## Foundational and mathematical uses of higher types (1999)

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Venue: | REFLECTIONS ON THE FOUNDATIONS OF MATHEMATICS: ESSAY IN HONOR OF SOLOMON FEFERMAN |

Citations: | 13 - 4 self |

### BibTeX

@TECHREPORT{Kohlenbach99foundationaland,

author = {Ulrich Kohlenbach},

title = {Foundational and mathematical uses of higher types},

institution = {REFLECTIONS ON THE FOUNDATIONS OF MATHEMATICS: ESSAY IN HONOR OF SOLOMON FEFERMAN},

year = {1999}

}

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### Abstract

In this paper we develop mathematically strong systems of analysis in higher types which, nevertheless, are proof-theoretically weak, i.e. conservative over elementary resp. primitive recursive arithmetic. These systems are based on non-collapsing hierarchies ( n -WKL+ ; n -WKL+ ) of principles which generalize (and for n = 0 coincide with) the so-called `weak' König's lemma WKL (which has been studied extensively in the context of second order arithmetic) to logically more complex tree predicates. Whereas the second order context used in the program of reverse mathematics requires an encoding of higher analytical concepts like continuous functions F : X ! Y between Polish spaces X;Y , the more exible language of our systems allows to treat such objects directly. This is of relevance as the encoding of F used in reverse mathematics tacitly yields a constructively enriched notion of continuous functions which e.g. for F : IN ! IN can be seen (in our higher order context)

### Citations

207 |
Subsystems of second order arithmetic
- Simpson
- 1998
(Show Context)
Citation Context ...of all finite types). Via appropriate representations and codings of higher objects (like continuous functions between Polish spaces) a great deal of mathematics can be developed already in ACA0 (see =-=[35]-=- for a comprehensive treatment). Feferman’s system, however, allows a more direct treatment of such objects and their mathematics and also contains a strong uniform (‘explicit’) version of arithmetica... |

154 |
Über eine bisher noch nicht benützte Erweiterung des finiten Standpunkts
- Gödel
- 1958
(Show Context)
Citation Context ...recursor constants Rρ (see [23]). E-PA ω is the extension of E-PRA ω obtained by the addition of the schema of full induction and all (impredicative) primitive recursive functionals in the sense of =-=[13]-=-. The schema of full choice is given by AC ρ,τ : ∀x ρ ∃y τ A(x, y) → ∃Y τ(ρ) ∀x ρ A(x, Y x), AC := ⋃ ρ,τ ∈T { AC ρ,τ }. The schema of quantifier-free choice QF-AC ρ,τ is defined as the restriction of ... |

127 |
Metamathematical investigation of intuitionistic Arithmetic andAnalysis
- Troelstra
- 1973
(Show Context)
Citation Context ...sts a Φ ∈ A2 such that Φ has no associate in A1. By (ii), A is a model of the restriction of E-PA ω +QF-AC 1,0 to the fragment with pure types only. Modulo the well-known reduction to pure types (see =-=[38]-=-(1.8.5-1.8.8)), E-PA ω +QF-AC 1,0 therefore has a model in which there exists a functional Φ 2 which has no associate and therefore – by the previous proposition – no r.m.-code f. Nevertheless, all fu... |

91 |
A language and axioms for explicit mathematics
- Feferman
- 1975
(Show Context)
Citation Context ...tructive µ-operator. These features hold in an even stronger form for theories with flexible (variable) types which were developed successively by Feferman in his framework of explicit mathematics in =-=[4]-=-,[6],[7] culminating in a formal system called W (where ‘W’ stands for ‘Weyl’) which was shown to be proof-theoretically reducible to and conservative over PA in [11]. The enormous mathematical power ... |

