# Lógos and Máthēma 2

## Studies in the Philosophy of Logic and Mathematics

#### Series:

## Roman Murawski

# The Status of Church’s Thesis (co-author: Jan Woleński)

# The Status of Church’s Thesis

## Extract

Co-authored by Jan Woleński

Thus (CT) identifies the class of effectively computable or calculable (we will treat these two categories as equivalent) functions with the class of partially recursive functions. This means that every element belonging to the former class is also a member of the latter class and reversely. Clearly, (CT) generates an extensional co-extensiveness of effective computability and partial recursivity. Since we have no mathematical tasks, the exact definition of recursive functions and their properties is not relevant here. On the other hand, we want to stress the property of being effective computable, which plays a basic role in philosophical thinking about (CT).31

A useful notion in providing intuitions concerning effectiveness is that of an algorithm. It refers to a completely specified procedure for solving problems of a given type. Important here is that an algorithm does not require creativity, ingenuity or intuition (only the ability to recognize symbols is assumed) and that its application is prescribed in advance and does not depend upon any empirical or random factors. Moreover, this procedure is performable in a finite number of steps. Thus a function f ∶ Nn → N is said to be effectively computable (briefly: computable) if and only if its values can be computed by an algorithm. Putting this in other words: a function f ∶Nn →N is computable if and only if there exists a mechanical method by which for any n-tuple (a1, . . . , an) of arguments, the value f (a1, . . . , an) can...

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