequate for deriving all logical consequences of a set deriving all logical consequences of a set of axioms. A computer incorporating only of axioms. A computer incorporating only those rules could thus carry out all such those rules could thus carry out all such derivations.derivations.

Page 22: Shaken Foundations or Groundbreaking Realignment? A Centennial Assessment of Kurt Gödels Impact on Logic, Mathematics and and Computer Science Computer.

2.2. The The inincompleteness theoremcompleteness theorem, recast by , recast by Alan Turing as the Alan Turing as the Halting ProblemHalting Problem, , established limits on what computers can established limits on what computers can do. But for computer science its do. But for computer science its proofproof was of greater significance, because:was of greater significance, because:

a.a. It provided a precise definition of the It provided a precise definition of the class of (what are now called) class of (what are now called) primitive primitive recursive functionsrecursive functions..

b.b. In it, In it, object languageobject language statements were statements were distinguished from distinguished from metalinguisticmetalinguistic ones. ones.

Page 23: Shaken Foundations or Groundbreaking Realignment? A Centennial Assessment of Kurt Gödels Impact on Logic, Mathematics and and Computer Science Computer.

c.c. It introduced the idea of It introduced the idea of representing one representing one data typedata type (sequences of strings) (sequences of strings) by by anotheranother (numbers). (numbers).

In addition, as Martin Davis has remarked, In addition, as Martin Davis has remarked,

the very the very structurestructure of the incompleteness of the incompleteness

proof strongly resembles that of a computerproof strongly resembles that of a computer

program.program.

Page 24: Shaken Foundations or Groundbreaking Realignment? A Centennial Assessment of Kurt Gödels Impact on Logic, Mathematics and and Computer Science Computer.

3.3. In his 1934 lectures Gödel defined the In his 1934 lectures Gödel defined the notion of notion of general recursive functiongeneral recursive function that that Alonzo Church adopted in the statement Alonzo Church adopted in the statement of his Thesis (1935).of his Thesis (1935).

4.4. The length-of-proof paper gave what is The length-of-proof paper gave what is apparently the earliest statement of a apparently the earliest statement of a ‘‘speed-upspeed-up’ ’ theoremtheorem..

5.5. In his Gibbs lecture, Gödel addressed In his Gibbs lecture, Gödel addressed the question the question whether the power of the whether the power of the human mind exceeds that of any finite human mind exceeds that of any finite machinemachine..

Page 25: Shaken Foundations or Groundbreaking Realignment? A Centennial Assessment of Kurt Gödels Impact on Logic, Mathematics and and Computer Science Computer.

6.6. In his 1956 letter to von Neumann (then In his 1956 letter to von Neumann (then terminally ill with spinal cancer), Gödel terminally ill with spinal cancer), Gödel posed a question equivalent to what is posed a question equivalent to what is now called the now called the P = NPP = NP problem problem — the — the central question in theoretical computer central question in theoretical computer science today. science today. No earlier statement of No earlier statement of that problem has been found. But the that problem has been found. But the letter had no influence on the develop-letter had no influence on the develop-ment of computer science, since it was ment of computer science, since it was not known until the 1990s.not known until the 1990s.

Page 26: Shaken Foundations or Groundbreaking Realignment? A Centennial Assessment of Kurt Gödels Impact on Logic, Mathematics and and Computer Science Computer.

So how do the assessments of So how do the assessments of Gödel’s impact quoted at the Gödel’s impact quoted at the

outsetoutsetbear up under critical scrutiny?bear up under critical scrutiny?

Page 27: Shaken Foundations or Groundbreaking Realignment? A Centennial Assessment of Kurt Gödels Impact on Logic, Mathematics and and Computer Science Computer.

Re the quotation from Popper:Re the quotation from Popper:

If the incompleteness theorems were per-If the incompleteness theorems were per-ceived by some as an “earthquake”, itsceived by some as an “earthquake”, its‘‘shock waves’ were felt most acutely byshock waves’ were felt most acutely bythose at its epicenter: the formalists of those at its epicenter: the formalists of Hilbert’s school (towards whom Gödel Hilbert’s school (towards whom Gödel tailored his proof, in which he eschewedtailored his proof, in which he eschewedsemantic methods). Elsewhere the reverbe-semantic methods). Elsewhere the reverbe-rations attenuated rapidly.rations attenuated rapidly.

