Results 1 
3 of
3
Parametricity as a Notion of Uniformity in Reflexive Graphs
, 2002
"... data types embody uniformity in the form of information hiding. Information hiding enforces the uniform treatment of those entities that dier only on hidden information. ..."
Abstract

Cited by 11 (3 self)
 Add to MetaCart
data types embody uniformity in the form of information hiding. Information hiding enforces the uniform treatment of those entities that dier only on hidden information.
Step By Recursive Step: Church's Analysis Of Effective Calculability
 BULLETIN OF SYMBOLIC LOGIC
, 1997
"... Alonzo Church's mathematical work on computability and undecidability is wellknown indeed, and we seem to have an excellent understanding of the context in which it arose. The approach Church took to the underlying conceptual issues, by contrast, is less well understood. Why, for example, was "Ch ..."
Abstract

Cited by 11 (0 self)
 Add to MetaCart
Alonzo Church's mathematical work on computability and undecidability is wellknown indeed, and we seem to have an excellent understanding of the context in which it arose. The approach Church took to the underlying conceptual issues, by contrast, is less well understood. Why, for example, was "Church's Thesis" put forward publicly only in April 1935, when it had been formulated already in February/March 1934? Why did Church choose to formulate it then in terms of G odel's general recursiveness, not his own #definability as he had done in 1934? A number of letters were exchanged between Church and Paul Bernays during the period from December 1934 to August 1937; they throw light on critical developments in Princeton during that period and reveal novel aspects of Church's distinctive contribution to the analysis of the informal notion of e#ective calculability. In particular, they allow me to give informed, though still tentative answers to the questions I raised; the char...
possible and for his help with this paper.
"... In the year before Zermelo published his proof of the wellordering principle from AC, the renowned Cambridge mathematician G. H. Hardy published a proof that there is an uncountable wellorderable subset of the real line [1], [2]. Hardy’s technique was surprisingly modern: he used a ladder system o ..."
Abstract
 Add to MetaCart
In the year before Zermelo published his proof of the wellordering principle from AC, the renowned Cambridge mathematician G. H. Hardy published a proof that there is an uncountable wellorderable subset of the real line [1], [2]. Hardy’s technique was surprisingly modern: he used a ladder system on ω1 to build an uncountable set of sequences of natural numbers wellordered by the preorder < ∗ of eventual domination. We now know that some form of the axiom of choice (AC) is needed for this, and even that the existence of Hardy’s uncountable set does not imply the existence of a ladder system on ω1 just assuming ZF. Definition 1. Given a limit ordinal α, a ladder at α is a strictly ascending sequence of ordinals less than α whose supremum is α. Given an ordinal γ, a ladder system on γ is a family {Lα: α ∈ γ, α is a limit ordinal of countable cofinality} where each Lα is a ladder at α. Theorem 1. In ZF, the following axioms are equivalent: 1. There is a ladder system on ω1. 2. There is a Hausdorff gap. 3a. There is a wellorderable special Aronszajn tree. 3b. There is a wellorderable special tree of height ω1. 3c. There is an Rspecial tree T of height ω1 with a choice function for the levels of