## PREDICATIVITY BEYOND Γ0 (2005)

### BibTeX

@MISC{Weaver05predicativitybeyond,

author = {Nik Weaver},

title = {PREDICATIVITY BEYOND Γ0},

year = {2005}

}

### OpenURL

### Abstract

Abstract. We reevaluate the claim that predicative reasoning (given the natural numbers) is limited by the Feferman-Schütte ordinal Γ0. First we comprehensively criticize the arguments that have been offered in support of this position. Then we analyze predicativism from first principles and develop a general method for accessing ordinals which is predicatively valid according to this analysis. We find that the Veblen ordinal φΩω(0), and larger ordinals, are predicatively provable. The precise delineation of the extent of predicative reasoning is possibly one of the most remarkable modern results in the foundations of mathematics. Building on

### Citations

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Citation Context ..., T, F). (b) Too weak. The Ref ∗ construction is described in [14] as a “closure” operation and the question of its significance is discussed in terms of Kripke’s theory of grounded truth outlined in =-=[34]-=-. A casual reading of §6 of [14] might leave the impression that the statements A such that Ref ∗ (PA(P)) proves T(�A�) are supposed to be precisely the grounded true statements of the language L(P, T... |

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Citation Context ...etic, say ACA0 (see [44]), then he should accept not only the theorems of the system itself, but also additional statements such as Con(ACA0) which reflect the fact that the axioms are true. Feferman =-=[9]-=- analyzed several such “reflection principles” and found the strongest of them to be the formalized ω-rule schema (∀n)[Prov(�A(n)�) → A(n)] , where �A(n)� is the Gödel number of A(n) and Prov formaliz... |

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Citation Context ...roof-theoretically important countable ordinals, this result is still significant. Probably the most celebrated example of an allegedly impredicative mainstream theorem, Kruskal’s theorem (see, e.g., =-=[21]-=-), is now seen to be predicatively justified. It is equivalent over a weak base system to the well-ordering of a notation for φΩω(0) [38]. It should not be difficult to strengthen Theorem 2.10 so as t... |

33 |
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Citation Context ... possibly the primary reason, for predicativism’s nearly universal unpopularity. 1 There do exist important mainstream theorems (most famously, the Cantor-Bendixson theorem [27] and Kruskal’s theorem =-=[21]-=-) which are known to in various senses require provability of Γ0, and in any case Γ0 is sufficiently tame that it is simply hard to take seriously any approach to foundations that prevents one from re... |

23 |
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Citation Context ...rtually every discussion of predicativism for the past forty years. Γ0 is now commonly referred to as “the ordinal of predicativity”. Some recent publications which explicitly make this assertion are =-=[1, 2, 3, 4, 6, 18, 19, 20, 23, 25, 26, 37, 38, 40, 45, 46]-=-. This achievement is notable both for its technical sophistication and for the insight it provides into an important foundational stance. Although predicativism is out of favor now, at one time it wa... |

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Citation Context ...tified. The former is provable in ATR0, whose consistency needs only the ordinals less than Γ0 [21], and the latter is equivalent over a weak base system to the well-ordering of a notation for φΩω(0) =-=[39]-=-. It should not be difficult to strengthen Theorem 2.10 so as to prove the wellfoundedness of a notation for the “large” Veblen ordinal φΩΩ(0). But I expect that substantially larger ordinals can be a... |

16 | Intuitionistic choice and classical logic
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Citation Context ...or any particular n ∈ ω and X ⊆ ω the atomic formula “n ∈ X” has a definite truth value. Thus, at the level of arithmetical statements our logic is classical. Similar considerations were discussed in =-=[8]-=-, leading to the suggestion that predicativists can adopt the numerical omniscience schema (∀n)(A(n) ∨ ¬A(n)) → [(∀n) A(n) ∨ (∃n) ¬A(n)] (where here A is any formula of second order arithmetic and n i... |

12 |
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Citation Context ...bably now a primary reason, possibly the primary reason, for predicativism’s nearly universal unpopularity. 1 There do exist important mainstream theorems (most famously, the Cantor-Bendixson theorem =-=[27]-=- and Kruskal’s theorem [21]) which are known to in various senses require provability of Γ0, and in any case Γ0 is sufficiently tame that it is simply hard to take seriously any approach to foundation... |

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Citation Context ...eation of the extent of predicative reasoning is possibly one of the most remarkable modern results in the foundations of mathematics. Building on ideas of Kreisel [28, 29], Feferman [10] and Schütte =-=[41, 42]-=- independently identified a countable ordinal Γ0 and argued that it is the smallest predicatively non-provable ordinal. (Throughout, I take “predicative” to mean “predicative given the natural numbers... |

