Re: incompleteness and inconsistency

Peter_Smith wrote:
I understand that you don't necessarily agree with Kreisel, but what
can one say in behalf of the idea induction on infinite ordinals as
finitistic? I don't understand how we can be finitistic using induction
on infinite ordinals. (And I don't necessarily disagree with the idea
that induction on infinite ordinals can be finitistic; I'm just saying
I don't understand why one would think it is finitistic.)

Well, it's not so daft as it sounds, if you recall the links between
induction over small ordinals and the use of multiple recursions. And
you can see why someone might think that definitions by simultaneous
recursion are finitistically kosher. (*If* I remember, the book by
Epstein and Carnielli has something quite accessible about this.)

Ah, so induction over certain ordinals is in some sense "reducible"
(for lack of my vocabulary here) to definition by simultaneous
recursion (on omega?). I can see that one might bring that in to
vindicate induction on certain ordinals.

I'll look for that book. I'll understand this better as I study more of
it and especially when I get a better grasp of what makes epsilon0 so
special.

> Isn't some infinitistic set theory needed even to FORMULATE the
language of PRA? .....
I mean, just to set up the axioms of PRA
don't we have to do a fair amount in the subject of recursive
functions, which requires having the set of natural numbers at our
disposal?

But "having the natural numbers at our disposal" is surely not the same
as having *infinitistic* set theory at our disposal. (If might help to
recall the folk-law result that Peano Arithmetic is equivalent to the
theory of hereditarily *finite* sets.)

But not just having natural numbers; don't we rely on having the SET of
natural numbers when we make our definition of 'is a recursive
function'? And maybe other set theory involving infinite sets
throughout the study of recursive functions that is prerequisite to
setting up PRA?

MoeBlee

.

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