78 | Gödel’s functional (‘Dialectica’) interpretation
- Avigad, Feferman
- 1977
(Show Context)
Citation Context ...one to extract at least a primitive recursive algorithm. In the other direction, e.g. our analysis of proofs in approximation theory (which used the principle of the attainment of the maximum of f ∈ C=-=[0, 1]-=-, see [20]) led us to an elimination procedure of weak König’s lemma WKL over a variety of subsystems of arithmetic in all finite types thereby contributing to ‘1)’ above (see [19]). Likewise our trea... |

56 | Analysing proofs in analysis, in
- Kohlenbach
- 1993
(Show Context)
Citation Context ...Dragalin translation are most useful (in particular compared to techniques which try to avoid any passage through higher types, see [28]). Whereas we have focused on ‘2)’ in several publications (see =-=[21]-=-,[20],[24] among others), this paper addresses ‘1),’ to which S. Feferman has contributed so profoundly. We study mathematical strong, but nevertheless PRA-reducible, systems in all finite types, emph... |

48 | Systems of explicit mathematics with non-constructive µ-operator
- Feferman, Jäger
- 1993
(Show Context)
Citation Context ...amework of explicit mathematics in [4],[6],[7] culminating in a formal system called W (where ‘W’ stands for ‘Weyl’) which was shown to be proof-theoretically reducible to and conservative over PA in =-=[11]-=-. The enormous mathematical power and flexibility of the system W led Feferman in [9] to the formulation of the thesis that all (or almost all) scientifically applicable mathematics can be developed i... |

46 |
Theories of finite type related to mathematical practice
- Feferman
- 1977
(Show Context)
Citation Context ... observed that large parts of analysis can be developed on the basis of arithmetical comprehension. This theme was further developed in the 50’s by P. Lorenzen among others. In the late 70’s Feferman =-=[5]-=- and G. Takeuti [37] independently designed formal systems based on arithmetical comprehension in the framework of higher order arithmetic which are conservative over PA. For this property it is impor... |

40 | Partial Realizations of Hilbert’s Program
- Simpson
- 1988
(Show Context)
Citation Context ...PRA and identify (following [36]) PRA with finitism, then the parts of mathematics which can be carried out in TPRA have a finitistic foundation (partial realization of D. Hilbert’s program, see e.g. =-=[34]-=-). 2) Mathematical relevance: here the guiding question is ‘What more do we know if we have proved a theorem by restricted means than if we merely know that it is true?’ (G. Kreisel) The aim is to get... |

38 |
Effective bounds from ineffective proofs in analysis: an application of functional interpretation and majorization
- Kohlenbach
- 1992
(Show Context)
Citation Context ...e maximum of f ∈ C[0, 1], see [20]) led us to an elimination procedure of weak König’s lemma WKL over a variety of subsystems of arithmetic in all finite types thereby contributing to ‘1)’ above (see =-=[19]-=-). Likewise our treatment of e.g. the Bolzano-Weierstraß principle in [26] via an elimination technique of Skolem functions yielded also new conservation results for comprehension principles ([27]). H... |

34 | Mathematically strong subsystems of analysis with low rate of growth of provably recursive functionals
- Kohlenbach
- 1996
(Show Context)
Citation Context ...etically reduced to and is Π0 2-conservative over PRA (H. Friedman (1976, unpublished) and [33]; for a historical discussion which in particular points out various errors in the literature on WKL see =-=[23]-=- (p.69)). In [19] we introduced an extension (in the spirit of Feferman’s PA-conservative system from [5] mentioned above) of WKL0 to all finite types and proved among other things that this extension... |

31 | Effective moduli from ineffective uniqueness proofs. An unwinding of de La Vallée Poussin’s proof for Chebycheff approximation
- Kohlenbach
- 1993
(Show Context)
Citation Context ...ct at least a primitive recursive algorithm. In the other direction, e.g. our analysis of proofs in approximation theory (which used the principle of the attainment of the maximum of f ∈ C[0, 1], see =-=[20]-=-) led us to an elimination procedure of weak König’s lemma WKL over a variety of subsystems of arithmetic in all finite types thereby contributing to ‘1)’ above (see [19]). Likewise our treatment of e... |

