Re: arithmetic in ZF

On May 2, 4:34 pm, Paul Holbach wrote:

PH>> Is Priest really that naive ...? <<PH

Paul
===
Not necessarily. It may simply reflect the possibility that, first,
Priest's hereticism could be confined to reasoning within the ambit of
standard interpretations of classical theory; and, second, that such
interpretations are yet to definitively address the issue of whether
the terms 'non-algorithmic' and 'non-constructive' are to be treated as
synonymous.

As I argue elsewhere, Goedel's reasoning can be taken to establish that
we can constructively, and in an intuitionistically unobjectionable
manner, establish that an arithmetical relation R(n) holds for any
given natural number n, but that there is no algorithmic way of
verifying this assertion (in other words, any Turing machine that
computes R(x), treated as a Boolean function, will go into a
verifiable, non-terminating, loop for some natural number n).

On this view, the particular argument between Priest and Cartwright,
Tennant, et al, becomes vacuous; we simply define any predicate [P] of
a formal language L as well-defined if, and only if, given any element
s in a domain D over which the variables of the language are
well-defined under an interpretation M, there is always an effective
method for determining whether P(s) holds or not in M for the
interpreted predicate P.

(Note that, given an interpretation M of a formal theory L, we can
always - according to a reasonable reading of Priest - define the
domain D as consisting of all elements that satisfy the axioms of L
under any interpretation. The fact that some D-elements are dissimilar
may need qualification when defining satisfaction in M but, prima
facie, this should not lead to any irresolvable issues.)

The existence of functions in real and complex analysis that are
continuous, but not uniformly continuous, and of sequences that are
convergent, but not uniformly convergent, indicates that such a
definition of a well-defined predicate can be mathematically
constructive, and intuitionistically unobjectionable.

We can then define [P] as well-defining a mathematical object (i.e., a
syntactic totality), say {x: P(x)}, in M if, and only if, there is an
algorithm (effective uniform method) for identifying elements such that
P(x) holds in M.

In the absence of such an algorithm, we may still have that, for any s
in D, it is effectively decidable whether or not P(s) holds in M, but
we can no longer treat all such elements as a syntactic totality that
can be referred to within, or by, the language without inviting
inconsistency.

On this view, the 'Domain' referred to by Priest need not be a
syntactic totality in the sense of being a mathematical object as
defined above. However, it can still be referred to, albeit loosely, as
a semantic totality for expressive, and philosophical, purposes, so
long as we do not associate any algorithmic properties with it without
a formal proof.

Regards,

Bhup

.

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