it relies on the timeless laws of physics and chemistry to produce new phenomena that transcend those laws without violating them. In the next four sections we discuss in more detail the role of time in the treatment of paradox, incompleteness, uncertainty, and emergence in order to justify the necessity of including natural time itself rather than formal time in our explanatory systems.

Paradox
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By the end of the 19th century the desire to show that mathematics is certain had led to Hilbert’s programme to show that mathematics could be recast as a syntactical system, one in which all operations were strictly “mechanical” and semantics played a minimal role. In this spirit, Frege, Russell and Whitehead reduced several branches of mathematics to logic, thus apparently justifying the belief that the programme was feasible. However, Russell’s work produced a paradox in set theory that ultimately raised doubts about it. Russell’s paradox asks if the set of all sets that do not contain themselves contains itself. Paradoxically, if it doesn’t, then it does; if it does, then it doesn’t. Several solutions have been proposed to rid logic of this paradox, but there is still some debate as to whether any of 