Re: primitive recursive: obsolete?

Aatu Koskensilta wrote:
On 2008-05-09, in sci.logic, MoeBlee wrote:
Do you have any thoughts on my question about the notion of
'finitistic'. Why do people limit 'finitistic' to primitive recursive?
Why not take finitistic as anything total recursive?

Because not all total recursive functions are recognisable as such by
finitistic principles.

The restriction to primitive recursive functions stems from the fact
that the principle of definition by primitive recursion is equivalent
to the principle of induction.

With my limited understanding I'd guess so.

Solely on basis of our basic understanding of the naturals as what
one obtains from 0 by repeatedly applying the "add one"-operation

So what happens to the prime numbers and the multiplication-operation?
Naturally, what's perceived as the natural numbers is more than just
a model of Presburger Arithmetic!

the principle of induction and the principle of definition by primitive
recursive induction are equally compelling.

As you've noted above the two are equivalent; if one is compelling, so would
be the other. But "compelling" in what sense?

This is the reason for taking primitive recursive
arithmetic as the canonical formalisation for finitism; if we further
allow that properties definable from primitive recursive properties by
means of the usual logical operations of first-order logic --
including unrestricted quantification --, in effect accepting the
totality of the naturals as something determinate, we get PA.

But at what cost are you willing to pay for accepting such "totality of
the naturals as something determinate"? For example, are you willing
to accept the distinct possibility that GC is arithmetically true, but
can't be arithmetically proven, or can't be proven in any theory
"as strong as arithmetic"? And suppose you're willing to pay such a
steep price, what would that even mean to FOL reasoning?


.

Relevant Pages

  • Re: primitive recursive: obsolete?
    ... that the principle of definition by primitive recursion is equivalent ... to the principle of induction. ... arithmetic as the canonical formalisation for finitism; ...
    (sci.logic)
  • Re: primitive recursive: obsolete?
    ... Because not all total recursive functions are recognisable as such by ... that the principle of definition by primitive recursion is equivalent ... to the principle of induction. ... arithmetic as the canonical formalisation for finitism; ...
    (sci.logic)
  • Re: primitive recursive: obsolete?
    ... to the principle of induction. ... arithmetic as the canonical formalisation for finitism; ... the principle of primitive recursion and induction ...
    (sci.logic)
  • Re: primitive recursive: obsolete?
    ... to the principle of induction. ... arithmetic as the canonical formalisation for finitism; ... the principle of primitive recursion and induction ...
    (sci.logic)