26 |
Fragments of arithmetic
- Sieg
- 1985
(Show Context)
Citation Context ...eatment of ordinary mathematics in WKL0). This fact is of foundational relevance since WKL0 can be prooftheoretically reduced to and is Π0 2-conservative over PRA (H. Friedman (1976, unpublished) and =-=[33]-=-; for a historical discussion which in particular points out various errors in the literature on WKL see [23] (p.69)). In [19] we introduced an extension (in the spirit of Feferman’s PA-conservative s... |

23 |
Recursion on the countable functionals
- Normann
- 1980
(Show Context)
Citation Context ...GHER TYPES 15 Theorem 4.6. E-PA ω +QF-AC 1,0 +QF-AC 0,1 does not prove that every continuous functional Φ 2 has an r.m.-code (i.e. that Φ is continuous in the sense of reverse mathematics). Proof: In =-=[32]-=-(6.4) a type-structure A = 〈Ak〉k∈IN over ω is constructed with the following properties: (i) E2| \ A1 /∈ A2, where E2(f 1 ) = 0 ↔ ∃x(fx = 0); (ii) A is closed under computation in the sense of Kleene’... |

22 |
Kritische Untersuchungen über die Grundlagen der Analysis, Veit
- Weyl, Kontinuum
- 1918
(Show Context)
Citation Context ...ll very briefly some of the history of research on ‘1)’. As Feferman pointed out in [7], ‘Hermann Weyl initiated a program for the arithmetical foundation of mathematics’ in his book ‘Das Kontinuum’ (=-=[40]-=-). In this book, Weyl observed that large parts of analysis can be developed on the basis of arithmetical comprehension. This theme was further developed in the 50’s by P. Lorenzen among others. In th... |

19 | On the no-counterexample interpretation
- Kohlenbach
- 1999
(Show Context)
Citation Context ...y combined with tools like negative translation and/or the Friedman-Dragalin translation are most useful (in particular compared to techniques which try to avoid any passage through higher types, see =-=[28]-=-). Whereas we have focused on ‘2)’ in several publications (see [21],[20],[24] among others), this paper addresses ‘1),’ to which S. Feferman has contributed so profoundly. We study mathematical stron... |

17 | Proof theory and computational analysis - Kohlenbach - 1998 |

16 |
Why a little bit goes a long way: Logical foundations of scientifically applicable mathematics
- Feferman
- 1993
(Show Context)
Citation Context ... (where ‘W’ stands for ‘Weyl’) which was shown to be proof-theoretically reducible to and conservative over PA in [11]. The enormous mathematical power and flexibility of the system W led Feferman in =-=[9]-=- to the formulation of the thesis that all (or almost all) scientifically applicable mathematics can be developed in W. In the late 70’s, H. Friedman observed that large parts of the mathematics that ... |

14 |
The Boolean prime ideal theorem does not imply the axiom of choice
- Halpern, Levy
- 1971
(Show Context)
Citation Context ...AC 0,1 in the proof of ‘→’ in the proposition above is unavoidable already for X = Y = IR since in this case the implication is known to be unprovable even in Zermelo–Fraenkel set theory ZF, see [16],=-=[15]-=- and [12]. We now discuss the indirect representation of continuous functions G : X → Y between Polish spaces X, Y via codes g as used in the context of reverse mathematics (see definition II.6.1 in [... |

14 | Pointwise hereditary majorization and some applications
- Kohlenbach
- 1992
(Show Context)
Citation Context ...oof of proposition 4.14 in [19] can easily be replaced by a modulus txk of uniform continuity on {y : y ≤1 sx}. For closed t ∈E-GnA ω such a modulus t can be constructed in E-GnA ω by the method of =-=[18]-=- since the majorization argument used there is available in E-GnA ω as was shown in [23]. ✷ Proposition 5.5. Let m, n ≥ 0. Over T :=E-GkA ω (k ≥ 3), E-PRA ω or E-PA ω the following principles are equi... |