Page 28: Shaken Foundations or Groundbreaking Realignment? A Centennial Assessment of Kurt Gödels Impact on Logic, Mathematics and and Computer Science Computer.

Nota beneNota bene::

Gödel Gödel himself himself started out as a started out as a followerfollower of ofHilbert’s program, Hilbert’s program, not not as an iconoclast. Heas an iconoclast. Hediscovered his incompleteness theorems in discovered his incompleteness theorems in the course of trying to give a consistencythe course of trying to give a consistencyproof for analysis relative to arithmetic proof for analysis relative to arithmetic ——in in furtherancefurtherance of Hilbert’s program. But he of Hilbert’s program. But hesoon realized that arithmetic truth, unlike soon realized that arithmetic truth, unlike provability, is provability, is not definable not definable in arithmetic.in arithmetic.

Page 29: Shaken Foundations or Groundbreaking Realignment? A Centennial Assessment of Kurt Gödels Impact on Logic, Mathematics and and Computer Science Computer.

Gödel to Yossef Balas (p. 1 of unsent draft)

Page 30: Shaken Foundations or Groundbreaking Realignment? A Centennial Assessment of Kurt Gödels Impact on Logic, Mathematics and and Computer Science Computer.

Gödel to Balas (p. 2)( truth is not expressible in the same language)

Page 31: Shaken Foundations or Groundbreaking Realignment? A Centennial Assessment of Kurt Gödels Impact on Logic, Mathematics and and Computer Science Computer.

Thus Gödel himself was among those Thus Gödel himself was among those

most affected by his own discoveries.most affected by his own discoveries.

How did he respondHow did he respond? ?

By adopting a Platonistic view of By adopting a Platonistic view of

mathematical truth while remainingmathematical truth while remaining

faithful to Hilbert’s belief that “In mathe-faithful to Hilbert’s belief that “In mathe-

matics there is no matics there is no Ignorabimus Ignorabimus ”.”.

Page 32: Shaken Foundations or Groundbreaking Realignment? A Centennial Assessment of Kurt Gödels Impact on Logic, Mathematics and and Computer Science Computer.

Specifically, Gödel shared the Specifically, Gödel shared the view expressed by Penroseview expressed by Penrose

But unlike Penrose, he did not claim that hisBut unlike Penrose, he did not claim that his

incompleteness theorems incompleteness theorems provedproved the super- the super-

iority of human over machine intelligence. iority of human over machine intelligence.

He only asserted that they implied a disjunc-He only asserted that they implied a disjunc-

tive conclusion:tive conclusion:

Page 33: Shaken Foundations or Groundbreaking Realignment? A Centennial Assessment of Kurt Gödels Impact on Logic, Mathematics and and Computer Science Computer.

(From Gödel’s Gibbs Lecture)(From Gödel’s Gibbs Lecture)

““Either Either … the human mind infinitely surpas-… the human mind infinitely surpas-

ses the powers of any finite machine, ses the powers of any finite machine, or elseor else

there exist absolutely unknowable Diophan-there exist absolutely unknowable Diophan-

tine problems.”tine problems.”

Gödel thought the former “more likely”.Gödel thought the former “more likely”.

Page 34: Shaken Foundations or Groundbreaking Realignment? A Centennial Assessment of Kurt Gödels Impact on Logic, Mathematics and and Computer Science Computer.

In contrast to the quotationIn contrast to the quotation from Webb: from Webb:

Late in his life Gödel published a brief Late in his life Gödel published a brief

remark entitled “A philosophical error inremark entitled “A philosophical error in

Turing’s work”. In it he disputed Turing’sTuring’s work”. In it he disputed Turing’s

contention that a machine was capable of contention that a machine was capable of

carrying out any humanly computable carrying out any humanly computable

procedure. Turing, he said, had disregardedprocedure. Turing, he said, had disregarded

that “mind, in its use, is not static, but con-that “mind, in its use, is not static, but con-

stantly developing.”stantly developing.”

