A correspondence between Martin-löf type theory, the ramified theory of types and pure type systems

In Russell''s Ramified Theory of Types RTT, two hierarchical concepts dominate:orders and types. The use of orders has as a consequencethat the logic part of RTT is predicative.The concept of order however, is almost deadsince Ramsey eliminated it from RTT. This is whywe find Church''s simple theory of types (which uses the type concept without the order one) at the bottom of the Barendregt Cube rather than RTT. Despite the disappearance of orders which have a strong correlation with predicativity, predicative logic still plays an influential role in Computer Science.An important example is the proof checker Nuprl, which is basedon Martin-Löf''s Type Theory which uses type universes. Those type universes,and also degrees of expressions in AUTOMATH, are closely related toorders. In this paper, we show that orders have not disappeared from modern logic and computer science, rather, orders play a crucial role in understanding the hierarchy of modern systems. In order to achieve our goal, we concentrate on a subsystem of Nuprl.The novelty of our paper lies in: (1) a modest revival of Russell's orders, (2) the placing of the historical system RTT underlying the famous Principia Mathematica in a context with a modern system of computer mathematics (Nuprl) and modern type theories (Martin-Löf''s type theory and PTSs), and (3) the presentation of acomplex type system (Nuprl) as a simple and compact PTS.
Keywords Nuprl  orders  Pure Type Systems  types
Categories (categorize this paper)
Reprint years 2004
DOI 10.1023/A:1011286100450
 Save to my reading list
Follow the author(s)
Edit this record
My bibliography
Export citation
Find it on Scholar
Mark as duplicate
Request removal from index
Revision history
Download options
Our Archive

Upload a copy of this paper     Check publisher's policy     Papers currently archived: 30,370
Through your library
References found in this work BETA

No references found.

Add more references

Citations of this work BETA
Types in Logic and Mathematics Before 1940.Fairouz Kamareddine, Twan Laan & Rob Nederpelt - 2002 - Bulletin of Symbolic Logic 8 (2):185-245.

Add more citations

Similar books and articles
Added to PP index

Total downloads
15 ( #322,972 of 2,193,784 )

Recent downloads (6 months)
1 ( #290,983 of 2,193,784 )

How can I increase my downloads?

Monthly downloads
My notes
Sign in to use this feature