We assume that ψ and χ preserve intuitability. Then we have for any numerical term s the inference from *CI*(*s*) to *CI*[ψ(*s*)] and by replacement to *CI*[φ(0, *s*)]. Suppose we can infer *CI*[φ(*b, s*)] from *CI*(*s*). Then since χ
preserves intuitability, we have *CI*[χ(*b*, φ(*b, s*))],^{43} and hence by replacement *CI*[φ(*Sb, s*)]. Hence we have by induction ∀x{*Nx* → *CI*[φ(*x, s*)]}. By
UII, for any term *t* such that *Nt* and *CI*(*t*), *CI*[φ(*t, s*)]. That is, φ preserves
intuitability.

But this is enough to show that *CI*(*t*) holds for any term of PRA.

If we allow more complicated inductions involving the predicate *CI*,
we can prove that functors defined by more complex recursions preserve
intuitability, for example Ackermann-type functors introduced by nested
double recursion. Particularly since these proofs will still be intuitionistic,
it is not obvious that they should be ruled out. But that they are possible
does indicate that the mathematical possibility of intuition is being conceived rather generously.

Does this give rise to an argument for Hilberts thesis? Suppose *s* and *t*
are such that I can intuit a string of length *s*, and I can intuit a string of
length *t*. Then, surely, by comparing them I can determine whether *s* = *t* is
true. It thus seems that any closed formula of PRA can be decided in an
intuitive way. But what follows about intuitive knowledge of generalizav
tions? The most this argument shows is that if a formula of PRA is true for
all values of its variables, then in each particular case this can be known
intuitively. But this does not yield intuitive knowledge of the generalization. Even if this obstacle can be circumvented, I see no way to get around
the fact that what these considerations yield is the *possibility* of intuitive
knowledge, according to a rather liberal kind of possibility. But Hilberts
thesis concerns actual intuitive knowledge, at least given a proof in PRA.
Even if the questions about the ideas involved can be resolved, the case for Hilberts thesis does not seem to be materially advanced. More generally,
the relevance to questions about what we know intuitively of questions
about what intuitions are possible in principle is not clear.

**APPENDIX**

For the conventional formulation of PRA, we can assume that the logic allowed is
minimal propositional logic, with ¬*A* defined as *A* → 0 = 1. For to obtain intuitionistic logic, it suffices to derive 0 = 1 → *A* for any *A*. If *A* is atomic it is an equation *s* = *t*. We introduce by primitive recursion a function symbol φ satisfying φ(0) = *s*

^{43}

*CI*(

*b*) follows from the assumption

*Nb*, since we have accepted the induction that yields ∀

*x*[

*Nx*→

*CI*(

*x*)].

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