The Incompleteness Theorems

Smoryñski, Craig

doi:10.1016/s0049-237x(08)71123-6

Cited by 245 publications

(87 citation statements)

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“…But more recently a number of leading foundational thinkers, e.g., Kreisel (1976), Smorynski (1977Smorynski ( , 1985Smorynski ( , 1988, Prawitz (1981) and Simpson (1988), have claimed that even Gödel's first theorem is enough to refute Hilbert's program. Michael Detlefsen, on the other hand, has argued vigorously against this interpretation (Detlefsen 1990).…”

Section: Hilbert's Program and Gödel's First Theoremmentioning

confidence: 99%

“…Namely, unlike in Gödel's first theorem, where it is enough that one has a predicate that extensionally represents provability, one must now require that the particular way one formalizes provability satisfies certain intensional extra conditions (see e.g., Smorynski 1977), most necessarily the so-called second derivability condition…”

Section: Gödel's Second Theorem and The Consistency Programmentioning

confidence: 99%

“…There is a widely accepted, rather sophisticated logical explication of Hilbert's program that is essentially due to Kreisel (1958Kreisel ( , 1968Kreisel ( , 1976) and endorsed by various distinguished logicians, e.g., by Feferman (1988Feferman ( , 1994, Sieg (1984Sieg ( , 1988Sieg ( , 1990aSieg ( , 1999, Prawitz (1981), Smorynski (1977Smorynski ( ,1985Smorynski ( , 1988, and Simpson (1988), as well as by philosophers such as e.g., Kitcher (1976) and Giaquinto (1983). This interpretation relates Hilbert's requirement of consistency proof closely to the soundness and conservativity properties.…”

Section: Reflection Principles -Logicians' Interpretations Of Hilbertmentioning

confidence: 99%

“…Indeed, in the early 1920s, it was understood very narrowly indeed (but see note 11). 6 For the derivability conditions, see e.g., Smorynski (1977). 7 This was emphasized by Michael Detlefsen during a discussion.…”

Section: Notesmentioning

confidence: 99%

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Hilbert's Program Revisited

2003

Synthese

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ABSTRACT. After sketching the main lines of Hilbert's program, certain well-known and influential interpretations of the program are critically evaluated, and an alternative interpretation is presented. Finally, some recent developments in logic related to Hilbert's program are reviewed.In its heyday in the 1920s, Hilbert's program was arguably the most sophisticated and progressive research program in the foundations of mathematics. However, after Gödel's celebrated incompleteness results it became an almost universally held opinion that Hilbert's program was dead and buried, and consequently interest in it diminished and the received picture of it became somewhat caricatured and unfair. But more recently, there has been lots of new serious interest in Hilbert's program. Consequently, there now exists some illuminating historical work on Hilbert's thought. Moreover, there are also new systematic interpretations of Hilbert's program, which argue -in various ways -that there is a sound core in the program which was not affected by Gödel's results.2 My aim in this paper is to critically evaluate these recent influential interpretations in the light of both historical (textual) and systematic logical facts and to hopefully settle some of the controversies. I shall first give a brief and relatively uncontroversial description of Hilbert's program, and then proceed to more detailed and controversial issues of interpretation.On the one hand, the roots of Hilbert's program go back to the foundational debates in the late 19th century, especially to Kroenecker's attack on Cantorian set theory and the abstract analysis just developed. This debate affected Hilbert permanently. On the other hand, Hilbert's own thought went through various important changes, and it would be an error to simply equate Hilbert's views in, say, 1900, and his mature program, which was formulated only in the early 1920s.Hilbert spoke about a consistency proof for arithmetic, or analysis, already in his famous 1900 talk on the open problems in mathematics (Hilbert 1900). This may give the wrong impression that Hilbert's program was already there. However, in 1900 Hilbert thought that this consistency Synthese 137: 157-177, 2003.

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Section: Hilbert's Program and Gödel's First Theoremmentioning

confidence: 99%

Section: Gödel's Second Theorem and The Consistency Programmentioning

confidence: 99%

Section: Reflection Principles -Logicians' Interpretations Of Hilbertmentioning

confidence: 99%

Section: Notesmentioning

confidence: 99%

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Hilbert's Program Revisited

2003

Synthese

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“…Later, relationships between reflection principles and some other true unprovable statements were established, including a well-known combinatorial principle due to Paris and Harrington. A survey of the early (until 1977) results on reflection principles is contained in Smoryński [85].…”

Section: Introductionmentioning

confidence: 99%

Reflection principles and provability algebras in formal arithmetic

Beklemishev

2005

Russ. Math. Surv.

128

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Heterologicality and Incompleteness

Cieśliński

2002

MLQ - Math. Log. Quart.

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We present a semantic proof of Gödel's second incompleteness theorem, employing Grelling's antinomy of heterological expressions. For a theory T containing ZF, we define the sentence HETT which says intuitively that the predicate "heterological" is itself heterological. We show that this sentence doesn't follow from T and is equivalent (provably in T ) to the consistency of T . Finally we show how to construct a similar incompleteness proof for Peano Arithmetic.Mathematics Subject Classification: 03C55, 03C62,03E35.

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The Incompleteness Theorems

Cited by 245 publications

References 10 publications

Hilbert's Program Revisited

Hilbert's Program Revisited

Reflection principles and provability algebras in formal arithmetic

Heterologicality and Incompleteness

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