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.