Re: primitive recursive: obsolete?

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. Solely on basis of our basic
understanding of the naturals as what one obtains from 0 by repeatedly
applying the "add one"-operation the principle of induction and the
principle of definition by primitive recursive induction are equally
compelling. 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.

--
Aatu Koskensilta (aatu.koskensilta@xxxxxxxxx)

"Wovon man nicht sprechen kann, daruber muss man schweigen"
- Ludwig Wittgenstein, Tractatus Logico-Philosophicus
.

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; ... totality of the naturals as something determinate, ...
    (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)
  • 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)
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