Question

This is really a second part to my first question, but I felt that this was different enough from the first part that it merited its own question.

So, using Church numerals, we define

$3 = {\lambda} f. {\lambda}x.f(f(f(x)))$,

and

$4 = {\lambda} f. {\lambda}x.f(f(f(f(x))))$.

We can then add with an expression like

$3\ g\ (4\ g\ z)$

And this reduces to:

$(g (g (g (g (g (g (g\ z)))))))$.

But, of course, this is not how we would define $7$ in the scheme above. $7$ would be

${\lambda}g.{\lambda}z.(g (g (g (g (g (g (g\ z)))))))$.

Why is it still legitimate to call the application $3\ g\ (4\ g\ z)$ "7" when we can no longer perform functions with it?

No correct solution

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