## Partial realizations of Hilbert’s program (1988)

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Venue: | JOURNAL OF SYMBOLIC LOGIC |

Citations: | 40 - 8 self |

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@ARTICLE{Simpson88partialrealizations,

author = {Stephen G. Simpson},

title = {Partial realizations of Hilbert’s program},

journal = {JOURNAL OF SYMBOLIC LOGIC},

year = {1988},

volume = {53},

number = {2},

pages = {349}

}

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### Citations

207 |
Subsystems of second order arithmetic
- Simpson
- 1998
(Show Context)
Citation Context ...h is provable in WKL0 7sis already provable in PRA and hence is witnessed by a primitive recursive Skolem function. Friedman’s proof of this result is model-theoretic and will be published by Simpson =-=[24]-=-. Subsequently Sieg [20] used a Gentzen-style method to give an alternative proof of Friedman’s result. Actually Sieg exhibited a primitive recursive proof transformation. Thus the reducibility of WKL... |

154 |
Über eine bisher noch nicht benützte Erweiterung des finiten Standpunkts
- Gödel
- 1958
(Show Context)
Citation Context ...ification over the domain of all subsets of the natural numbers. At this APA-ASL symposium, Feferman referred to predicative reductionism as a “relativized” form of Hilbert’s Program. Similarly Gödel =-=[10]-=- has proposed an “extension” of the finitistic standpoint, by way of primitive recursive functionals of higher type. Also Bernays [1], p. 502, has discussed a program of intuitionistic reductionism wh... |

133 |
On formally undecidable propositions of Principia Mathematica and related systems I. Translated by B. Meltzer. <Web page: http://home.ddc.net/ygg/etext/godel/index.htm
- Gödel
- 1931
(Show Context)
Citation Context ... of affairs by saying that Z2 is sentences. This would constitute conservative over PRA with respect to Π 0 1 a precise and definitive realization of Hilbert’s Program. Unfortunately, Gödel’s Theorem =-=[9]-=- shows that any such realization of step 2.3 is impossible. There are plenty of Π0 1 sentences which are provable in 5sZ2 but not in PRA. (An example of such a sentence is the one which asserts the co... |

119 | The unreasonable effectiveness of mathematics in the natural sciences
- Wigner
- 1960
(Show Context)
Citation Context ...nalysis, the only way to demonstrate that mathematics is valid is to show that it refers to reality. And make no mistake about it-the validity of mathematics is under siege. In a widely cited article =-=[28]-=-, Wigner declares that there is no rational explanation for the usefulness of mathematics in the physical sciences. He goes on to assert that all but the most elementary parts of mathematics are nothi... |

64 |
Mathematics: The Loss of Certainty
- Kline
- 1980
(Show Context)
Citation Context ...ysical sciences. He goes on to assert that all but the most elementary parts of mathematics are nothing but a miraculous formal game. Kline, in his influential book Mathematics: The Loss of Certainty =-=[17]-=-, deploys a wide assortment of mathematical arguments and historical references to show that “there is no truth in mathematics.” Kline’s book was published by the Oxford University Press and reviewed ... |

47 |
Systems of predicative analysis
- Feferman
- 1964
(Show Context)
Citation Context ...ly one of many possible reductionist schemes. In the aftermath of Gödel’s Theorem, several authors have proposed reductionist programs which are quite different from Hilbert’s. For instance, Feferman =-=[5]-=- has developed an elaborate program of predicative reductionism. (See also Simpson [22], pp. 152–154.) Certainly Feferman’s predicative standpoint is very far away from finitism. It accepts full class... |

41 |
On Number Choice Schema and its Relation to Induction
- Parsons
- 1970
(Show Context)
Citation Context ...bited a primitive recursive proof transformation. Thus the reducibility of WKL0 to PRA is itself provable in PRA. (These conclusions due to Sieg [20] could also have been derived from work of Parsons =-=[19]-=- and Harrington [12].) The above results of Friedman and Sieg may be summarized as follows. Any mathematical theorem which can be proved in WKL0 is finitistically reducible in the sense of Hilbert’s P... |

35 | Heijenoort (editor), From Frege to Gödel: A sourcebook in mathematical logic, 1879–1931 - van - 1967 |

