[ozb]

You can implement the "nonstandard arithmetic" suggestion using bignum integers backed by an infinite tape (subject to availability of said infinite tape). Finite integers have a Halt symbol, non-finite ones simply don't. Arithmetic on non-finite integers is not computable, but individual digits generally are. Any finite integer is computably less than any non-finite integer. "Less than" between two non-finite integers is not generally computable, therefore not defined. "Not equals" is semidecidable, so generally all "not equals" statements between two non-finite integers are defined, but "equals" mostly isn't. printf on a non-finite integer will simply print out infinite digits one by one. You can also define and generate non-finite integers from any computable sequence of integers. size(void*) can be defined as eg 1111111... (Repeating forever, in an arbitrary base).

If you demand that you can always computably do arithmetic on size_t's, allow storing arbitrary arithmetic or even logical expressions in your bignum integers, and call those bignum as well. Define "less than" on infinite integers based on "alphabetical" order. Then the only thing that is non-computable is (in)equality between two expressions for which non-finite-ness is unprovable under First order logic. Given Godel's Completeness (note, not Incompleteness) Theorem, that should probably meet the definition of a C implementation, though I haven't read the standard.