11 | Higher order reverse mathematics - Kohlenbach |

10 |
Remarks on Herbrand normal forms and Herbrand realizations
- Kohlenbach
- 1992
(Show Context)
Citation Context ...A0(f, y) of A (we may assume that A is in prenex normal form) is provable there a-fortiori. Hence by theorem 3.7 E-PA ω ⊢ ∀f A0(f, Ψ(f)) for suitable closed terms Ψ of E-PA ω . Thus E-PA ω ⊢ A H . By =-=[17]-=-(thm.4.1) we can conclude that 17 PA ⊢ A. 1) and 2): For Π 0 2-sentences A the argument above applies equally to E-G3A ω (resp. E-PRA ω ) yielding E-G3A ω ⊢ A (resp. E-PRA ω ⊢ A). The conclusion now f... |

10 | On uniform weak König’s lemma
- Kohlenbach
- 2002
(Show Context)
Citation Context ...analogously to the uniform version of arithmetical comprehension given by µ), the strength of the6 ULRICH KOHLENBACH resulting system crucially depends on the amount of extensionality available (see =-=[30]-=-). §2. Description of the theories E-GnA ω , E-PRA ω and E-PA ω . The set T of all finite types is defined inductively by (i) 0 ∈ T and (ii) ρ, τ ∈ T ⇒ τ(ρ) ∈ T. Terms which denote a natural number ha... |

10 |
Note on the fan theorem
- Troelstra
- 1974
(Show Context)
Citation Context ... x1 ≤τρ x2 :≡ ∀y ρ (x1y ≤τ x2y); 2) minρτ(x ρτ 1 , xρτ 2 ) := λyτ . minρ(x1y, x2y), with min0 from above. In the following we will need the definition of the binary (‘weak’) König’s lemma as given in =-=[39]-=-: Definition 2.3 (Troelstra(74)). ⎧ ⎨ ∀f WKL:≡ ⎩ 1( T(f) ∧ ∀x0∃n0 (lth n =0 x ∧ fn =0 0) → ∃b ≤1 λk.1∀x0 (f(bx) =0 0) ) , where Tf :≡ ∀n 0 , m 0 (f(n ∗ m) =0 0 → fn =0 0) ∧ ∀n 0 , x 0 (f(n ∗ 〈x〉) =0 0... |

9 |
Weyl vindicated: Das Kontinuum seventy years later
- Feferman
- 1998
(Show Context)
Citation Context ...orrespond to their ordinary mathematical definitions and to develop a general theory of representations. Let us recall very briefly some of the history of research on ‘1)’. As Feferman pointed out in =-=[7]-=-, ‘Hermann Weyl initiated a program for the arithmetical foundation of mathematics’ in his book ‘Das Kontinuum’ ([40]). In this book, Weyl observed that large parts of analysis can be developed on the... |

9 |
Models of ZF-set theory
- Felgner
- 1971
(Show Context)
Citation Context ... the proof of ‘→’ in the proposition above is unavoidable already for X = Y = IR since in this case the implication is known to be unprovable even in Zermelo–Fraenkel set theory ZF, see [16],[15] and =-=[12]-=-. We now discuss the indirect representation of continuous functions G : X → Y between Polish spaces X, Y via codes g as used in the context of reverse mathematics (see definition II.6.1 in [35]). Sin... |

9 | On the arithmetical content of restricted forms of comprehension, choice and general uniform boundedness. Ann. Pure and Applied Logic 95
- Kohlenbach
- 1998
(Show Context)
Citation Context ...see [19]). Likewise our treatment of e.g. the Bolzano-Weierstraß principle in [26] via an elimination technique of Skolem functions yielded also new conservation results for comprehension principles (=-=[27]-=-). However, there are also important differences due to the different points emphasized in 1) and 2): Whereas there are hardly foundational (understood in the sense of Hilbert) reasons to study system... |

