We present the information-theoretic incompleteness theorems that arise in a theory of program-size complexity based on something close to real LISP. The complexity of a formal axiomatic system is defined to be the minimum size in characters of a LISP definition of the proofchecking function associated with the formal system. Using this concrete and easy to understand definition, we show (a) that it is difficult to exhibit complex S-expressions, and (b) that it is difficult to determine the

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... of the LISP halting probability Ω LISP . We also construct improved versions Ω LISP and Ω LISP of the LISP halting probability that asymptotically have maximum possible LISP complexity.