## Unfolding finitist arithmetic (2010)

Citations: | 3 - 3 self |

### BibTeX

@MISC{Feferman10unfoldingfinitist,

author = {Solomon Feferman and Thomas Strahm},

title = {Unfolding finitist arithmetic},

year = {2010}

}

### OpenURL

### Abstract

The concept of the (full) unfolding U(S) of a schematic system S is used to answer the following question: Which operations and predicates, and which principles concerning them, ought to be accepted if one has accepted S? The program to determine U(S) for various systems S of foundational significance was previously carried out for a system of non-finitist arithmetic, NFA; it was shown that U(NFA) is prooftheoretically equivalent to predicative analysis. In the present paper we work out the unfolding notions for a basic schematic system of finitist arithmetic, FA, and for an extension of that by a form BR of the so-called Bar Rule. It is shown that U(FA) and U(FA + BR) are proof-theoretically equivalent, respectively, to Primitive Recursive Arithmetic, PRA, and to Peano Arithmetic, PA.

### Citations

419 |
Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme, I.Monatshefte für Mathematik und Physik 38
- Gödel
- 1931
(Show Context)
Citation Context ...plicitly, since he accepted the proof (by Ackermann, among others) of consistency of a system corresponding to PRA necessarily making use of stronger principles. Gödel’s second incompleteness theorem =-=[7]-=- led von Neumann to the conclusion that Hilbert’s program could not succeed for PA; Gödel thought at first that it might, but within a few years he came around to the same opinion. In order for that t... |

91 |
A language and axioms for explicit mathematics
- Feferman
- 1975
(Show Context)
Citation Context .... , sn): ¯s ∈ N := N(s1) ∧ · · · ∧ N(sn), (∃x ∈ N)A := (∃x)(x ∈ N ∧ A), (∀x ∈ N)A := (∀x)(x ∈ N → A). The logic of U0(NFA) is the classical logic of partial terms LPT of Beeson [1], cf. also Feferman =-=[2]-=-. We recall that LPT embodies strictness axioms saying that all subterms of a defined compound term are defined as well. Moreover, if (s = t) holds then both s and t are defined, and s is defined prov... |

72 |
Foundations of constructive mathematics, Metamathematical Studies
- Beeson
- 1985
(Show Context)
Citation Context ... numbers (¯s = s1, . . . , sn): ¯s ∈ N := N(s1) ∧ · · · ∧ N(sn), (∃x ∈ N)A := (∃x)(x ∈ N ∧ A), (∀x ∈ N)A := (∀x)(x ∈ N → A). The logic of U0(NFA) is the classical logic of partial terms LPT of Beeson =-=[1]-=-, cf. also Feferman [2]. We recall that LPT embodies strictness axioms saying that all subterms of a defined compound term are defined as well. Moreover, if (s = t) holds then both s and t are defined... |

41 |
On Number Choice Schema and its Relation to Induction
- Parsons
- 1970
(Show Context)
Citation Context ... arithmetic with Σ1 induction in the language L1 without ¯ P , then we immediately have that Σ + 1 -IA is a conservative extension of Σ1-IA. Moreover, it is well-known (cf. e.g. Mints [17] or Parsons =-=[18]-=-) that Σ1-IA is a conservative extension of primitive recursive arithmetic PRA in the following sense: if Σ1-IA proves (∀¯x)(∃y)A[¯x, y] for A a quantifier-free formula in the language L1, then there ... |

33 |
Elements of mathematical logic
- Kreisel, Krivine
- 1976
(Show Context)
Citation Context ...rguments to be, or not be, finitistic. Gödel’s own thoughts on this will be described below. The first proposed formal characterization was made by Kreisel [14], then in a revised form in his article =-=[15]-=-, with further discussion in [16]; according to that, finitism is equivalent in strength to PA. The second proposed formal characterization was made by Tait in [23] and [24], the latter reprinted in [... |

21 |
Vortrag bei Zilsel
- Gödel
- 1938
(Show Context)
Citation Context ...sed at length in Feferman [4], where much of the evidence rests on his posthumously published notes for a 1933 lecture in Cambridge, Massachusetts [8] and a 1938 lecture to Zilsel’s seminar in Vienna =-=[9]-=-, as well as on extended correspondence with Bernays, reproduced in [11]. In both the 1933 and 1938 lectures Gödel informally describes several levels of constructivity, and equates finitist reasoning... |

16 | Gödel’s program for new axioms: Why, where, how and what
- Feferman
- 1996
(Show Context)
Citation Context ...itist arithmetic, nonfinitist arithmetic, Bar Rule, Primitive Recursive Arithmetic, Peano Arithmetic, predicative analysis. 1 Introduction This is a continuation of the program introduced in Feferman =-=[3]-=-, to determine the unfolding of the principal foundational schematic systems S, from ∗ Department of Mathematics, Stanford University, Stanford CA 94305, USA. Email: feferman@stanford.edu ∗∗ Institut ... |