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Citation Context ...rtually every discussion of predicativism for the past forty years. Γ0 is now commonly referred to as “the ordinal of predicativity”. Some recent publications which explicitly make this assertion are =-=[1, 2, 3, 4, 6, 18, 19, 20, 23, 25, 26, 37, 38, 40, 45, 46]-=-. This achievement is notable both for its technical sophistication and for the insight it provides into an important foundational stance. Although predicativism is out of favor now, at one time it wa... |

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Citation Context ...l” Veblen ordinal φΩω(0). The development is similar to that in §2.8 and will be presented here in slightly less detail. Let κ = φΩΩ(0) and fix a notation system for κκ (e.g., see the introduction to =-=[35]-=-). In the following I will identify ordinals with their notations and I will use α, β, γ to range over ordinals ≺ κ and a, b, c to range over ordinals ≺ κκ . Every nonzero a can be uniquely written in... |

8 |
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Citation Context ...catively provable. The precise delineation of the extent of predicative reasoning is possibly one of the most remarkable modern results in the foundations of mathematics. Building on ideas of Kreisel =-=[27, 28]-=-, Feferman [10] and Schütte [40, 41] independently identified a countable ordinal Γ0 and argued that it is the smallest predicatively non-provable ordinal. (Throughout, I take “predicative” to mean “p... |

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(Show Context)
Citation Context ...rtually every discussion of predicativism for the past forty years. Γ0 is now commonly referred to as “the ordinal of predicativity”. Some recent publications which explicitly make this assertion are =-=[1, 2, 3, 4, 6, 18, 19, 20, 23, 25, 26, 37, 38, 40, 45, 46]-=-. This achievement is notable both for its technical sophistication and for the insight it provides into an important foundational stance. Although predicativism is out of favor now, at one time it wa... |

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4 |
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4 | Internal finite tree embeddings - Friedman - 1998 |

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4 |
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Citation Context ...rtually every discussion of predicativism for the past forty years. Γ0 is now commonly referred to as “the ordinal of predicativity”. Some recent publications which explicitly make this assertion are =-=[1, 2, 3, 4, 6, 18, 19, 20, 23, 25, 26, 36, 37, 39, 44, 45]-=-. This achievement is notable both for its technical sophistication and for the insight it provides into an important foundational stance. Although predicativism is out of favor now, at one time it wa... |

3 |
of proof and ordinals implicit in given concepts
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Citation Context ...the proof that γn+1 is well-founded is supposed to legitimate passage to Nγn+1. This takes us to Kreisel’s final argument. (c) Kreisel’s third answer. Kreisel’s most sophisticated analysis appears in =-=[33]-=-. Here he rightly addresses the central question of exactly how a predicativist would infer soundness of Sa once I(a) has been proven. On my reading, the novel idea is that this inference (or somethin... |

3 |
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Citation Context ... important foundational stance. Although predicativism is out of favor now, at one time it was advocated by such luminaries as Poincaré, Russell, and Weyl. (Historical overviews are given in [18] and =-=[36]-=-.) Its central principle — that sets have to be “built up from below” — is, on its face, reasonable and attractive. With its rejection of a metaphysical set concept, predicativism also provides a coge... |

3 |
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Citation Context ...eation of the extent of predicative reasoning is possibly one of the most remarkable modern results in the foundations of mathematics. Building on ideas of Kreisel [28, 29], Feferman [10] and Schütte =-=[41, 42]-=- independently identified a countable ordinal Γ0 and argued that it is the smallest predicatively non-provable ordinal. (Throughout, I take “predicative” to mean “predicative given the natural numbers... |

2 |
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Citation Context ...able from those axioms. (b) Kreisel’s second answer. A second argument in response to something like the objection raised above was made by Kreisel ([32], §3.631) and cited with approval by Feferman (=-=[11]-=-, p. 134). Unfortunately, the cited passage is rather inscrutable, so it is hard to be sure what Kreisel had in mind. I think it is something like this. Predicativists are at any given moment only abl... |

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2 |
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Citation Context ...y n, an is an ordinal notation implies an+1 is an ordinal notation. It therefore becomes hard to believe that someone who is presumed to grasp induction on ω (and even, allegedly, in “schematic” form =-=[14, 16, 19]-=-) would not be able to infer the single assertion that an is an ordinal notation for all n. It is reasonable to expect that if a predicativist understands how to go from an to an+1 for any single valu... |

