## Gödel on Intuition and on Hilbert’s finitism

### BibTeX

@MISC{Tait_gödelon,

author = {W. W. Tait},

title = {Gödel on Intuition and on Hilbert’s finitism},

year = {}

}

### OpenURL

### Abstract

There are some puzzles about Gödel’s published and unpublished remarks concerning finitism that have led some commentators to believe that his conception of it was unstable, that he oscillated back and forth between different accounts of it. I want to discuss these puzzles and argue that, on the contrary, Gödel’s writings represent a smooth evolution, with just one rather small double-reversal, of his view of finitism. He used the term “finit ” (in German) or “finitary ” or “finitistic ” primarily to refer to Hilbert’s conception of finitary mathematics. On two occasions (only, as far as I know), the lecture notes for his lecture at Zilsel’s [Gödel, 1938a] and the lecture notes for a lecture at Yale [Gödel, *1941], he used it in a way that he knew—in the second case, explicitly—went beyond what Hilbert meant. Early in his career, he believed that finitism (in Hilbert’s sense) is openended, in the sense that no correct formal system can be known to formalize all finitist proofs and, in particular, all possible finitist proofs of consistency of first-order number theory, P A; but starting in the Dialectica paper

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Citation Context ... evidence rests on what is intuitive” (anschaulichen) [Gödel, 1958, p. 281], he was expressing what he had always taken to be Hilbert’s conception. As far as I know, Hilbert’s “ Über das Unendliche” [=-=Hilbert, 1926-=-] is the only source in Hilbert’s writings on finitism or proof theory that he ever cited, and surely his characterization of finitary mathematics accurately reflects what he would have read there. Bu... |

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Citation Context ... the system A presently.) Literally, Gödel is mistaken about this: As Richard Zach has noted [Zach, 2003], Hilbert had approved as finitist Ackermann’s use of induction up to ωωω in his dissertation [=-=Ackermann, 1924-=-]. In fact, though, the only instance of transfinite induction for which Ackermann gave any justification in his paper was induction up to ω2 and his argument was essentially just the reduction of thi... |

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Citation Context ...stem 3 These lecture notes, together with those for his lecture in 1933 at a joint meeting of the AMA and the AMS, play a central role in my discussion. In neither case does the introductory note in [=-=Gödel, 1995-=-] indicate any other source of information about the actual content of the lecture. Given the nature of lecture notes in general and in particular— despite the heroic efforts of the editors to clarify... |

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Citation Context ...order arithmetic. Certainly if he were identifying what is finitistically provable with to the conclusion that even primitive recursion, in particular, exponentiation, cannot be founded on intuition [=-=Parsons, 1998-=-, p. 265]. He believes that not even the elaboration of the finitist point of view in [Bernays, 1930–31] avoids this conclusion (p. 263); but that is not clear to me. If one takes as the basis of fini... |

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Citation Context ...stem 3 These lecture notes, together with those for his lecture in 1933 at a joint meeting of the AMA and the AMS, play a central role in my discussion. In neither case does the introductory note in [=-=Gödel, 1995-=-] indicate any other source of information about the actual content of the lecture. Given the nature of lecture notes in general and in particular— despite the heroic efforts of the editors to clarify... |

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Citation Context ...stem 3 These lecture notes, together with those for his lecture in 1933 at a joint meeting of the AMA and the AMS, play a central role in my discussion. In neither case does the introductory note in [=-=Gödel, 1995-=-] indicate any other source of information about the actual content of the lecture. Given the nature of lecture notes in general and in particular— despite the heroic efforts of the editors to clarify... |

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Citation Context |

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Citation Context ...ithmetic is in rough agreement with the intuitionism of Poincaré and Weyl. For the former the principle of iteration is given in intuition and is the one synthetic a priori truth of mathematics (see [=-=Poincaré, 1900-=-]) and for the latter, at least in his intuitionistic phase, it is the basis of all arithmetic, the one principle that need not and indeed cannot be proved. See [Weyl, 1921], the end of Part II §1,“Th... |

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Citation Context ...for pointing this out to me so that I avoided error (at least about this). As to the precise relationship between Gödel and Turing on computability and the issues involved, the reader should consult [=-=Sieg, 2006-=-]. 20of recursion up to α.) But I don’t see why an arbitrary finitary consistency proof for P A should translate into Gentzen’s. Nevertheless, the argument that recursion on ordinals < ɛ0 is not fini... |

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Citation Context ...nal sense, it is a fundament of his philosophy 5 It is true that intuition of may be suggested by “in this kind of perception, i.e., in mathematical intuition” in [1964, p. 271]. But, as I argued in [=-=Tait, 1986-=-, fn. 3], the context makes it clear that it is propositional knowledge—namely axioms—that the intuition is to be yielding. 9to distinguish sensibility, the faculty of intuition, from understanding, ... |

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Citation Context ...Mathematik, Volume 2. But this argument is simply a nicer proof of (essentially) the result just mentioned from [Tait, 1961] and, again, does not yield a finitary proof of induction up to < ɛ0. (See [=-=Tait, 2006-=-, pp. 90-91] for a discussion of this.) But none of these changes concern Gödel’s conception of finitism as the mathematics whose evidence rests on concrete intuition. They only concern the possibilit... |

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Citation Context ...e University of Pittsburgh.) It of course also led to a more structural conception of mathematics itself, in which computation and construction yielded to logic and existence axioms and proofs. (See [=-=Stein, 1988-=-].) 10construct the three lines joining them and thereby, assuming that they are non-collinear, construct the triangle ABC. Kant recognized that we cannot be speaking of empirical construction here, ... |

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(Show Context)
Citation Context .... Gödel himself in [1972, p. 274, fn.4 and fn.f] explicitly recognizes that this goes beyond finitary reasoning in Hilbert’s sense. A second proof Gödel cites [2003a, letter # 68b (7/25/69)] is mine [=-=Tait, 1961-=-], which proves in this connection only that recursion on ω α is reducible to nested recursion on ω × α. The third [Gödel, 2003a, letter #68b (7/25/69)] is Bernays’ argument for induction up to < ɛ0 i... |

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