Background

At the heart of CT is the notion of an algorithm, characterized in traditional fashion by Mendelson as

an effective and completely specified procedure for solving a whole class of problems. $\ldots$ An algorithm does not require ingenuity; its application is prescribed in advance and does not depend upon any empirical or random factors. ([27], p. 225)

An effectively computable function is then said to be the computing of a function by an idealized ``worker" or ``computist" following an algorithm.⁴ (Without loss of generality, we can for present purposes view all functions as taking natural numbers into natural numbers; i.e., for some arbitrary f, $f : \mbox{{\bf N}} \rightarrow \mbox{{\bf N}}$). CT also involves a more formal notion, typically that of a so-called Turing-computable function (or, alternatively, and equivalently, that of a $\mu$-recursive function, or, $\ldots$). Mendelson employs Turing's approach, and Turing machines will be familiar to most readers; we'll follow him: a function $f : \mbox{{\bf N}} \rightarrow \mbox{{\bf N}}$ is Turing-computable iff there exists a TM M which, starting with n on its tape (perhaps represented by n $\mid$s), leaves f(n) on its tape after processing. (The details of the processing are harmlessly left aside for now.) Given this definition, CT amounts to

CT

A function is effectively computable if and only if it's Turing-computable.³

Now what exactly is Mendelson's aim? He tells us:

Here is the main conclusion I wish to draw: it is completely unwarranted to say that CT is unprovable just because it states an equivalence between a vague, imprecise notion (effectively computable function) and a precise mathematical notion (partial-recursive function). ([27], p. 232)

From this it follows that Mendelson's target is the traditional argument for the unprovability of CT. And the line of reasoning he means to attack runs as follows.

Arg1
	EQU	If some thesis T states an equivalence between a vague, imprecise notion and a precise, mathematical notion, T is unprovable.
.$^\cdot$ .	(1)	If CT states an equivalence between a vague, imprecise notion and a precise, mathematical notion, CT is unprovable.
	(2)	CT states an equivalence between a vague, imprecise notion and a precise, mathematical notion.
.$^\cdot$ .	(3)	CT is unprovable.