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    <p>The problem with lambda expressions is that they cannot be interpreted as functions in a mathematical sense, i.e. mappings from one set to another.</p> <p>The reason is the cardinality of the set of functions from a set <code>A</code> on itself is always larger than the cardinality of <code>A</code>, so not all functions from <code>A</code> to <code>A</code> can be an element of <code>A</code>. That is, there is a function <code>f: A -&gt; A</code> for which the expression <code>f(f)</code> does not make sense.</p> <p>This is like the "set of all sets not containing itself", which does not make sense logically.</p> <p>JavaScript is not a model of the lambda calculus.</p> <p>The problem with your example is that</p> <pre><code>(lambda x.g(x x)) (lambda x.g(x x)) </code></pre> <p>should be equivalent to</p> <pre><code>g((lambda x.g(x x)) (lambda x.g(x x))) </code></pre> <p>but it is not in your JavaScript program where <code>g</code> is the indicator function of <code>0</code>.</p> <p><code>x x</code> is always <code>undefined</code>. Hence the first line evaluates to <code>g (undefined) = 0</code>. The second line evaluates to <code>g (g (undefined)) = g (0) = 1</code>. This means that your JavaScript model of the lambda calculus is, in fact, not really a model.</p> <p>Since for each non-empty set <code>D</code> there is a function from <code>D</code> to <code>D</code> without a fixed point, obviously there can be no model of the lambda calculus. I think it should be even possible to prove that there cannot be an implementation of the Y-combinator in any Turing-complete language.</p>
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    1. COI would amend your last paragraph. Just because |D^D| > |D| in a set-theoretical sense does not mean lambda calculus has no models. See http://mathoverflow.net/questions/16752/scott-on-the-consistency-of-the-lambda-calculus
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