What is the history of the Y-combinator? Inspired by the comments to  this question, I wonder if someone can explain the history of the fixed point combinator (often called the Y combinator) in lambda calculus.  
Where did it first appear?  Was it directly inspired by the Arithmetic Fixed Point Theorem?  The two are very similar in spirit.  
Based on the dates of Church's introduction of lambda calculus and Goedel's incompleteness theorem, it seems to me the Arithmetic Fixed Point Theorem must have come first.
 A: Turing's 1937 paper (of just one page!) which defines $\Theta$, his own fixed point combinator, refers to a 1936 paper by Kleene in which a "function L with a property similar to the essential property of $\Theta$" is defined.
References:
A M Turing, The P function in $\lambda$-K conversion, Journal of Symbolic Logic Vol 4 No 2 December 1937 p. 164
S C Kleene, $\lambda$ definability and recursiveness, Duke Mathematical Journal, Vol 2 1936 p. 346
A: Although I don't have historical information, let me comment on the suggestion that the $Y$ combinator may have been inspired by the arithmetical fixed-point theorem.  I think a more likely inspiration would be Russell's paradox.  If you think of application (of functions to arguments) as analogous to membership in sets (i.e., think if $x(y)$ as analogous to $y\in x$), then the $Y$ combinator builds a fixed point $Yf$ for a function $f$ in the same way that Russell built a fixed-point for negation.  
A: The paper History of Lambda-calculus and combinatory logic by F. Cardone and J.R. Hindley is a good starting point for answering such a question, and many others (it has 38 pages of bibliography). There’s a brief account on fixed-point combinators on page 8, although it doesn’t seem to settle the question completely.
