# Scott on the consistency of the lambda calculus

I have twice heard it attributed to Dana Scott that he said something to the effect that the consistency of the lambda-calculus was an accident.

Does anyone have a reasonable-sounding source for this? I find it hard to believe that Scott would talk about the Church-Rosser theorem in this way; I guess that this a mangling of something else he said, or some context is hidden.

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## 1 Answer

You could ask him directly, but the story he told me was that he was working on domain theory because he wanted to give a denotational semantics of typed lambda calculus, or more generally typed programming languages. (He had been telling people they should design typed languages, rather than untyped ones, and so he wanted to show how a mathematical theory of typed programming languages would work.) But it turned out that his theory of domains also provides a model of the untyped lambda calculus. In this sense it was an accident.

I also asked him once why he thought it was important to give a model of the untyped lambda calculus when it had been known by the Church-Rosser theorem the calculus was consistent. I cannot reporoduce the exact answer, but in effect he said that it was important to understand what models of a theory looked like, not just that it was consistent. I think this reveals a certain "semantic" view of mathematics.

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Which suggests "domain-model-ability" is the property in question, and not "consistency". I think your suggestion is sound, and I probably should ask directly. The obvious advantage of domain models over, say, Böhm-tree models is that they allow you to solve fixpoint equations, and so are useful in the semantics of programming languages. Which might point to the concrete influence of Strachey in his choice of research goals, rather than a high-level matter of perspective. – Charles Stewart Mar 1 '10 at 14:53
There's also the philosophical angle, in that the asymmetry of observability of termination induces the Sierpinski topology on the 2-point space. A great deal of basic domain theory can be derived from that intuition. – Neel Krishnaswami Mar 1 '10 at 15:19
Professor Scott always mentions Strachey when he speaks about the history of domain theory. – Andrej Bauer Mar 1 '10 at 22:15