8 |
On effectively discontinuous type-2 objects
- Grilliot
(Show Context)
Citation Context ...ually is implied by F − so that in this context (which we use throughout this paper) the F − -elimination applies to proofs based on F as well. The proof of this fact uses an argument due to Grilliot =-=[14]-=-. The result allows one to construct a PRA-reducible finite type system T ∗ which is based on Σ0 1-UB. The relevance of this is due to fact that T ∗ allows one to develop the analysis of continuous fu... |

8 | The use of a logical principle of uniform boundedness in analysis
- Kohlenbach
- 1995
(Show Context)
Citation Context ...oves a strong principle of uniform boundedness Σ 0 1 -UB which allows one to give short proofs of the usual WKL-applications in analysis relative to very weak (polynomially bounded) systems (see [23],=-=[25]-=-) but does not contribute to the growth of provably recursive functionals. This axiom as well as the principle of uniform boundedness is ‘non-standard’ in the sense that it is not true in the full set... |

6 | growth in standard parts of analysis. Habilitationsschrift - Kohlenbach, Real - 1995 |

5 |
Infinity in mathematics: Is Cantor necessary
- Feferman
- 1987
(Show Context)
Citation Context ...be a conservative extension of first order Peano Arithmetic PA, then that part of mathematics has an arithmetical foundation (partial realization of H. Weyl’s program, see S. Feferman’s discussion in =-=[8]-=-). If we work in a system TPRA which can be shown (finitistically) even to be conservative over Primitive Recursive Arithmetic PRA and identify (following [36]) PRA with finitism, then the parts of ma... |

4 |
Foundations of Constructive Mathematics. Springer Ergebnisse der Mathematik und ihrer Grenzgebiete 3.Folge
- Beeson
- 1985
(Show Context)
Citation Context ... of convergence), elements of X can be represented by number-theoretic functions x1 and, moreover, every such function can be considered as a representative of a uniquely determined element of X (see =-=[2]-=- and [20] for details). On these representatives we have a pseudo metric dX. The elements of X can be identified with the equivalence classes w.r.t. x =Y x :≡ (dX(x, y) =IR 0). Functions G : X → Y bet... |

4 |
Things that can and things that can’t be done
- Kohlenbach
(Show Context)
Citation Context ...itive recursive functionals of type 2, in particular it does not contain Φit. As a consequence of this, E-G∞Aω +QF-AC0,0 does not prove the schema of Σ0 1-induction. As we have shown in [26],[27] and =-=[29]-=-, one can add to E-G∞Aω +QF-AC1,0+ QFAC0,1 function parameter-free schematic forms of e.g. Π0 1-comprehension, the Bolzano-Weierstraß principle for sequences in [0, 1] d , the Arzela-Ascoli lemma etc.... |

1 | Working foundations. Synthese 62 - Feferman - 1985 |

1 |
The axiom of choice and two definitions of continuity
- Jägermann
- 1965
(Show Context)
Citation Context ...f QF-AC 0,1 in the proof of ‘→’ in the proposition above is unavoidable already for X = Y = IR since in this case the implication is known to be unprovable even in Zermelo–Fraenkel set theory ZF, see =-=[16]-=-,[15] and [12]. We now discuss the indirect representation of continuous functions G : X → Y between Polish spaces X, Y via codes g as used in the context of reverse mathematics (see definition II.6.1... |

1 |
A conservative extension of Peano arithmetic. Part II of ‘Two applications of logic to mathematics’, Publ
- Takeuti
- 1978
(Show Context)
Citation Context ...e parts of analysis can be developed on the basis of arithmetical comprehension. This theme was further developed in the 50’s by P. Lorenzen among others. In the late 70’s Feferman [5] and G. Takeuti =-=[37]-=- independently designed formal systems based on arithmetical comprehension in the framework of higher order arithmetic which are conservative over PA. For this property it is important that the schema... |