TITLE: What is the history of the Y-combinator?
QUESTION [17 upvotes]: 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.

REPLY [12 votes]: 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.