10 |
Hilbert’s program. In the Stanford encyclopedia of philosophy, http://plato.stanford.edu/entries/hilbert-program
- Zach
(Show Context)
Citation Context ... informal and formal, which has its source in Hilbert’s consistency program. This must necessarily be comparatively brief and we shall just cite a few references; the online encyclopedia article Zach =-=[29]-=- provides an excellent introduction and many further key references. Hilbert viewed reasoning about the actual infinite as the source of possible inconsistencies in mathematics. He thus proposed to es... |

9 |
Quantifier-free and one-quantifier systems
- Mints
- 1973
(Show Context)
Citation Context ... usual system of arithmetic with Σ1 induction in the language L1 without ¯ P , then we immediately have that Σ + 1 -IA is a conservative extension of Σ1-IA. Moreover, it is well-known (cf. e.g. Mints =-=[17]-=- or Parsons [18]) that Σ1-IA is a conservative extension of primitive recursive arithmetic PRA in the following sense: if Σ1-IA proves (∀¯x)(∃y)A[¯x, y] for A a quantifier-free formula in the language... |

9 |
Finitism and intuitive knowledge
- Parsons
- 1998
(Show Context)
Citation Context ...m at least includes PRA. 6 It is a matter of some historical discussion whether Hilbert accepted 5 As Gödel showed by his arithmetization of syntax, the former can be reduced to the latter. 6 Parsons =-=[19]-=- has argued that the ideas of concrete intuition expressed by Hilbert do not allow one to go beyond what can be obtained by addition, multiplication and bounded quantification; if that is granted, not... |

9 |
Fragments of arithmetic. Annals of Pure and Applied Logic, 28:33–71
- Sieg
- 1985
(Show Context)
Citation Context ... , An[ ¯ B/ ¯ P ] ⇒ A[ ¯ B/ ¯ P ] is a derivable rule of Σ + 1 -IA. collecProof. The first assertion of our lemma is a simple consequence of Σ + 1 tion, which is available in Σ + 1 -IA (cf. e.g. Sieg =-=[20]-=-, p. 53). For the second assertion let us assume that A1, A2, . . . , An ⇒ A is derivable in Σ + 1 -IA and ¯ B is in Σ + 1 . Then we have a proof of A in Σ + 1 -IA plus A1, A2, . . . , An. Replacing ¯... |

8 |
Ordinal logics and the characterization of informal concepts of proof
- Kreisel
- 1960
(Show Context)
Citation Context ...t Hilbert and his circle judged particular arguments to be, or not be, finitistic. Gödel’s own thoughts on this will be described below. The first proposed formal characterization was made by Kreisel =-=[14]-=-, then in a revised form in his article [15], with further discussion in [16]; according to that, finitism is equivalent in strength to PA. The second proposed formal characterization was made by Tait... |

8 |
Principles of proof and ordinals implicit in given concepts
- Kreisel
- 1970
(Show Context)
Citation Context ...stic. Gödel’s own thoughts on this will be described below. The first proposed formal characterization was made by Kreisel [14], then in a revised form in his article [15], with further discussion in =-=[16]-=-; according to that, finitism is equivalent in strength to PA. The second proposed formal characterization was made by Tait in [23] and [24], the latter reprinted in [25]; according to that, finitism ... |

6 |
Constructive Reasoning
- Tait
- 1968
(Show Context)
Citation Context ...en in a revised form in his article [15], with further discussion in [16]; according to that, finitism is equivalent in strength to PA. The second proposed formal characterization was made by Tait in =-=[23]-=- and [24], the latter reprinted in [25]; according to that, finitism is equivalent in strength to PRA. We shall take up these formulations in reverse order. Both agree that it makes sense to character... |

6 |
The Provenance of Pure Reason
- Tait
- 2005
(Show Context)
Citation Context ...], with further discussion in [16]; according to that, finitism is equivalent in strength to PA. The second proposed formal characterization was made by Tait in [23] and [24], the latter reprinted in =-=[25]-=-; according to that, finitism is equivalent in strength to PRA. We shall take up these formulations in reverse order. Both agree that it makes sense to characterize the objects and methods of finitism... |

5 | The unfolding of non-finitist arithmetic
- Feferman, Strahm
(Show Context)
Citation Context ...he basic operations given on the universe of discourse of S together with the basic logical operations used to construct the predicates of S. The preceding article in this series, Feferman and Strahm =-=[5]-=-, provided the first example of these notions worked out in detail, namely for a schematic system of classical non-finitist arithmetic, NFA. Its basic operations on individuals with the constant 0 are... |

