## On the Arithmetical Content of Restricted Forms of Comprehension, Choice and General Uniform Boundedness (1997)

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Venue: | PURE AND APPLIED LOGIC |

Citations: | 9 - 4 self |

### BibTeX

@INPROCEEDINGS{Kohlenbach97onthe,

author = {Ulrich Kohlenbach},

title = {On the Arithmetical Content of Restricted Forms of Comprehension, Choice and General Uniform Boundedness},

booktitle = {PURE AND APPLIED LOGIC},

year = {1997},

publisher = {}

}

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### Abstract

In this paper the numerical strength of fragments of arithmetical comprehension, choice and general uniform boundedness is studied systematically. These principles are investigated relative to base systems T n in all finite types which are suited to formalize substantial parts of analysis but nevertheless have provably recursive function(al)s of low growth. We reduce the use of instances of these principles in T n -proofs of a large class of formulas to the use of instances of certain arithmetical principles thereby determining faithfully the arithmetical content of the former. This is achieved using the method of elimination of Skolem functions for monotone formulas which was introduced by the author in a previous paper. As

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Mathematical logic
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Citation Context ...imply G n A ! ` A (see [13] for a detailed discussion of this phenomenon), which is in contrast to the first-order case where the derivability of A H follows from that of A by Herbrand's theorem (see =-=[20]-=-). If however A is monotone then this rule is valid also for G n A ! (but for very different reasons): Theorem 2.7 ([13]) Let A be as in thm.2.5 and \Delta be a set of sentences 8x ffi 9ysae sx8z j G ... |

154 |
Über eine bisher noch nicht benützte Erweiterung des finiten Standpunkts
- Gödel
- 1958
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Citation Context ...n2IN G n A ! . PA ! , PA ! i are the extensions of G n A ! , G n A ! i obtained by the addition of the schema of full induction and all (impredicative) primitive recursive functionals in the sense of =-=[5]-=-. E--T ! (i) denotes the theory which results from T ! (i) when the quantifier--free rule of extensionality is replaced by the axioms of extensionality (E) 8x ae ; y ae ; zsae (x = ae y ! zx =szy) for... |

127 |
Metamathematical investigation of intuitionistic Arithmetic andAnalysis
- Troelstra
- 1973
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Citation Context ...elation `maj' from [6] which is due to [2]. For more details see [8]. Let A(a) be a formula of GnA ! (a are all free variables of A) and 9x8yAD (x; y; a) its Godel functional interpretation (see e.g. =-=[25]-=- for details on Godel's functional interpretation). We say that a tuple of closed terms t realizes the monotone functional interpretation of A(a) if 7 () 9x \Gamma t s-maj xs8a; y AD (x a; y; a) \Delt... |

73 |
Provably recursive functionals of analysis: a consistency proof of analysis by an extension of principles formulated in current intuitionistic mathematics
- Spector
- 1962
(Show Context)
Citation Context ... 8~g \Pi 0 k -UB \Gamma j n (~g): Hence the assumption of the rule to be proved yields G n A ! + \Delta+ AC-qf + 8g \Pi 0 1 -CA(g) ` F \Gamma ! 8u 1 8vstu9w fl B 0 (u; v; w): From the work of Spector =-=[23]-=- it follows that G n A ! + AC-qf +8g \Pi 0 1 -CA(g) has (via negative translation) a Godel functional interpretation in G n A ! i + (BR 0;1 ) by terms 2 G n R ! [B 0;1 ]. In [2] it is shown that the t... |

56 | Analysing proofs in analysis, in
- Kohlenbach
- 1993
(Show Context)
Citation Context ...functional interpretation of A(a) if 7 () 9x(t s-maj xs8a; y AD (x a; y; a)) (Monotone functional interpretation which directly extracts a tuple t satisfyings() from a proof of A(a) was introduced in =-=[9]-=-. See also [12] for details.) Definition 2.3 ([13]) Let A 2 L(G n A ! ) be a formula having the form A j 8u 1 8vstu9y 0 1 8x 0 1 : : : 9y 0 k 8x 0 k 9w fl A 0 (u; v; y 1 ; x 1 ; : : : ; y k ; x k ; w)... |

