Basis for Lambda Functions

>>From time to time I get annoyed with lambda-functions, at least as
an approach to math foundations, on the ground that they are in fact
lacking a proper foundation themselves. Specifically, it is to be
understood that the functions involved may take other functions
as input and give out yet others as output. But it then seems always
to be glibly assumed that there are NO OTHER sorts of functions,
no "foundation functions", and worse, no foundation objects.

I find this very unsatisfactory, intuitively speaking.

Now I am aware that exactly the same sort of complaint could be made
against ZF as a foundation, in that (without urelelments) there is
nothing to start on, nothing to be members of the basic sets.
Except, OC, the empty set, which *does* in fact fit in very
neatly with the remaining ideas of set theory.

But OC most working mathies still think of there being urelements
of one sort or another, partly depending on context, as it might be
natural numbers, or strings, or reals, or even functions of those.
Then they happily go on from there, still using the full apparatus
of set theory whenever they might need it; (seldom, probably).


So my query is:- is there some basic standard "material" for lambda
functions, some basic objects that can act as inputs and outputs.
We may not need them, like we don't need urelelments, but it would be
nice to have some such things to think about, maybe less elegant,
but definitely more satisfying.

So what could/should act as a "nitty-gritty" basis for lambda fns?


Does the question even make sense? What *should* I be asking?

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Bill Taylor W.Taylor@xxxxxxxxxxxxxxxxxxxxx
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relation between A and B == function from A to P(B)
== multi-valued partial function from A to B
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