## 7.2.2 Bottom and meta-equality

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A program may loop indefinitely. As an example, take the factorial function:

If a computer is asked to compute the value of the expression then the computer will loop indefinitely. For that reason we shall say that the value of is where [bottom] reads bottom.

We shall say that is meta-equal to and write . Meta-equality is equality in the theory behind Logiweb. The theory behind Logiweb is called map theory.

The base page defines another equality named . We shall refer to as computable equality. As an example, computable equality appears in the definition of the factorial function above.

Readers with a mathematical background should note this: meta-equality is a genuine equality relation with all the properties one expects from equality, including the rule of substitution of equals. Meta-equality lives in the meta-theory, which is map theory. In contrast, computable equality is an equivalence relation over a certain domain. Computable equality lives in Logiweb in the sense that the Logiweb Abstract Machine can compute it.

In short, is equality and is an equivalence relation, which may confuse mathematicians.

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Copyright © 2010 Klaus Grue, GRD-2010-01-05