45 |
Zur intuitionistischen Arithmetik und Zahlentheorie, Ergebnisse eines Mathematischen Kolloquiums 4
- Gödel
- 1933
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Citation Context ...0 0)); since the latter can be reduced to the former (relative to G n A ! for ns2) by coding l; x together and applying comprehension without number parameters to this pair. 8 Here we can use Godel's =-=[4]-=- translation or any other of the various negative translations. For a systematical treatment of negative translations see [16]. 9 This last assertion is not stated in the formulation of the theorem in... |

44 |
Hereditarily majorizable functionals of finite type, Metamathematical investigation of intuitionistic Arithmetic and Analysis
- Howard
- 1973
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Citation Context ... x 2 :j 8y ae (x 1 ysx 2 y); 8 ? ! ? : x s--maj 0 x :j xs0 x; x s--majsae x :j 8y ae ; y ae (y s--maj ae y ! x y s--majsx y; xy): Remark 2.2 `s--maj' is a variant of W.A. Howard's relation `maj' from =-=[6]-=- which is due to [2]. For more details see [8]. Let A(a) be a formula of G n A ! (a are all free variables of A) and 9x8yAD (x; y; a) its Godel functional interpretation (see e.g. [24] for details on ... |

41 |
On Number Choice Schema and its Relation to Induction
- Parsons
- 1970
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Citation Context ...A 2 + AC 0;0 -qf +\Pi 0 1 -AC \Gamma proves every function parameter-free instance of \Pi 0 1 -CP, i.e. EA +\Pi 0 1 -CP is a subsystem of EA 2 + AC 0;0 -qf +\Pi 0 1 -AC \Gamma . It is well-known (see =-=[18]-=-) that there exists an instance A of \Pi 0 1 -CP which is not provable in EA +\Sigma 0 1 -IA. On the other hand EA +\Pi 0 1 -CP is \Pi 0 3 - conservative over EA +\Sigma 0 1 -IA by a result due to H. ... |

36 |
Strongly majorizable functionals of finite type: a model of bar recursion containing discontinuous functionals
- Bezem
- 1985
(Show Context)
Citation Context ...sx 2 y); 8 ? ! ? : x s--maj 0 x :j xs0 x; x s--majsae x :j 8y ae ; y ae (y s--maj ae y ! x y s--majsx y; xy): Remark 2.2 `s--maj' is a variant of W.A. Howard's relation `maj' from [6] which is due to =-=[2]-=-. For more details see [8]. Let A(a) be a formula of G n A ! (a are all free variables of A) and 9x8yAD (x; y; a) its Godel functional interpretation (see e.g. [24] for details on Godel's functional i... |

34 | Mathematically strong subsystems of analysis with low rate of growth of provably recursive functionals
- Kohlenbach
- 1996
(Show Context)
Citation Context ...ension, choice and uniform boundedness relative to weak base systems, formulated in the language of all finite types, which are suited to formalize substantial parts of analysis. In a previous paper (=-=[12]-=-) we have introduced a hierarchy G n A ! of systems where the definable functions correspond to the well-known Grzegorczyk hierarchy. These systems extended by the schema of full quantifier-free choic... |

29 |
On n-quantifier induction
- Parsons
- 1972
(Show Context)
Citation Context ...terms 2 T k\Gamma1 . The latter follows from the proof that the negative translation of \Sigma 0 k -IA has a functional interpretation in T k\Gamma1 (provable in (a subsystem of) PA ! i ) as given in =-=[19]-=- and the fact that every (closed) term of T k\Gamma1 can be majorized (in the sense of definition 2.1) by a suitable term in T k\Gamma1 which follows from Howard's proof of this fact for full T as giv... |

28 | Extensional Gödel Functional Interpretation - Luckhardt |

26 |
Fragments of arithmetic
- Sieg
- 1985
(Show Context)
Citation Context ...ystems G n A ! + AC-qf +\Delta instead of EA 2 + AC 0;0 -qf which we are treating in this paper. In particular we show the following result: 4 Both aspects are not taken into account appropriately in =-=[21]-=- where \Pi 0 k -CA \Gamma and \Pi 0 k -AC \Gamma are studied systematically for the first time. As a consequence of this, theorems 5.8,5.13 and corollaries 5.9,5.14 in [21] are not correct (see [11] a... |

