## A most artistic package of a jumble of ideas

Citations: | 2 - 2 self |

### BibTeX

@MISC{Ferreira_amost,

author = {Fernando Ferreira},

title = {A most artistic package of a jumble of ideas},

year = {}

}

### OpenURL

### Abstract

In the course of ten short sections, we comment on Gödel’s seminal “Dialectica ” paper of fifty years ago and its aftermath. We start by suggesting that Gödel’s use of functionals of finite type is yet another instance of the realistic attitude of Gödel towards mathematics and in tune with his defense of the postulation of ever increasing higher types in foundational studies. We also make some observations concerning Gödel’s recasting of intuitionistic arithmetic via the “Dialectica ” interpretation, discuss the extra principles that the interpretation validates, and comment on extensionality and higher order equality. The latter sections focus on the role of majorizability considerations within the “Dialectica ” and related interpretations for extracting computational information from ordinary proofs in mathematics. I Kurt Gödel’s realism, a stance “against the current ” of his time, is now wellknown

### Citations

216 |
Mathematical Logic
- Shoenfield
- 1967
(Show Context)
Citation Context ...ation it is also possible to deal with systems of classical arithmetic and analysis. Alternatively, one can proceed directly, as Joseph Shoenfield does for Peano arithmetic in his well-known textbook =-=[Sho67]-=-. In the classical setting, the only characteristic principle for the Dialectica is the quantifier-free axiom of choice (in all types). 32 A similar situation happens with the bounded functional inter... |

211 |
Intensional interpretations of functionals of finite type I
- Tait
- 1967
(Show Context)
Citation Context ...tification of the truth-functional interpretation of T lies just in this fact. But how can we decide whether t = q, even for t and q closed terms of the type 0 (the type of N)? William Tait showed in =-=[Tai67]-=- that these terms reduce mechanically to numerals (the terms normalize to numerals). Numerals are the same, or not, and this decides the equation. Nevertheless, as aptly observed by Georg Kreisel in p... |

143 |
Über eine bisher noch nicht benützte Erweiterung des finiten Standpunktes
- Gödel
- 1958
(Show Context)
Citation Context ...is the royal road to the study of the universe of sets. It is significant that in a lecture of 1941 (posthumously published in [Göd41]), and that both in the Dialectica paper and its 1972 descendant (=-=[Göd58]-=- and [Göd72], respectively), Gödel suggests extending the notion of effective rule to (constructive) transfinite types. Effective rules in higher types give rise to new effective rules which compute a... |

76 |
Interpretation of analysis by means of constructive functionals of finite types
- Kreisel
- 1958
(Show Context)
Citation Context ... of Howard that the majorizable functionals form a model of Gödel’s T. Another model of T, defined independently by Stephen Kleene (in [Kle59] under the name of countable functionals) and Kreisel (in =-=[Kre59]-=-), is the model ECF ω of the extensional continuous functionals. 18 A compactness argument shows that the type 2 continuous functional are majorizable. This is no longer the case for type 3 functional... |

72 |
Provably recursive functionals of analysis: a consistency proof of analysis by an extension of principles in current intuitionistic mathematics
- Spector
- 1962
(Show Context)
Citation Context ... bounds can still be obtained. This is made possible by Clifford Spector’s 1962 deep generalization of Gödel’s interpretation to second-order classical arithmetic using bar-recursive functionals (see =-=[Spe62]-=-). NB the reason for the existence of uniform bounds rests, ultimately, on the fact that Spector’s bar-recursive functionals are majorizable. One must emphasize again that the important issue here is ... |

68 | Gödel’s functional (‘Dialectica’) interpretation
- Avigad, Feferman
- 1998
(Show Context)
Citation Context ...972 (unpublished in Gödel’s lifetime) for interesting discussions and, of course, the proper technical definitions and pointers to the relevant material. An alternative (and more inclusive) option is =-=[AF98]-=-. 3This feature is usually described by saying that T is impredicative. It is better described as a lack of purity of methods within the type-hierarchy of computable functionals. 4The intuitionistic m... |

54 |
Applied Proof Theory: Proof Interpretations and their Use
- Kohlenbach
- 2008
(Show Context)
Citation Context ...the compact spaces. Many striking uniform bounds in the field of numerical functional analysis have been obtained in the recent past by Kohlenbach and his co-workers using these ideas (see [KO03] and =-=[Koh08]-=-). Of course, the a priori theoretical knowledge that such bounds must exist is instrumental in the applied work. The recent introduction of new base types by Kohlenbach (normed, metric and other spac... |

