[FOM] Inconsistent systems.

A.P. Hazen a.hazen at philosophy.unimelb.edu.au
Sun Oct 1 23:42:55 EDT 2006

Bill Taylor makes a:
>A brief inquiry.
>In the Lucas/Penrose thread, Panu Raatikainen said:
>>  such distinguished logicians as Frege, Curry, Church, Quine, Rosser
>>  and Martin-Lof have seriously proposed... theories that... turned out
>>  to be inconsistent.
>I am aware of the cases of Frege and Quine, but not the others.

------Church published "A set of postulates for the foundations of 
logic" in "Annals of Mathematics" vol. 33 (1932): inconsistent 
system.  (The lambda calculus was a consistent fragment.)  Revised 
version, ibid., vol. 34 (1933): inconsitent again.  Finally "A proof 
of freedom  from contradiction," in "P.N.A.S." vol. 21 (1935), 
proving that he had  finally got it right.  Gödel reviewed all three 
papers,  and there  is a BIT more information in Kleene's 
introduction to Gödel's reviews in vol. I of G's "Collected Works."
     ((((The final, consistent, version seems to me to bear comparison
	with Myhill's idea  of "levels  of implication": cf. Myhill's
	articles in "Synthese" vol. 60 (1984) and in the Fitch festsschrifft,
	"The Logical  Way of Doing Things" (I think that's the title) ed.
	by  A.R. Anderson, R.B. Marcus &....))))
------Curry had presented something inconsistent  based on 
combinators; it's  inconsistency (and that of Church's original 
version) was proven by Kleene and Rosser in "The inconsistency of 
certain formal logics," in "Annals of Mathematics" vol. 36 (1935); 
Curry's own paper of that title, in  "JSL" vol. 7 (1942), extracts 
the logical coreof their argument from the combinatory machinery: the 
Curry Paradox.  But see notes in Curry's "Combinatory  Logic."
------Matin-Löf at some stage proposed  an extension of his type 
theory, with something like a "type of types": this fell to what is 
known  as  Girard's Paradox.
------Rosser  I'm not sure of.

Allen Hazen
Philosophy Department
University of Melbourne

More information about the FOM mailing list