28 |
Systems of second order arithmetic with restricted induction
- Friedman
- 1976
(Show Context)
Citation Context ...ecent investigations have revealed that the answer to the above question is: quite a large part. The purpose of this section is to explain these recent discoveries. I shall now do so. First, Friedman =-=[6]-=- has defined a certain interesting subsystem of Z2 known as WKL0. The language of WKL0 isthesameasthatofZ2. The logic of WKL0 is full classical logic including the unrestricted law of the excluded mid... |

27 |
Countable algebra and set existence axioms
- Friedman, Simpson, et al.
- 1983
(Show Context)
Citation Context ...4. The Hahn–Banach Theorem and Alaoglu’s Theorem for separable Banach spaces (Brown–Simpson [3], Brown [2]). 4.5. The existence of prime ideals in countable commutative rings (Friedman– Simpson–Smith =-=[8]-=-). 4.6. Existence and uniqueness of the algebraic closure of a countable field (Friedman–Simpson–Smith [8]). 8s4.7. Existence and uniqueness of the real closure of a countable formally real field (Fri... |

20 |
private communication
- Friedman, Stepnosky, et al.
(Show Context)
Citation Context ...nce, compactness of the closed unit interval [0, 1] within WKL0. Second, it has been shown that WKL0 is conservative over PRA with respect to Π0 1 sentences. This result is originally due to Friedman =-=[7]-=- who in fact obtained a stronger result: WKL0 is conservative over PRA with respect to Π0 2 sentences. This means that any Π02 sentence which is provable in WKL0 7sis already provable in PRA and hence... |

17 |
Which Set Existence Axioms are Needed to Prove Cauchy/Peano Theorem for Ordinary Differential Equations?’, The
- Simpson
- 1984
(Show Context)
Citation Context ... is mathematically rather strong. For example, the following mathematical theorems are provable in WKL0. 4.1. The Heine–Borel covering theorem for closed bounded subsets of Euclidean n-space (Simpson =-=[21, 24]-=-) or for closed subsets of a totally bounded complete separable metric space (Brown–Simpson [3], Brown [2]). 4.2. Basic properties of continuous functions of several real variables. For instance, any ... |

14 |
Which set existence axioms are needed to prove the separable Hahn-Banach theorem
- Brown, Simpson
- 1986
(Show Context)
Citation Context ...KL0. 4.1. The Heine–Borel covering theorem for closed bounded subsets of Euclidean n-space (Simpson [21, 24]) or for closed subsets of a totally bounded complete separable metric space (Brown–Simpson =-=[3]-=-, Brown [2]). 4.2. Basic properties of continuous functions of several real variables. For instance, any continuous real-valued function on a closed bounded rectangle in R n is uniformly continuous an... |

12 |
Functional analysis in weak subsystems of second order arithmetic
- Brown
- 1987
(Show Context)
Citation Context ...he Heine–Borel covering theorem for closed bounded subsets of Euclidean n-space (Simpson [21, 24]) or for closed subsets of a totally bounded complete separable metric space (Brown–Simpson [3], Brown =-=[2]-=-). 4.2. Basic properties of continuous functions of several real variables. For instance, any continuous real-valued function on a closed bounded rectangle in R n is uniformly continuous and Riemann i... |

11 |
The unreasonable effectiveness of mathematics
- Wigner
- 1960
(Show Context)
Citation Context ...lysis, the only way to demonstrate that mathematics is valid is to show that it refers to reality. And make no mistake about it — the validity of mathematics is under siege. In a widely cited article =-=[28]-=-, Wigner declares that there is no rational explanation for the usefulness of mathematics in the physical sciences. He goes on to assert that all but the most elementary parts of mathematics are nothi... |

9 | Uber eine bisher noch nicht ben"utzte Erweiterung des finiten Standpunktes - Godel - 1958 |

8 |
1985b, ‘Friedman’s Research on Subsystems of Second Order Arithmetic
- Simpson
(Show Context)
Citation Context ...eral authors have proposed reductionist programs which are quite different from Hilbert’s. For instance, Feferman [5] has developed an elaborate program of predicative reductionism. (See also Simpson =-=[22]-=-, pp. 152–154.) Certainly Feferman’s predicative standpoint is very far away from finitism. It accepts full classical logic and allows the set of all natural numbers as a completed infinite totality. ... |