2 |
on “Predicativity as a philosophical position” by G.R
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Citation Context ...language that contains symbols for all primitive recursive functions and reword the arguments — here and below — to ensure that all recursive functions in use are actually primitive recursive.) 5. In =-=[17]-=- Feferman refers to “the argument that the characterization of predicativity requires one to go beyond predicative notions and principles” ([17], footnote 6), which sounds like it could be a version o... |

2 |
An extension of Schütte’s Klammersymbols
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Citation Context ... ≺ αn}; and if βn = 1 and αn = ˜αn + 1 then it is {h(a) + κ˜αn γ : γ ≺ κ}. In the last case (βn = 1 and αn a successor) we say that a is of type 1, and otherwise it is of type 0. (Cf. Definition 1 of =-=[22]-=-.) We consider 0 to be of type 0 and we let its canonical sequence be empty. Let Typ0 be a formulaPREDICATIVITY BEYOND Γ0 31 such that Typ 0(a) holds if and only if a is of type 0, and let Seq be a f... |

2 |
Predicativism as a philosophical position, Review Internationale de Philosophie 229
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2 |
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Citation Context ...edly and forcefully defended by two major figures, Feferman and Kreisel. Many current authors simply assert it as a known fact. The only substantial published criticism of which I am aware appears in =-=[24]-=-, but even that is somewhat ambivalent and seems to conclude in favor of the thesis. Therefore, I take it that I have a burden not only to positively demonstrate the power of predicative reasoning, bu... |

2 |
logics and the characterization of informal concepts of proof
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(Show Context)
Citation Context ...catively provable. The precise delineation of the extent of predicative reasoning is possibly one of the most remarkable modern results in the foundations of mathematics. Building on ideas of Kreisel =-=[28, 29]-=-, Feferman [10] and Schütte [41, 42] independently identified a countable ordinal Γ0 and argued that it is the smallest predicatively non-provable ordinal. (Throughout, I take “predicative” to mean “p... |

2 |
of intuitionistic logic
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(Show Context)
Citation Context ...The difficulties involved with schematic predicates shed light on the predicative unacceptability of some formal systems which superficially have a strong predicative flavor. For example, in [29] and =-=[31]-=- the possibility is raised that under intuitionistic logic theories of generalized inductive definitions might be predicatively valid, and this idea does have superficial appeal. However, on close exa... |

2 |
Transfinite dependent choice and ω-model reflection
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2 |
the outer limits
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2 |
Mathematical conceptualism, manuscript
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Citation Context ...ualism” for the brand of predicativism considered here and make a case that it is cogent, rigorous, attractive, and better suited to ordinary mathematical practice than all other foundational stances =-=[47]-=-. 1. A critique of the Γ0 thesis At issue is the assertion that there are well-ordered sets of all order types less than Γ0 and of no order types greater than or equal to Γ0 which can be proven to be ... |

1 |
On the relationship between ATR0 and fID<ω
- Avigad
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(Show Context)
Citation Context ...rtually every discussion of predicativism for the past forty years. Γ0 is now commonly referred to as “the ordinal of predicativity”. Some recent publications which explicitly make this assertion are =-=[1, 2, 3, 4, 6, 18, 19, 20, 23, 25, 26, 36, 37, 39, 44, 45]-=-. This achievement is notable both for its technical sophistication and for the insight it provides into an important foundational stance. Although predicativism is out of favor now, at one time it wa... |

1 |
on “Predicativity as a philosophical position” by G.R. Hellman, Revue Internationale de Philosophie 229
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(Show Context)
Citation Context ...language that contains symbols for all primitive recursive functions and reword the arguments — here and below — to ensure that all recursive functions in use are actually primitive recursive.) 5. In =-=[17]-=- Feferman refers to “the argument that the characterization of predicativity requires one to go beyond predicative notions and principles” ([17], footnote 6), which sounds like it could be a version o... |

1 |
Predicativism as a philosophical position, Revue Internationale de Philosophie 229
- Hellman
- 2004
(Show Context)
Citation Context ...rtually every discussion of predicativism for the past forty years. Γ0 is now commonly referred to as “the ordinal of predicativity”. Some recent publications which explicitly make this assertion are =-=[1, 2, 3, 4, 6, 18, 19, 20, 23, 25, 26, 36, 37, 39, 44, 45]-=-. This achievement is notable both for its technical sophistication and for the insight it provides into an important foundational stance. Although predicativism is out of favor now, at one time it wa... |