Combinatory logic

Hi, here is a quote from this website

http://www.earlham.edu/~peters/courses/logsys/nonstbib.htm#combinatory

Combinatory logic

Logics that replace variables with functions in order to clarify
intuitive operations on variables such as substitution. Systems of
arithmetic built from combinatory logic can contain all partial
recursive functions and avoid Gödel incompleteness. <<

Is the last statement true?

.

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