Re: Why an inconsistent ZF may be desirable, and should be welcome.

From: Bhupinder Singh Anand (re_at_alixcomsi.com)
Date: 03/25/05 

Date: 25 Mar 2005 00:49:22 -0800

On Mar 16, 3:03 pm, Ross A. Finlayson wrote:

RAF>> I had misunderstood your point about the primitive and composite
recursive functions having the same representation. If they have the
same representation I think they're the same thing. <<R

Ross
===
I have not understood your reference to 'composite recursive
functions'. Do you mean arithmetical functions?

RAF>> In addressing Goedel's incompleteness results, vis-a-vis the
completeness result which I think leads to the contradiction you
address about the functions with the same representation having
different ranges due to completeness... <<RAF

Actually, the argument is not that they have different ranges. It is,
rather, about the nature of their common range, and about if, and when,
such a range can be treated as a completed totality (in other words, as
a valid 'set' in an axiomatic set theory).

I argue that although a primitive recursive relation, and an
arithmetical relation, may be instantiationally equivalent - in the
sense that both either hold, or not hold, for any substitution of
natural numbers for their free variables - we cannot always conclude
from this that their closure under quantification necessary yields two
propositions that have the same meaning under any interpretation.

To give a crude analogy, if A has only one child, B, and B has only one
child, C, then the two relations 'x is a grandchild of A', and 'x is a
child of B', either hold, or do not hold, together. However, we cannot,
reasonably, assert that the two propositions, '(E!x) x is a grandchild
of A', and '(E!x) x is a child of B', have the same meaning, even
though both are true.

(Note: '(E!x)' interprets as 'There is a unique x such that'.)

RAF>> ... basically I got to thinking that the Goedel sentence has an
infinitely long chain of supporting sentences ... <<RAF

I don't know in what sense you see the Goedel sentence as being
'supported' by an infinitely long chain of sentences.

Goedel's seminal 1931 paper is, no doubt, an extremely tough read, but
one that I feel is well worth mastering. I believe that it is still the
best - and most reliable - exposition of his reasoning, particularly
since it contains some persisting grey areas that are obscured by
subsequent expositions (including his own).

In this paper, Goedel, first, defines a formal system of Peano
Arithmetic, P, and codes the formulas of P, and finite sequences of
P-formulas, uniquely by the natural numbers.

He then defines 45 primitive recursive number-theoretic (i.e.,
involving only elementary logical concepts and the successor concept,
which is taken to be the most basic property of the intuitive natural
numbers) functions and relations from first principles in a
constructive, and intuitively unobjectionable way.

He finally constructs the primitive recursive relation ~xB(Sb(y
19|Z(y))), say Q(x, y), which, when interpreted intuitively as a
relation between natural numbers, states that:

It is not true that x is the Goedel-number of a formal proof sequence
of P-formulas, where the last member of the sequence is the P-formula
that is obtained by substituting y for the variable whose Goedel-number
is 19 in the P-formula whose Goedel-number is y.

Now, in this paper, Goedel also shows that every recursive relation is
instantiationally equivalent to some arithmetical relation (defined
only by the concepts +, * (addition and multiplication of natural
numbers), and the logical constants v (or), ~ (negation), (Ax), =
(equality).

Thus there is some arithmetical relation R(x, y) that holds if, and
only if, Q(x, y) holds.

Since R(x, y) is arithmetical, it can also be treated as a P-formula,
say [R(x, y)].

Goedel then considers the P-formula [(Ax)R(x, y)], which has some
Goedel-number, say p.

The Goedel sentence, is, then, the P-formula, [(Ax)R(x, p)].

As you can see, there is no infinite chain involved in this definition.

Further, due to their equivalence, both the arithmetical relation, R(x,
p), and the primitive recursive relation, Q(x, p) hold if, and only if:

It is not true that x is the Goedel-number of a formal proof sequence
of P-formulas, where the last member of the sequence is the P-formula
that is obtained by substituting p for the variable whose Goedel-number
is 19 in the P-formula whose Goedel-number is p.

So, the standard interpretation of the Goedel-sentence, [(Ax)R(x, p)],
translates as:

It is not true for any x that x is the Goedel-number of a formal proof
sequence of P-formulas, where the last member of the sequence is the
P-formula that is obtained by substituting p for the variable whose
Goedel-number is 19 in the P-formula whose Goedel-number is p.

In other words, the standard interpretation of the Goedel-sentence,
[(Ax)R(x, p)], translates as:

There is no formal proof sequence of P for the P-formula that is
obtained by substituting p for the variable whose Goedel-number is 19
in the P-formula whose Goedel-number is p.

Since, in Goedel's coding, the variable y has the Goedel-number 19, it
means that the standard interpretation of the Goedel-sentence,
[(Ax)R(x, p)], translates as:

There is no formal proof sequence of P for the P-formula [(Ax)R(x, p)].

Again, as you can see, the reasoning is finitary, and there is no
infinite chain involved.

RAF>> ... self-referential statements are validated by themselves
representing true statements about the objects in the realm of
discourse via tautology and the free propositional calculus, binary
truth tables over predicates, in being self-referential eternally via
self-referential induction ... <<RAF

As you can see, although the Goedel sentence can be termed as
semantically self-referential, it is not so syntactically. In his
paper, Goedel took pains to stress this distinction, and to restrict
his reasoning to syntactic arguments only.

RAF>> ... if you say ZF is inconsistent the immediate conclusion of
that statement is that ZF is inconsistent. <<RAF

Well, yes, if one keeps the explicit premises - under which the
inconsistency is derived - in mind.

My apologies for the delayed response, and thanks for your interest,
and your generous comments.

Regards,

Bhup