Jan M. Smith. The Independence of Peano's Fourth Axiom from Martin-Lof's Type Theory without Universes. Journal of Symbolic Logic, 53(3), 1988.  Home/Search   Document Details and Download   Summary   Related Articles   Check

.... z instead of CBB:BB Prop(CBB tt) CBB ff) z:BBCBB z This is the only instance of a so called large elimination rule in the entire formalization, and the extra strength is used only to prove Goal ttnotff: not (Q tt ff) which is not provable in ECC without a large elimination rule [Smi88] 1 . BB has the usual classical boolean operators, conjunction andd, disjunction orr and conditional if, together with the lifting functions istt and isff, which convert booleans to (decidable) propositions: istt [b:BB] Q b tt] isff [b:BB] Q b ff] if [a:BB] D Prop] d,e:D] ....

Jan Smith. The independence of Peano's fourth axiom from Martin-L of's type theory without universes. Journal of Symbolic Logic, 53(3), 1988.

.... Pi Id N (0; Suc(0) 0 type corresponding to Peano s fourth axiom 0 6= 1. Hint: define using R N a function f : N U such that Pi f0 = 1 : U and Pi f(Suc(0) 0 : U . Later on we will show by a semantic argument that the above type is not inhabited in the absence of a universe (Smith 1988). E2.6 (Troelstra and van Dalen 1988) Show that in type theory without the empty type 0 such an empty type can be defined as Id N (0; Suc(0) The elimination operator R 0 oe must then be defined by induction on the structure of oe. Notice that in view of the semantic result anticipated in the ....

....2 Ty Gamma. Hofmann E3.6 Check that equations (Cons L) Cons Id) hold in the set theoretic model. E3.7 This exercise will be taken up in later sections and will lead us up to Jan Smith s proof of the independence of Peano s fourth axiom from MartinL of s type theory without universes (Smith 1988). Let P be the poset of truth values fff; ttg where ff tt viewed as a category. Show that P has a terminal object, viz. tt. Extend P to a CwF by putting Ty P (tt) Ty P (ff) fff; ttg and Tm P ( Gamma; oe) Gamma oe] Hint: define comprehension by Gamma:oe def = Gamma oe. An intuition ....

Smith, J. (1988). The independence of Peano's fourth axiom from MartinL of's type theory without universes. Journal of Symbolic Logic 53(3).

....of Zermelo Fraenkel set theory CZF can be interpreted [3] But in the standard formulations of type theory universes are needed also for the more basic purpose of defining families of sets by structural recursion. For example, the predicate Z (as used in the type theoretic proof that 0 6= s(n) [29, 43]) with the recursion equations Z(0) Z(s(n) is defined in terms of universes and the rule of N elimination. Martin Lof [27] introduced an infinite tower of universes U 0 : U 1 : U 2 : Delta Delta Delta. These were formulated a la Russell [29] which means that there is no ....

J. Smith. The independence of Peano's fourth axiom from Martin-Lof's type theory without universes. Journal of Symbolic Logic, 49(3), 1988.

....Case of Higman s Lemma. Personal communication, 1994. A Lambda Calculus Model of Martin Lof s Theory of Types with Explicit Substitution Abstract This paper presents a proof irrelevant model of Martin Lof s theory of types with explicit substitution; that is, a model in the style of [Smi88], in which types are interpreted as truth values and objects (or proofs) are irrelevant. The fundamental difference here is the need to cope with a formal system which in addition to types has sets and substitutions. This difference leads us to a whole reformulation of the model which consists in ....

....by Martin Lof and intended for formalizing intuitionistic mathematics. The progressive development of type theory by successive modifications introduced by Martin Lof has led to several formulations of the system, see for instance [Mar75] and [Mar84] For these presentations of type theory Smith [Smi88] defined a model in which every type is interpreted either as the empty set or as a singleton. As a consequence of this interpretation of types any two objects of a type are never distinguishable in the model. Hence, if one of them satisfies a predicate, so does the other, which means that objects ....

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J. Smith. The Independence of Peano's Fourth Axiom from Martin-Lof's Type Theory without Universes. Journal of Symbolic Logic, 53:840--845, 1988.

.... lambda calculus model of Martin Lof s theory of types with explicit substitution Daniel Fridlender September 1997 Abstract This paper presents a proof irrelevant model of Martin Lof s theory of types with explicit substitution; that is, a model in the style of [Smi88], in which types are interpreted as truth values and objects (or proofs) are irrelevant. The fundamental difference here is the need to cope with a formal system which in addition to types has sets and substitutions. This difference leads us to a whole reformulation of the model which consists in ....