44 | Some logical metatheorems with applications in functional analysis
- Kohlenbach
(Show Context)
Citation Context ...l concerns (only now are we beginning to understand that the use of extensionality in mathematical proofs may pose problems for the extraction of numerical bounds; see the discussions in section 3 of =-=[Koh05]-=-). It is not an option for Gödel’s par5ticular interpretation (extended to HA ω ) to have extensional equality. Already the above instance of extensionality is not interpretable. It is worthwhile to ... |

41 |
Hereditarily majorizable functionals of finite type
- Howard
- 1973
(Show Context)
Citation Context ...n (extended to HA ω ) to have extensional equality. Already the above instance of extensionality is not interpretable. It is worthwhile to see why this is so. The argument is due to William Howard in =-=[How73]-=-. We introduce a variant of Howard’s majorizability relations due to Mark Bezem in [Bez85] (strong majorizability). These are defined inductively on the types according to the following clauses: (a) x... |

36 |
Strongly majorizable functionals of finite type: a model for bar recursion containing discontinuous functionals
- Bezem
- 1985
(Show Context)
Citation Context ...nality is not interpretable. It is worthwhile to see why this is so. The argument is due to William Howard in [How73]. We introduce a variant of Howard’s majorizability relations due to Mark Bezem in =-=[Bez85]-=- (strong majorizability). These are defined inductively on the types according to the following clauses: (a) x ≤ ∗ 0 y := x ≤ y (b) x ≤ ∗ ρ→σ y := ∀u ρ , v ρ (u ≤ ∗ ρ v → xu ≤ ∗ σ yv ∧ yu ≤ ∗ σ yv) A ... |

33 | Mathematically strong subsystems of analysis with low rate of growth of provably recursive functionals. archive
- Kohlenbach
- 1996
(Show Context)
Citation Context ...ies of this principle with L. E. J. Brouwer’s fan theorem (for c of type 1 and y of type 0). 29 The collection scheme subsumes the so-called uniform boundedness principles introduced by Kohlenbach in =-=[Koh96]-=-. 30,31 The other principle is the intensional bounded contra-collection scheme: ∀b∃x ✂ c∀y ✂ b Abd(x, y) → ∃x ✂ c∀y Abd(x, y), where b, c are of any type and Abd is a bounded formula. Of course, this... |

31 | Effective moduli from ineffective uniqueness proofs. An unwinding of de La Vallée Poussin’s proof for Chebycheff approximation
- Kohlenbach
- 1993
(Show Context)
Citation Context ...s article will concentrate on the role of majorizability within the Dialectica and related interpretations for extracting computational information from ordinary proofs in mathematics. V In 1993 (see =-=[Koh93]-=-), Ulrich Kohlenbach introduced a slight modification of Gödel’s functional interpretation, dubbed the monotone functional interpretation. It is based on the same assignment of formulas but it weakens... |

26 |
On the interpretation of non-finitist proofs, part I
- Kreisel
- 1951
(Show Context)
Citation Context ... new conclusion only demands that t is a majorant of a witness, i.e., that HA ω ⊢ ∃x ≤∗ t ∀yAD(x, y). What is accomplished by weakening the conclusion of Gödel’s theorem? In his first published paper =-=[Kre51]-=-, Kreisel stressed that (for sensible theories) the addition of true (e.g., all true) Π0 1-sentences does not affect the class of provably recursive functions. In the context of the Dialectica interpr... |

17 |
On an extension of finitary mathematics which has not yet been used
- Gödel
(Show Context)
Citation Context ... road to the study of the universe of sets. It is significant that in a lecture of 1941 (posthumously published in [Göd41]), and that both in the Dialectica paper and its 1972 descendant ([Göd58] and =-=[Göd72]-=-, respectively), Gödel suggests extending the notion of effective rule to (constructive) transfinite types. Effective rules in higher types give rise to new effective rules which compute additional fu... |

15 | Proof mining: a systematic way of analysing proofs in mathematics
- Kohlenbach, Oliva
- 2003
(Show Context)
Citation Context ... values in the compact spaces. Many striking uniform bounds in the field of numerical functional analysis have been obtained in the recent past by Kohlenbach and his co-workers using these ideas (see =-=[KO03]-=- and [Koh08]). Of course, the a priori theoretical knowledge that such bounds must exist is instrumental in the applied work. The recent introduction of new base types by Kohlenbach (normed, metric an... |

13 |
2005), Bounded functional interpretation
- Ferreira, Oliva
(Show Context)
Citation Context ...ional interpretation only scratches the surface. A new functional interpretation – dubbed bounded functional interpretation – injects uniformities all the way in, via a new assigment of formulas (see =-=[FO05]-=-). Many of the basic theoretic results of Kohlenbach have very perspicuous proofs using the new interpretation. The nature of the injected uniformities can be best described via the characteristic pri... |