8 |
Subsystems of Z2 and Reverse Mathematics, appendix to
- Simpson
- 1986
(Show Context)
Citation Context ...yields precise answers to special cases of this question. A fairly detailed survey of Reverse Mathematics will be found in my appendix to the forthcoming second edition of Takeuti’s proof theory book =-=[23]-=-. Here I must confine myself to a very brief summary. Most of the work on Reverse Mathematics has been carried out in the context of subsystems of Z2. There are a great many different subsystems of Z2... |

8 | Grundlagen der Mathematik, vols - Hilbert, Bernays |

7 |
Hilbert’s Epistemology
- Kitcher
- 1976
(Show Context)
Citation Context ...for the big system. It would then follow that any Π 0 1 sentence provable in the big system is finitistically true. (For an explanation of the role of Π0 1 sentences in Hilbert’s Program, see Kitcher =-=[16]-=- and Tait [25].) Thus the big system as a whole would be finitistically justified. The infinite objects of the big system would find meaning as valid auxiliary devices used to prove theorems about phy... |

7 |
Fragments of arithmetic, Annals of Pure and
- Sieg
- 1985
(Show Context)
Citation Context ...is already provable in PRA and hence is witnessed by a primitive recursive Skolem function. Friedman’s proof of this result is model-theoretic and will be published by Simpson [24]. Subsequently Sieg =-=[20]-=- used a Gentzen-style method to give an alternative proof of Friedman’s result. Actually Sieg exhibited a primitive recursive proof transformation. Thus the reducibility of WKL0 to PRA is itself prova... |

5 |
David," in: Encyclopedia of Philosophy
- Bernays, Hilbert
- 1967
(Show Context)
Citation Context ... is compact, i.e. enjoys the Heine-Borel covering property for sequences of basic open sets. Friedman pointed out that compactness of 2N implies, for instance, compactness of the closed unit interval =-=[0, 1]-=- within WKLo. Second, it has been shown that WKLo is conservative over PRA with respect to Ho sentences. This result is originally due to Friedman [7] who is fact obtained a stronger result: WKLo is c... |

4 | Subsystems of Second Order Arithmetic, in preparation [25 - Simpson - 1981 |

2 | Heijenoort (editor), From Frege to G"odel: A Source Book - van - 1967 |

1 |
in: Encyclopedia of Philosophy, vol.3, edited by P
- Bernays, “Hilbert
- 1967
(Show Context)
Citation Context ...ism as a “relativized” form of Hilbert’s Program. Similarly Gödel [10] has proposed an “extension” of the finitistic standpoint, by way of primitive recursive functionals of higher type. Also Bernays =-=[1]-=-, p. 502, has discussed a program of intuitionistic reductionism which he regards as a “broadening” or “enlarging” of proof theory. In his introductory remarks to this symposium, Sieg interpreted Bern... |

1 |
Review of [17
- Corcoran
(Show Context)
Citation Context ... show that “there is no truth in mathematics.” Kline’s book was published by the Oxford University Press and reviewed favorably in the New York Times. (For a much more insightful review, see Corcoran =-=[4]-=-.) Neither Wigner nor Kline is viewed as an enemy of mathematics. But with friends like these, who needs enemies? Arguments like those of Kline and Wigner turn up with alarming frequency in coffee-roo... |

1 |
What is Cantor's Continuum Problem?, in: Philosophy of Mathematics: Selected Readings, 2nd edition, edited by
- Godel
- 1983
(Show Context)
Citation Context ... cumulative hierarchy bear any resemblance to external reality. The rest are a huge extrapolation based on a crude model of abstract thought processes. Gödel himself comes close to admitting as much (=-=[11]-=-, pp. 483–484). Arguing in favor of the cumulative hierarchy, Gödel ([11], pp. 477 and 485) proposes a validation in terms of testable number-theoretic consequences. Unfortunately such tests seem hard... |

1 |
personal communication to
- Harrington
- 1977
(Show Context)
Citation Context ...cursive proof transformation. Thus the reducibility of WKL0 to PRA is itself provable in PRA. (These conclusions due to Sieg [20] could also have been derived from work of Parsons [19] and Harrington =-=[12]-=-.) The above results of Friedman and Sieg may be summarized as follows. Any mathematical theorem which can be proved in WKL0 is finitistically reducible in the sense of Hilbert’s Program. In particula... |

