Re: arithmetic in ZF


Bhupinder Singh Anand wrote:

> As I argue elsewhere, Goedel's reasoning
> can be taken to establish

Can, only if you're stupid enough to keep beating
your head against that particular stone wall.
This has been refuted 69 times now.

> that we can constructively, and in an
> intuitionistically unobjectionable
> manner, establish that an arithmetical
> relation R(n) holds for any
> given natural number n,

No, you can't, and more to the point, THIS
IS WHY PEOPLE SHOULD NOT attempt this AT HOME!
By At Home, I mean, IN NATURAL LANGUAGE!
The point being that YOU HAVE GOT YOUR QUANTIFIERS
BACKWARDS.

GIVEN ANY NATURAL NUMBER n, we can establish
intuitionistically and constructively that
R(n) holds, i.e., that n does not encode a
proof of a contradiction from PA.
This IS NOT the same as saying that we
can establish intuitionistically that
"R(n) holds for any natural number n"!
We CAN establish all the INDIVIDUAL pieces
constructively but there is NOTHING intuitionistic
or constructive about calling INFINITELY many
int/con tasks ONE int/con task! Intuitionistic
and constructive logic are every bit as bereft
of INfinitary inference rules as classical!

> but that there is no algorithmic way of
> verifying this assertion

In that case, there was no algorithmic way
OF DOING IT IN THE FIRST PLACE. There are
infinitely many DIFFERENT algorithms for confirming
R(n) but there is no ONE algorithm that does them
all.

> (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).

That loop is NOT verifiable for ALL n by any particular
TM! For every n, there are ALL 3 of: a) a TM that
decides R(n) (This is trivial when you think about it;
either the TM that always/immediately halts false or the
one that always/immediately halts true MUST be right;
this is why applying TMs to individual problems as
OPPOSED to families of problems is silly to begin with);
b) a TM that verifies that some-TM-other-than-(a) Loops
INSTEAD OF deciding R(n), and c) a TM that loops while
trying(and failing) to determine (b).

Everybody KNOWS already that every INDIVIDUAL
finite n can be treated constructively. THE ONLY
thing that matters is whether you can have ONE FINITE
thing handling INFINITELY MANY DIFFERENT infinite
cases, and there is NO interpretation of ANYthing
that Godel says that suggests that this is possible when
the axiom-set is PA and the problems are proofs of non-
contradiction.
>
> On this view,

WHAT view?
You have not coherently articulated any 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,

This is IDIOTIC.
You canNOT HOPE to generalize over "formal languages"
in general! ANYthing can be a formal language! IF we
are talking about classical first-order languages (which
is SURELY the case you BETTER get right FIRST before
moving on to anything more complicated), ALL predicates
are well-defined! What makes something a predicate is
PURELY syntactic! If the predicate is NAMED then it
is defined!

> 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,

This is, I repeat, IDIOTIC: whether the PREDICATE
is well-defined has ABSOLUTELY NOTHING to do with
any possible DOMAIN OR INTERPRETATION! You can vary
the domain and the interpretation ARBITRARILY! IF the
predicate was well-defined then it will NOT STOP being
so JUST Because you moved to a different domain!
If it was NOT well-defined then it will not START being
so just because you decided to think about it under
a different interpretation!

> there is always an effective
> method for determining whether P(s) holds
> or not in M for the
> interpreted predicate P.

This surely has a HELL Of a lot more to do with M
than it does with 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.

That is idiotic.
I don't know what Priest was smoking but that
is simply incoherent, unless he is defining
"interpretation" a hell of a lot more narrowly
than everybody else. ABSOLUTELY EVERYTHING
can be an element of the domain under SOME interpretation.
Absolutely NOTHING can satisfy the axioms of L under
ANY interpretation: OBVIOUSLY, NO MATTER WHAT an element
is, I can define an interpretation over a domain of
those elements that DOESN'T satisfy L, unless L
is tautologus (i.e. empty).

> 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.)

Prima facie, the issue is that you still haven't
gotten it through your head that interpretations
are unconstrained. All this talk about domains is
irrelevant in this context.

>
> 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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