## Groundwork for weak analysis (2002)

Venue: | the Journal of Symbolic Logic |

Citations: | 10 - 4 self |

### BibTeX

@ARTICLE{Fernandes02groundworkfor,

author = {António M. Fernandes and Fernando Ferreira},

title = {Groundwork for weak analysis},

journal = {the Journal of Symbolic Logic},

year = {2002},

volume = {67},

pages = {557--578}

}

### OpenURL

### Abstract

Abstract. This paper develops the very basic notions of analysis in a weak secondorder theory of arithmetic BTFA whose provably total functions are the polynomial time computable functions. We formalize within BTFA the real number system and the notion of a continuous real function of a real variable. The theory BTFA is able to prove the intermediate value theorem, wherefore it follows that the system of real numbers is a real closed ordered field. In the last section of the paper, we show how to interpret the theory BTFA in Robinson’s theory of arithmetic Q. This fact entails that the elementary theory of the real closed ordered fields is interpretable in Q. §1. Introduction. The formalization of mathematics within second-order arithmetic has a long and distinguished history. We may say that it goes back to Richard Dedekind, and that it has been pursued by, among others, Hermann Weyl, David Hilbert, Paul Bernays, Harvey Friedman, and Stephen Simpson and his students (we may also mention the insights of Georg Kreisel, Solomon Feferman,

### Citations

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Citation Context ...It is well known that the four basic processes of arithmetic, namely: addition, subtraction, multiplication, and division have polynomial time computable algorithms (see, for instance, section 4.3 of =-=[15]-=-). Moreover, these algorithms provably do what they are supposed to do in Σb 1−NIA. However, to formally define the algorithms and to formally check that they work is a tedious and unsavory task. The ... |

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Citation Context ...otal functions (with appropriate graphs) are the polynomial time computable functions. Plainly, the work of Harvey Friedman and Ker-I Ko on the complexity of computations on the reals – as exposed in =-=[16]-=- – is bound to be of great importance for the pursuit of weak analysis, comparable to the importance of recursive analysis as a well-spring of ideas and constructions for the development of analysis o... |

68 | Bounded Arithmetic
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Citation Context ... with parameters. The theory Σb 1−NIA is interpretable in Buss’ well-known theory S1 2 , and vice-versa. With one such smooth interpretation in mind, it is possible to appeal to Buss’ main theorem of =-=[3]-=- and infer the following property: whenever Σb 1−NIA proves the sentence ∀x∃yφ(x, y), where φ(x, y)aΣb 1-formula, there is a polynomial time computable function f :2 <ω ↦→ 2 <ω such that, for all σ ∈ ... |

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Citation Context ...x<y∨ x = y ∨ y<x Ax. 11 x<y→∃z(x + z = y) Ax. 12 x<y→ x + z<y+ z Ax. 13 0 <z∧ x<y→ x · z<y· z Ax. 14 0 < 1 Ax. 15 ∀x(0 <x→ x =1∨ 1 <x) Ax. 16 ∀x(x =0∨ 0 <x) We refer the reader to Richard Kaye’s book =-=[14]-=- for the discussion of the fundamental consequences these axioms. Of course, the structure (N2,0,1,+, ·,<) satisfies much more than the above, since it inherits the Σb 1-induction on notation scheme f... |

31 | Mathematically strong subsystems of analysis with low rate of growth of provably recursive functionals
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Citation Context ...c Logic 0022-4812/00/0000-0000/$00.00s2 ANTÓNIO M. FERNANDES AND FERNANDO FERREIRA single-handed work in “proof mining” where concerned with theories related to the various Grzegorczyk classes (e.g., =-=[17]-=-, [18] and [19]). This work is an essay on a subject that we may call weak analysis: the formalization and study of mathematics in weak (i.e., mainly sub-exponential) subsystems of second-order arithm... |

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Citation Context ...s on algebraic questions within a second-order theory related to the elementary functions (see [25], [12] and [24]), Ferreira’s work on second-order theories related to polynomial time computability (=-=[5]-=- and [8]), and subsequent papers by Andrea Cantini [4] and Takeshi Yamazaki [29], [28]. In a different setting, viz. finite type arithmetic, we should also mention Ulrich Kohlenbach’s 2000 Mathematics... |

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(Show Context)
Citation Context ...related to the elementary functions (see [25], [12] and [24]), Ferreira’s work on second-order theories related to polynomial time computability ([5] and [8]), and subsequent papers by Andrea Cantini =-=[4]-=- and Takeshi Yamazaki [29], [28]. In a different setting, viz. finite type arithmetic, we should also mention Ulrich Kohlenbach’s 2000 Mathematics Subject Classification. Primary: 03F35; Secondary: 03... |

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Citation Context ...ils that RCOF is interpretable in Q (Harvey Friedman has also claimed this latter result – albeit without giving a proof – in a well-known web discussion forum for the foundations of mathematics: see =-=[10]-=-). Prima facie, this is a somewhat surprising result. After all, RCOF is a theory whose intended model is the continuum of real numbers, strong enough for the development of all analytic geometry, whe... |

1 |
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Citation Context ...rty: whenever Σb 1−NIA proves the sentence ∀x∃yφ(x, y), where φ(x, y)aΣb 1-formula, there is a polynomial time computable function f :2 <ω ↦→ 2 <ω such that, for all σ ∈ 2 <ω , φ(σ, f(σ)) (see [6] or =-=[2]-=- for a direct proof of this result). This is the precise sense of saying that the provably total functions of Σb 1−NIA are the polynomial time computable functions. It is possible to extend by definit... |

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Citation Context ... bounded arithmetical theory I∆0 in a natural way: zero is given by ɛ, successor ′ by concatenation with 1, addition + by concatenation, multiplication by ×, and the less than or equal relation by ⊆ (=-=[11]-=- is a good reference for first-order theories of arithmetic, including bounded theories). In general, one has the following form of induction, called tally induction: φ(ɛ) ∧∀x(x is tally ∧ φ(x) → φ(x ... |

1 |
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Citation Context ...mplexity. The exceptions that we found in the literature are some papers of Simpson and his students on algebraic questions within a second-order theory related to the elementary functions (see [25], =-=[12]-=- and [24]), Ferreira’s work on second-order theories related to polynomial time computability ([5] and [8]), and subsequent papers by Andrea Cantini [4] and Takeshi Yamazaki [29], [28]. In a different... |