19 | On the no-counterexample interpretation
- Kohlenbach
- 1999
(Show Context)
Citation Context ...d extract a uniform bound \Phi on `9w' which now of course is only in GnR ! [B 0;1 ] (instead of GnR ! ) and its verification can be carried out in GnA ! + ~ \Delta + (BR 0;1 ) + \Pi 0 1 -(DC 0 ). By =-=[16]-=- (proposition 4.2) it follows (since deg(fl1) = 2) that \Phi can be written as a primitive recursive functional ~ \Phi such that PA ! +BR 0;1 ` \Phi = fl1 ~ \Phi: The final claim follows using again t... |

17 |
Which Set Existence Axioms are Needed to Prove Cauchy/Peano Theorem for Ordinary Differential Equations?’, The
- Simpson
- 1984
(Show Context)
Citation Context ...ch that t(v; k; z; a) = 0 v. In [15] we have shown that every single (sequence of) instance(s) of the Bolzano-Weierstra principle for bounded sequences in IR d and of the Ascolilemma (in the sense of =-=[22]-=-) follows from suitable instances of \Pi 0 1 -UB \Gamma j n and used this to calibrate precisely the contribution of such instances to the growth of extractable bounds. This indicates the mathematical... |

14 | Pointwise hereditary majorization and some applications
- Kohlenbach
- 1992
(Show Context)
Citation Context ...j 0 x :j xs0 x; x s--majsae x :j 8y ae ; y ae (y s--maj ae y ! x y s--majsx y; xy): Remark 2.2 `s--maj' is a variant of W.A. Howard's relation `maj' from [6] which is due to [2]. For more details see =-=[8]-=-. Let A(a) be a formula of G n A ! (a are all free variables of A) and 9x8yAD (x; y; a) its Godel functional interpretation (see e.g. [24] for details on Godel's functional interpretation). We say tha... |

12 | Elimination of Skolem functions for monotone formulas in analysis
- Kohlenbach
- 1998
(Show Context)
Citation Context ...maj xs8a; y AD (x a; y; a)) (Monotone functional interpretation which directly extracts a tuple t satisfyings() from a proof of A(a) was introduced in [9]. See also [12] for details.) Definition 2.3 (=-=[13]-=-) Let A 2 L(G n A ! ) be a formula having the form A j 8u 1 8vstu9y 0 1 8x 0 1 : : : 9y 0 k 8x 0 k 9w fl A 0 (u; v; y 1 ; x 1 ; : : : ; y k ; x k ; w); where A 0 is quantifier--free and contains only ... |

8 | The use of a logical principle of uniform boundedness in analysis
- Kohlenbach
- 1995
(Show Context)
Citation Context ...al of classical analysis even for n = 2; 3. The axioms \Delta and AC-qf do not contribute to the growth of extractable uniform bounds which in the particular case of G 2 A ! are polynomials (see [12],=-=[14]-=- and in particular [10] for more information). In contrast to this, fragments of arithmetical comprehension and choice as well as generalizations of our principle of uniform \Sigma 0 1 -boundedness (f... |

6 |
growth in standard parts of analysis. Habilitationsschrift
- Kohlenbach, Real
- 1995
(Show Context)
Citation Context ...s even for n = 2; 3. The axioms \Delta and AC-qf do not contribute to the growth of extractable uniform bounds which in the particular case of G 2 A ! are polynomials (see [12],[14] and in particular =-=[10]-=- for more information). In contrast to this, fragments of arithmetical comprehension and choice as well as generalizations of our principle of uniform \Sigma 0 1 -boundedness (from [12]) to more compl... |

5 |
A proof-theoretic analysis of collection
- Beklemishev
- 1998
(Show Context)
Citation Context ...ied in [12]. 6 In [14] we showed that \Sigma 0 1 -UB \Gamma proves already relative to G 2 A ! + AC-qf many important analytical theorems (like Dini's theorem, the attainment of the maximum for f 2 C(=-=[0; 1]-=- d ; IR), the sequential Heine-Borel property for [0; 1] d , the existence of an inverse function for every strictly monotone function f 2 C[0; 1] and others) but does not contribute to the growth of ... |