5 |
The present situation in the foundations of mathematics
- Gödel
- 1995
(Show Context)
Citation Context ...views on the limits of finitism. This has been discussed at length in Feferman [4], where much of the evidence rests on his posthumously published notes for a 1933 lecture in Cambridge, Massachusetts =-=[8]-=- and a 1938 lecture to Zilsel’s seminar in Vienna [9], as well as on extended correspondence with Bernays, reproduced in [11]. In both the 1933 and 1938 lectures Gödel informally describes several lev... |

5 | Hilbert’s Program Then and Now
- Zach
- 2006
(Show Context)
Citation Context ...nd bounded quantification; if that is granted, not even exponentiation would be accepted as a finitist 5as finitist certain operations and inferences going beyond PRA; the evidence according to Zach =-=[30]-=-, p. 425, is that he did, at least implicitly, since he accepted the proof (by Ackermann, among others) of consistency of a system corresponding to PRA necessarily making use of stronger principles. G... |

4 |
et al (eds
- Helal, S
- 2003
(Show Context)
Citation Context ...posthumously published notes for a 1933 lecture in Cambridge, Massachusetts [8] and a 1938 lecture to Zilsel’s seminar in Vienna [9], as well as on extended correspondence with Bernays, reproduced in =-=[11]-=-. In both the 1933 and 1938 lectures Gödel informally describes several levels of constructivity, and equates finitist reasoning with the lowest level, given by means of 7a system A that is to meet s... |

3 | Gödel’s correspondence on proof theory and constructive mathematics
- Tait
(Show Context)
Citation Context ...s presented in the appendix to this paper. From there it will be seen that the argument can be directly formalized in U0(FA + BR); some more specific comments are to be found in the appendix. 15 15In =-=[26]-=-, sec. 6 Tait gives an analysis of a proposed proof of Bernays in [13], pp. 533–535 of induction up to ε0. Tait’s critical inspection of Bernays’ argument reveals that the result indeed only shows tha... |

2 | The non-constructive µ operator, fixed point theories with ordinals, and the bar rule
- Strahm
- 1961
(Show Context)
Citation Context ...Subst) does not increase the proof-theoretic strength of all three unfolding systems for NFA, as a rather straightforward adaptation of the upper bound arguments in Feferman and Strahm [5] and Strahm =-=[21]-=- reveals. On the other hand, we need to take a form of (Subst′) as the basic substitution rule in the formulation of the unfolding systems for finitist arithmetic in the following sections. 153 The u... |

1 |
Gödel on finitism, constructivity and Hilbert’s program
- Feferman, Gödel
- 2008
(Show Context)
Citation Context ...it follows that the union of the autonomous systems Tα is proof-theoretically equivalent to PA. Let us return to Gödel’s views on the limits of finitism. This has been discussed at length in Feferman =-=[4]-=-, where much of the evidence rests on his posthumously published notes for a 1933 lecture in Cambridge, Massachusetts [8] and a 1938 lecture to Zilsel’s seminar in Vienna [9], as well as on extended c... |

1 | Unfolding of finitist arithmetic (abstract - Feferman, Strahm |

1 |
Grundlagen der Mathematik, 2nd edition
- Hilbert, Bernays
- 1970
(Show Context)
Citation Context ...he positive integers. 5 Given that idea of its subject matter, what are the allowed finitistic methods of definition and proof? Even in the great collaboration with Bernays, Grundlagen der Mathematik =-=[12, 13]-=-, there is no detailed explanation of that. Given Hilbert’s great optimism about the prospects for his program without limit it may be that he thought people would recognize any piece of reasoning use... |

1 |
Wellfoundedness of exponentiation
- Tait
- 2006
(Show Context)
Citation Context ...s ordinary recursion on ωα or, more useful in our setting, nested recursion on ωα entails NDS(f, ωα ). A very concrete and compact presentation of the argument in [22] has recently been given by Tait =-=[27]-=- in a personal communication with the first author. That is presented in the appendix to this paper. From there it will be seen that the argument can be directly formalized in U0(FA + BR); some more s... |

1 |
Hilbert’s Program, 2003. Stanford Encyclopedia of Philosophy, http://plato.stanford.edu/entries/hilbert-program
- Zach
(Show Context)
Citation Context ... informal and formal, which has its source in Hilbert’s consistency program. This must necessarily be comparatively brief and we shall just cite a few references; the online encyclopedia article Zach =-=[29]-=- provides an excellent introduction and many further key references. Hilbert viewed reasoning about the actual infinite as the source of possible inconsistencies in mathematics. He thus proposed to es... |