....by Martin Lof and intended for formalizing intuitionistic mathematics. The progressive development of type theory by successive modifications introduced by Martin Lof has led to several formulations of the system, see for instance [Mar75] and [Mar84] For these presentations of type theory Smith [Smi88] defined a model in which every type is interpreted either as the empty set or as a singleton. As a consequence of this interpretation of types any two objects of a type are never distinguishable in the model. Hence, if one of them satisfies a predicate, so does the other, which means that objects ....

[Article contains additional citation context not shown here]

J. Smith. The Independence of Peano's Fourth Axiom from Martin-Lof's Type Theory without Universes. Journal of Symbolic Logic, 53:840--845, 1988.

....is left unchanged upon substitution. So in view of the first equation oe set there would have to be a function : oe set for every type which contradicts the second equation if = 0. A major application of such families of types is that they allow to derive Peano s fourth axiom [15, 18]. On the level of propositions we are still able to interpret this axiom, that is we have 0 L = Suc(0) ff, provided this holds in the target type theory. The difference is that Prf(ff) is weaker than the empty type because in the presence of an element of Prf(ff) every proposition is true, but ....

Jan Smith. The independence of Peano's fourth axiom from Martin-Lof's type theory without universes. Journal of Symbolic Logic, 53(3), 1988.

....hereafter called the scheme. We restrict ourselves to this scheme rather than making free use of Coquand s pattern matching [Coq92] or appealing to the use of universes, as explained in [NPS90] The result is a very limited theory, which admits a proof irrelevant model in the style of Smith [Smi88], and consequently consistency becomes elementary. It is sometimes impossible to prove intuitively true statements but these situations can often be solved by giving the statements appropriate reformulations. 3.1 An extension to type theory The formal system used to prove Higman s lemma is ....

....also the usual definition of propositional equality. In large parts of the proof in Section 4 we avoided using equality. However, eventually its inclusion became necessary. 3. 2 Limitations of the formal system The theory presented in Section 3 admits a proof irrelevant model like the one given in [Smi88]. This model provides an elementary proof of consistency. It also shows that many intuitively true statements are not provable. Such statements appeared very frequently during the development of the proof of Higman s Star (S (A; A)Set; w 1 w 2 List(A) Set starb Star(S, starm ....

J. Smith. The Independence of Peano's Fourth Axiom from Martin-Lof's Type Theory without Universes. Journal of Symbolic Logic, 53:840--845, 1988.

....developed more slowly and is still incomplete. There is a trivial model got by interpreting types as either the empty or one point set. While from a proof theoretic view there may be some use in this when the one point set represents true and the empty set false (e.g. to prove consistency as in [Smi88]) it is clearly inadequate as a model of polymorphism. In essence, the difficulty of providing nontrivial models arises from the impredicative nature of the calculus; in the abstraction of a universal type Piff:oe the type variable ff is understood to range over all types including the universal ....

J. Smith. The independence of peano's fourth axiom from Martin-Lof's type theory without universes. Journal of Symbolic Logic, 53:840--845, 1988.

....F (n) Set(natrec(n; b ; x; y) b T) where natrec is the recursion operator on the set of natural numbers. Assuming Id(N; 0; 1) it is easy, using tt 2 F (1) to show that F (0) is nonempty. Since F (0) we then obtain Id(N; 0; 1) i.e. by definition, Id(N; 0; 1) In Smith [22] it is shown that in type theory without a universe, no negated equalities at all can be proved. 4 The logical framework The main reason to introduce a type level, more basic than the level of sets, is to have a framework in which sets can be introduced by simple declarations. This is important ....

Jan M. Smith. The Independence of Peano's Fourth Axiom from Martin-Lof's Type Theory without Universes. Journal of Symbolic Logic, 53(3), 1988.

.... = S(Bool U ) Bool Iszero(succ(0) S(fg U ) fg Martin L#f s Type Theory 33 and therefore true 2 Bool = Iszero(0) subst(x; true) 2 Iszero(succ(0) fg Finally, we have the element we are looking for (Id(N; 0; succ(0) fg; x]subst(x; true) 2 Id(N; 0; succ(0) fg It is shown in Smith [37] that without a universe no negated equalities can be proved. 6 ALF, an interactive editor for type theory At the department of Computing Science in G#teborg, we have developed an interactive editor for objects and types in type theory. The editor is based on direct manipulation, that is, the ....

Jan M. Smith. The Independence of Peano's Fourth Axiom from Martin-L#f's Type Theory without Universes. Journal of Symbolic Logic, 53(3), 1988.

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