Inspired by the comments to  [this](https://mathoverflow.net/questions/30874/arithmetic-fixed-point-theorem) 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.



What is the history of the Y-combinator?
The paper [History of Lambda-calculus and combinatory logic](http://www.users.waitrose.com/~hindley/SomePapers_PDFs/2006CarHin,HistlamRp.pdf) 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.