11 | Kreisel’s ‘Unwinding Program
- Feferman
- 1996
(Show Context)
Citation Context ...ation to branches of mathematics other than arithmetic in p. 18 of his 1941 lecture. 25The name is due to Dana Scott. 26Solomon Feferman gives a similar assessment of Kreisel’s “unwinding” program in =-=[Fef96]-=-. 27On p. 26 of his 1941 lecture, Gödel says that with the help of the Dialectica interpretation it can be shown that LPO is independent from intuitionistic logic. This can be seen, for instance, by i... |

7 | Intuitionistic Choice and Restricted Classical Logic
- Kohlenbach
- 2000
(Show Context)
Citation Context ...re of 1941 when he wrote the wrong clause for disjunction. (On finding errors in Gödel, see the last paragraph of [Kre87].) 21This analysis of LLPO is related to, but not quite the same as the one in =-=[Koh01]-=-. 22The principle LLPO can be put in this form, since it is equivalent to (⋆) within the theory HAω + MPω + ACω + IPω ∀ . Sometimes, the difficult part is to show that a given principle can be put in ... |

7 |
Intuitionistic analysis and Gödel’s interpretation
- Yasugi
- 1963
(Show Context)
Citation Context ...n proof needs the full apparatus of HA in order to show that all concrete closed terms normalize. 10The extended soundness theorem is implicit in [Kre59], and the next theorem is due to Mariko Yasugi =-=[Yas63]-=-. 11 A curious, if misguided proposal, would be to propose a notion of intensional equality for type 0 as well. In this case, the problem of deciding equations between closed terms of type 0 becomes s... |

3 |
what sense is intuitionistic logic constructive
- In
- 1941
(Show Context)
Citation Context ...ion of ever increasing types in set theory – via large cardinal axioms – is the royal road to the study of the universe of sets. It is significant that in a lecture of 1941 (posthumously published in =-=[Göd41]-=-), and that both in the Dialectica paper and its 1972 descendant ([Göd58] and [Göd72], respectively), Gödel suggests extending the notion of effective rule to (constructive) transfinite types. Effecti... |

3 |
Gödel’s Excursions Into Intuitionistic Logic
- Kreisel
- 1987
(Show Context)
Citation Context ...se terms reduce mechanically to numerals (the terms normalize to numerals). Numerals are the same, or not, and this decides the equation. Nevertheless, as aptly observed by Georg Kreisel in p. 112 of =-=[Kre87]-=-, the normalization of terms is not apparent by the inspection of the notion of computable functional of finite type. 8,9 There is an intolerable gap between the obviousness of the acceptance of Gödel... |

2 | Injecting uniformities into Peano arithmetic
- Ferreira
- 2009
(Show Context)
Citation Context ...of choice (in all types). 32 A similar situation happens with the bounded functional interpretation. An interpretation that directly injects uniformities into Peano arithmetic was recently defined in =-=[Fer07]-=-. Three characteristic principles (which embody the injected uniformities) are necessary in this case. One is an unsound form of choice for bounded matrices which, nevertheless, includes the sound AC ... |

1 |
Gödel’s interpretation of intuitionism. Philosophia Mathematica
- Tait
- 1973
(Show Context)
Citation Context ...er statement assures that an unbounded search procedure will eventually halt with a natural number n such that ¬Aqf(n). One cannot but agree with Tait in his assessment given in the last paragraph of =-=[Tai06]-=-: “I would rather view Markov’s principle as an example of why, if one is looking for methods of proof which automatically yield algorithms for computing a witness for existential theorems, intuitioni... |

1 |
Introductory note to [Göd58] and [Göd72
- Troelstra
- 1990
(Show Context)
Citation Context ...d03a]). No precise proposal was advanced. A good candidate is to say that two terms are definitionally equal if they have the same normal form (thus obtaining the term model of T, described in 4.3 of =-=[Tro90]-=-). Another interpretation is considered by Gödel in note g of his 1972 paper: to identify functionals as arguments or values of higher-type functionals with the code numbers of their Turing machines. ... |

1 |
Introductory note to [Göd41
- Troelstra
- 1995
(Show Context)
Citation Context ...elp of the Dialectica interpretation it can be shown that LPO is independent from intuitionistic logic. This can be seen, for instance, by interpreting T by the hereditarily recursive operations (see =-=[Tro95]-=- for details). The new bounded functional interpretation yields this independence result directly because it refutes LPO. Other independence results can be obtained in this manner. 28The soundness the... |