4 |
n Collection Schema in Arithmetic
- Paris, Kirby
- 1978
(Show Context)
Citation Context ... 1 -CP which is not provable in EA +\Sigma 0 1 -IA. On the other hand EA +\Pi 0 1 -CP is \Pi 0 3 - conservative over EA +\Sigma 0 1 -IA by a result due to H. Friedman and (implicitly) J.Paris/L.Kirby =-=[17]-=- (see e.g. [7] for details). The universal closure of the instance A of \Pi 0 1 -CP can be shown to be equivalent to a \Pi 0 4 -sentence in EA +\Sigma 0 1 -IA. Hence EA 2 + AC 0;0 -qf +\Pi 0 1 -AC \Ga... |

3 |
On some formalized conservation results in arithmetic
- Clote, Hájek, et al.
- 1990
(Show Context)
Citation Context ... the Friedman-Paris-Kirby result was first given in [21]. However the proof in [21] contains a serious gap. See [1] for a correction of Sieg's proof. Another proof-theoretic treatment can be found in =-=[3]-=-. 6 Whereas we generally use the superscript `\Gamma' to denote the restriction S \Gamma of a schema S to function parameter-free instances of S, this superscript has a different meaning in the contex... |

3 | A note on the Π0 2 –induction rule
- Kohlenbach
(Show Context)
Citation Context ...appropriately in [21] where Π 0 k -CA − and Π 0 k -AC− are studied systematically for the first time. As a consequence of this, theorems 5.8,5.13 and corollaries 5.9,5.14 in [21] are not correct (see =-=[11]-=- and in particular chapter 12 of [10] for a thorough investigation of this matter). 5sLet t, ξ1,ξ2 be closed terms of GnAω and B :≡ ∀u1∀v ≤τtu Bar(u, v) a sentence of GnAω where Bar(u, v) ∈ Π0 ∞ . The... |

2 |
Arithmetizing proofs in analysis. To appear in
- Kohlenbach
(Show Context)
Citation Context ...se systems. In [13] we developed a general method to calibrate faithfully this contribution and applied it to instances of \Pi 0 1 - comprehension and \Pi 0 1 -choice. These results were then used in =-=[15]-=- to determine the arithmetical strength of single sequences of instances of the BolzanoWeierstrastheorem for bounded sequences in IR d , the Ascoli-lemma and others. In this paper we give a systematic... |

1 |
Models of Peano Arithmetic. Oxford Logic Guides 15
- Kay
- 1991
(Show Context)
Citation Context ... not provable in EA +\Sigma 0 1 -IA. On the other hand EA +\Pi 0 1 -CP is \Pi 0 3 - conservative over EA +\Sigma 0 1 -IA by a result due to H. Friedman and (implicitly) J.Paris/L.Kirby [17] (see e.g. =-=[7]-=- for details). The universal closure of the instance A of \Pi 0 1 -CP can be shown to be equivalent to a \Pi 0 4 -sentence in EA +\Sigma 0 1 -IA. Hence EA 2 + AC 0;0 -qf +\Pi 0 1 -AC \Gamma is not \Pi... |

1 |
A note on the \Pi 2 -induction rule
- Kohlenbach
- 1995
(Show Context)
Citation Context ...n [21] where \Pi 0 k -CA \Gamma and \Pi 0 k -AC \Gamma are studied systematically for the first time. As a consequence of this, theorems 5.8,5.13 and corollaries 5.9,5.14 in [21] are not correct (see =-=[11]-=- and in particular chapter 12 of [10] for a thorough investigation of this matter). 5 Let t;s1 ;s2 be closed terms of G n A ! and B :j 8u 1 8vstu B ar (u; v) a sentence of G n A ! where B ar (u; v) 2 ... |

1 |
Σ 0 n-collection schema in arithmetic
- Paris, Kirby
- 1978
(Show Context)
Citation Context ... exists an instance A of Π0 1-CP which is not provable in EA +Σ0 1-IA. On the other hand EA +Π0 1-CPisΠ0 3conservative over EA +Σ0 1-IA by a result due to H. Friedman and (implicitly) J.Paris/L.Kirby =-=[17]-=- (see e.g. [7] for details). The universal closure of the instance A of Π0 1-CP can be shown to be equivalent to a Π04 -sentence in EA -conservative over EA +Σ0 1-IA. Hence EA2 +AC0,0-qf +Π0 1-AC− is ... |