1 |
On the infinite, translated by S
- Hilbert
(Show Context)
Citation Context ...e point in the fortress of mathematics was the infinite. In order to defend the foundations of mathematics, it 2swas above all necessary to clarify and justify the mathematician’s use of the infinite =-=[13]-=-. Actually Hilbert saw the issue as having supramathematical significance. Mathematics is not only the most logical and rigorous of the sciences but also the most spectacular example of the power of “... |

1 |
The foundations of mathematics, translated by S
- Hilbert
(Show Context)
Citation Context ...out the manipulation of finite strings of symbols. 2.2. The second step is to reconstitute infinitistic mathematics as a big, elaborate formal system. This big system (more fully described in Hilbert =-=[14]-=-) contains unrestricted classical logic, infinite sets galore, and special variables ranging over natural numbers, functions from natural numbers to natural numbers, countable ordinals, etc. The formu... |

1 |
Grundlagen der Mathematik, vols.IandII, 2nd edition
- Hilbert, Bernays
- 1968
(Show Context)
Citation Context ...responding to infinitistic mathematics, is already sufficiently precise. For my purposes here I shall identify the big system as Z2, i.e. second order arithmetic. Supplement IV of Hilbert and Bernays =-=[15]-=- shows that Z2 is more than adequate for the formal development of classical analysis, etc. It would not matter if we replaced Z2 by Z3, Z4, orevenZFC. The unacceptable imprecision occurs in Hilbert’s... |

1 |
Aristotelian infinity
- Lear
- 1980
(Show Context)
Citation Context ...and potential infinity. According to Aristotle, there is no actual infinity, but potential infinity exists and first manifests itself to us in the continuous, via infinite divisibility. See also Lear =-=[18]-=-.) Hilbert accepts the picture of the world which is presented by contemporary physics. The atomic theory tells us that matter is not infinitely divisible. The quantum theory tells us that energy is l... |

1 |
Recent topics on proof theory (in Japanese
- Takeuti
- 1984
(Show Context)
Citation Context ...rogram. It certainly is not. I only assert the existence of a certain mutually reinforcing relationship between these two lines of research. I hope that I have adequately addressed Takeuti’s concerns =-=[26]-=- about the connection between Hilbert’s Program and Reverse Mathematics. 6 Answers to Some Possible Objections In this section I shall rebut some possible objections which might be raised against the ... |

1 |
Review of [17], Mathematical Reviews 82c:03013. This content downloaded from 146.186.134.137
- CORCORAN
(Show Context)
Citation Context ... show that "there is no truth in mathematics." Kline's book was published by the Oxford University Press and reviewed favorably in the New York Times. (For a much more insightful review, see Corcoran =-=[4]-=-.) Neither Wigner nor Kline is viewed as an enemy of mathematics. But with friends like these, who needs enemies? Arguments like those of Kline and Wigner turn up with alarming frequency in coffee-roo... |

1 |
eine bisher noch nicht bentitzte Erweiterung des finiten Standpunktes
- Uber
- 1958
(Show Context)
Citation Context ...ert's program. This content downloaded from 146.186.134.137 on Thu, 11 Jul 2013 09:54:22 AM All use subject to JSTOR Terms and ConditionsPARTIAL REALIZATIONS OF HILBERT'S PROGRAM 353 Similarly Gddel =-=[10]-=- has proposed an "extension" of the finitistic standpoint, by way of primitive recursive functionals of higher type. Also Bernays [1, p. 502] has discussed a program of intuitionistic reductionism whi... |

1 | is Cantor's continuum problem?, Philosophy of mathematics: selected readings - What - 1983 |

1 | research on subsystems of second order arithmetic, Harvey Friedman's research in the foundations of mathematics - Friedman's - 1985 |

1 |
of Z2 and reverse mathematics, appendix to
- Subsystems
- 1987
(Show Context)
Citation Context ...yields precise answers to special cases of this question. A fairly detailed survey of reverse mathematics will be found in my appendix to the forthcoming second edition of Takeuti's proof theory book =-=[23]-=-. Here I must confine myself to a very brief summary. Most of the work on reverse mathematics has been carried out in the context of subsystems of Z2. There are a great many different subsystems of Z2... |

1 | HEIJENOORT (editor), From Frege to Gidel: a source book in mathematical logic - VAN - 1967 |

1 | Review of [17], Math. Reviews 1982c, #03013. [5] S. Feferman, Systems of predicative analysis - Corcoran - 1964 |