Page 35: Shaken Foundations or Groundbreaking Realignment? A Centennial Assessment of Kurt Gödels Impact on Logic, Mathematics and and Computer Science Computer.

As for Morris Kline’s startling assertion, weAs for Morris Kline’s startling assertion, we

can perhaps agree that we no longer harborcan perhaps agree that we no longer harbor

some of the “certainties and complacenciessome of the “certainties and complacencies

of the past”. In particular, I suspect thereof the past”. In particular, I suspect there

are few today who share Hilbert’s idealism, are few today who share Hilbert’s idealism,

or Kline’s apparent belief that there shouldor Kline’s apparent belief that there should

be a single overarching theory embracingbe a single overarching theory embracing

all of mathematics. But mathematics isall of mathematics. But mathematics is

hardly plagued by “uncertainty and doubt”hardly plagued by “uncertainty and doubt”

about its future.about its future.

Page 36: Shaken Foundations or Groundbreaking Realignment? A Centennial Assessment of Kurt Gödels Impact on Logic, Mathematics and and Computer Science Computer.

Contrary to KlineContrary to Kline Gödel’s results altered the mathematical Gödel’s results altered the mathematical

landscape, but they did landscape, but they did notnot “produce a “produce a debacle”.debacle”.

There is There is lessless controversy today over controversy today over mathematical foundations than there was mathematical foundations than there was beforebefore Gödel’s work. Gödel’s work.

There is There is nono reason to regard mathematics reason to regard mathematics as any as any less secureless secure than before, and recent than before, and recent progress in settling long-standing ques-progress in settling long-standing ques-tions gives no cause for lamentation over tions gives no cause for lamentation over its future prospects.its future prospects.

Page 37: Shaken Foundations or Groundbreaking Realignment? A Centennial Assessment of Kurt Gödels Impact on Logic, Mathematics and and Computer Science Computer.

References: BooksReferences: Books

Kurt Gödel, Kurt Gödel, Collected WorksCollected Works (5 vols.), ed. (5 vols.), ed. Solomon Feferman et al. Oxford University Solomon Feferman et al. Oxford University Press, 1986Press, 1986─2003.─2003.

Torkel Franzén, Torkel Franzén, Inexhaustibility: A Non-Inexhaustibility: A Non-exhaustive Treatment. exhaustive Treatment. Association for Association for Symbolic Logic (A K Peters), 2004Symbolic Logic (A K Peters), 2004

Torkel Franzén, Torkel Franzén, Gödel’s Theorem: An Gödel’s Theorem: An Incomplete Guide to its Use and Abuse. Incomplete Guide to its Use and Abuse.

A K Peters, 2005.A K Peters, 2005.

Page 38: Shaken Foundations or Groundbreaking Realignment? A Centennial Assessment of Kurt Gödels Impact on Logic, Mathematics and and Computer Science Computer.

References: ArticlesReferences: Articles J. W. Dawson, “The reception of Gödel’s J. W. Dawson, “The reception of Gödel’s

incompleteness theorems,” pp. 74incompleteness theorems,” pp. 74–95 in S.G. –95 in S.G. Shanker (ed.), Shanker (ed.), Gödel’s Theorem in Focus, Gödel’s Theorem in Focus, Croom Croom Helm, 1988.Helm, 1988.

P. Mancosu, “Between Vienna and Berlin: The P. Mancosu, “Between Vienna and Berlin: The immediate reception of Gödel’s incompleteness immediate reception of Gödel’s incompleteness theorems”, theorems”, History and Philosophy of LogicHistory and Philosophy of Logic 20 20 (1999)(1999), , 33–45.33–45.

I. Grattan-Guinness, I. Grattan-Guinness, The reception of Gödel's The reception of Gödel's 1931 incompletability theorems by mathemati-1931 incompletability theorems by mathemati-cians, and some logicians, up to the early 1960s. cians, and some logicians, up to the early 1960s. (Vienna conference proceedings, to appear)(Vienna conference proceedings, to appear)