50 research outputs found
Proof Theory of Martin-Löf Type Theory. An overview
We give an overview over the historic development of proof theory and the main techniques used in ordinal theoretic proof theory. We argue, that in a revised Hilbertâs program, ordinal theoretic proof theory has to be supplemented by a second step, namely the development of strong equiconsistent constructive theories. Then we show, how, as part of such a programe, the proof theoretic analysis of Martin-Löf type theory with W-type and one microscopic universe containing only two finite sets is carried out. Then we look at the analysis Martin-Löf type theory with W-type and a universe closed under the W-type, and consider the extension of type theory by one Mahlo universe and its proof-theoretic analysis. Finally, we repeat the concept of inductive-recursive definitions, which extends the notion of inductive definitions substantially. We introduce a closed formalization, which can be used in generic programming, and explain, what is known about its strength.Nous donnons une vue dâensemble du dĂ©veloppement historique de la thĂ©orie de la preuve et des principales techniques utilisĂ©es dans la thĂ©orie ordinale de la preuve. Nous soutenons que, dans une forme rĂ©visĂ©e du programme dâHilbert, la thĂ©orie ordinale de la preuve doit ĂȘtre complĂ©tĂ©e par une seconde Ă©tape, Ă savoir le dĂ©veloppement de thĂ©ories constructives fortes et Ă©quiconsistantes. Comme partie dâun tel programme, nous prĂ©sentons ensuite lâanalyse, en thĂ©orie de la preuve, de la thĂ©orie des types de Martin-Löf avec un univers microscopique ne contenant que deux types finis. Nous examinons ensuite lâanalyse de la thĂ©orie des types de Martin-Löf avec type W et un univers clos pour ce type, puis nous Ă©tendons la thĂ©orie des types par un univers de Mahlo et considĂ©rons son analyse en thĂ©orie de la preuve. Enfin, nous prĂ©sentons le concept de dĂ©finition inductive-rĂ©cursive, qui Ă©tend de façon substantielle la notion de dĂ©finition inductive. Nous introduisons une formalisation close, qui peut ĂȘtre employĂ©e en programmation gĂ©nĂ©rique, et expliquons ce que nous savons de sa force ordinale
Constructive set theory and Brouwerian principles
The paper furnishes realizability models of constructive Zermelo-Fraenkel set theory, CZF, which also validate Brouwerian principles such as the axiom of continuous choice (CC), the fan theorem (FT), and monotone bar induction (BIM), and thereby determines the proof-theoretic strength of CZF augmented by these principles. The upshot is that CZF+CC+FT possesses the same strength as CZF, or more precisely, that CZF+CC+FTis conservative over CZF for 02 statements of arithmetic, whereas the addition of a restricted version of bar induction to CZF (called decidable bar induction, BID) leads to greater proof-theoretic strength in that CZF+BID proves the consistency of CZF
Admissible closures of polynomial time computable arithmetic
We propose two admissible closures and of Ferreira's system PTCA of polynomial time computable arithmetic and of full bounded arithmetic (or polynomial hierarchy computable arithmetic) PHCA. The main results obtained are: (i) is conservative over PTCA with respect to sentences, and (ii) is conservative over full bounded arithmetic PHCA for sentences. This yields that (i) the definable functions of are the polytime functions, and (ii) the definable functions of are the functions in the polynomial time hierarch
Type-Two Well-Ordering Principles, Admissible Sets, and Pi^1_1-Comprehension
This thesis introduces a well-ordering principle of type two, which we call the Bachmann-Howard principle. The main result states that the Bachmann-Howard principle is equivalent to the existence of admissible sets and thus to Pi^1_1-comprehension. This solves a conjecture of Rathjen and MontalbĂĄn. The equivalence is interesting because it relates "concrete" notions from ordinal analysis to "abstract" notions from reverse mathematics and set theory.
A type-one well-ordering principle is a map T which transforms each well-order X into another well-order T[X]. If T is particularly uniform then it is called a dilator (due to Girard). Our Bachmann-Howard principle transforms each dilator T into a well-order BH(T). The latter is a certain kind of fixed-point: It comes with an "almost" monotone collapse theta:T[BH(T)]->BH(T) (we cannot expect
full monotonicity, since the order-type of T[X] may always exceed the order-type of X). The Bachmann-Howard principle asserts that such a collapsing structure exists. In fact we define three variants of this principle: They are equivalent but differ in the sense in which the order BH(T) is "computed".
On a technical level, our investigation involves the following achievements: a detailed discussion of primitive recursive set theory as a basis for set-theoretic reverse
mathematics; a formalization of dilators in weak set theories and second-order arithmetic; a functorial version of the constructible hierarchy; an approach to deduction chains (SchĂŒtte) and beta-completeness (Girard) in a set-theoretic context; and a beta-consistency proof for Kripke-Platek set theory.
Independently of the Bachmann-Howard principle, the thesis contains a series of results connected to slow consistency (introduced by S.-D. Friedman, Rathjen and Weiermann): We present a slow reflection statement and investigate its consistency strength, as well as its computational properties. Exploiting the latter, we show that instances of the Paris-Harrington principle can only have extremely long proofs in certain fragments of arithmetic
Recommended from our members
Mathematical Logic: Proof theory, Constructive Mathematics
The workshop âMathematical Logic: Proof Theory, Constructive Mathematicsâ was centered around proof-theoretic aspects of current mathematics, constructive mathematics and logical aspects of computational complexit
On Relating Theories: Proof-Theoretical Reduction
The notion of proof-theoretical or finitistic reduction of one theory to another has a long tradition. Feferman and Sieg (Buchholz et al., Iterated inductive definitions and subsystems of analysis. Springer, Berlin, 1981, Chap. 1) and Feferman in (J Symbol Logic 53:364â384, 1988) made first steps to delineate it in more formal terms. The first goal of this paper is to corroborate their view that this notion has the greatest explanatory reach and is superior to others, especially in the context of foundational theories, i.e., theories devised for the purpose of formalizing and presenting various chunks of mathematics.
A second goal is to address a certain puzzlement that was expressed in Fefermanâs title of his Clermont-Ferrand lectures at the Logic Colloquium 1994: âHow is it that finitary proof theory became infinitary?â Hilbertâs aim was to use proof theory as a tool in his finitary consistency program to eliminate the actual infinite in mathematics from proofs of real statements. Beginning in the 1950s, however, proof theory began to employ infinitary methods. Infinitary rules and concepts, such as ordinals, entered the stage.
In general, the more that such infinitary methods were employed, the farther did proof theory depart from its initial aims and methods, and the closer did it come instead to ongoing developments in recursion theory, particularly as generalized to admissible sets; in both one makes use of analogues of regular cardinals, as well as âlargeâ cardinals (inaccessible, Mahlo, etc.). (Feferman 1994).
The current paper aims to explain how these infinitary tools, despite appearances to the contrary, can be formalized in an intuitionistic theory that is finitistically reducible to (actually Î 02
-conservative over) intuitionistic first order arithmetic, also known as Heyting arithmetic. Thus we have a beautiful example of Hilbertâs program at work, exemplifying the Hilbertian goal of moving from the ideal to the real by eliminating ideal elements
Worms and Spiders: Reflection calculi and ordinal notation systems
We give a general overview of ordinal notation systems arising from
reflection calculi, and extend the to represent impredicative ordinals up to
those representable using Buchholz-style collapsing functions