(Notations, non-standard concepts, and definitions used commonly in these investigations are detailed in this post.)

In this post I address two critical issues, as raised in private correspondence with researchers, which may illuminate some objections to Gödel’s reasoning and conclusions that have been raised elsewhere by Wittgenstein, Floyd, Putnam et al.:

(i) By Rosser’s reasoning, doesn’t simple consistency suffice for defining an undecidable arithmetical proposition?

(ii) Doesn’t Gödel’s undecidable formula assert its own unprovability?

NOTE: The following correspondence refers copiously to this paper that was presented in June 2015 at the workshop on Hilbert’s Epsilon and Tau in Logic, Informatics and Linguistics, University of Montpellier, France.

Subsequently, most of the cited results were detailed formally in the following paper due to appear in the December 2016 issue of ‘Cognitive Systems Research‘:

The Truth Assignments That Differentiate Human Reasoning From Mechanistic Reasoning: The evidence-based argument for Lucas’ Gödelian thesis‘.

A: Doesn’t simple consistency suffice for defining Rosser’s undecidable arithmetical proposition?

You claim that the PA system is \omega-inconsistent, and that Gödel’s first theorem holds vacuously. But by Rosser’s result, simple consistency suffices.

Well, it does seem surprising that Rosser’s claim—that his ‘undecidable’ proposition only assumes simple consistency—has not been addressed more extensively in the literature. Number-theoretic expositions of Rosser’s proof have generally remained either implicit or sketchy (see, for instance, this post).

Note that Rosser’s proposition and reasoning involve interpretation of an existential quantifier, whilst Gödel’s proposition and reasoning only involve interpretation of a universal quantifier.

The reason why Rosser’s claim is untenable is that—in order to interpret the existential quantifier as per Hilbert’s \epsilon-calculus—Rosser’s argument needs to assume his Rule C (see Elliott Mendelson, Introduction to Mathematical Logic, 1964 ed., p.73), which implicitly implies that Gödel’s arithmetic P—in which Rosser’s argumentation is grounded—is \omega-consistent .

See, for instance, this analysis of (a) Wang’s outline of Rosser’s argument on p.5, (b) Beth’s outline of Rosser’s argument on p.6, and (c) Mendelson’s exposition of Rosser’s argument in Section 4.2 on p.8.

Moreover, the assumption is foundationally fragile, because Rule C invalidly assumes that we can introduce an ‘unspecified’ formula denoting an ‘unspecified’ numeral into PA even if the formula has not been demonstrated to be algorithmically definable in terms of the alphabet of PA.

See Theorem 8.5 and following remarks in Section 8, pp.7-8 of this paper that was presented in June 2015 at the workshop on Hilbert’s Epsilon and Tau in Logic, Informatics and Linguistics, University of Montpellier, France.

B: As I see it, rule C is only a shortcut.

As I see it, rule C is only a shortcut; it is totally eliminable. Moreover, it is part of predicate logic, not of the Peano’s arithmetic.

Assuming that Rule C is a short cut which can always be eliminated is illusory, and is tantamount to invalidly (see Corollary 8.6, p.17 of the Epsilon 2015 paper) claiming that Hilbert’s \epsilon calculus is a conservative extension of the first-order predicate calculus.

Reason: Application of Rule C invalidly (see Theorem 8.5 and following remarks in Section 8, pp.7-8 of the Epsilon 2015 paper) involves introduction of a new individual constant, say [d], in a first-order theory K (see Mendelson 1964, p.74, I(iv)); ‘invalidly’ since Rule C does not qualify that [d] must be algorithmically computable from the alphabet of K—which is necessary if K is first-order.

Notation: We use square brackets to indicate that the expression within the brackets denotes a well-formed formula of a formal system, say K, that is to be viewed syntactically merely as a first-order string of K—i.e, one which is finitarily constructed from the alphabet of the language of K—without any reference to its meaning under any interpretation of K.

Essentially, Rule C mirrors in K the intuitionistically objectionable postulation that the formula [(\exists x)F(x)] of K can always be interpreted as:

F'(a) holds for some element a

in the domain of the interpretation of K under which the formula [F(x)] interprets as the relation F'(x).

The Epsilon 2015 paper shows that this is not a valid interpretation of the formula [(\exists x)F(x)] under any finitary, evidence-based, interpretation of K.

That, incidentally, is a consequence of the proof that PA is not \omega-consistent; which itself is a consequence of (Theorem 7.1, p.15, of the Epsilon 2015 paper):

Provability Theorem for PA: A PA formula [F(x)] is provable if, and only if, [F(x)] interprets as an arithmetical relation F'(x) that is algorithmically computable as always true (see Definition 3, p.7, of the Epsilon 2015 paper) over the structure \mathbb{N} of the natural numbers.

Compare with what Gödel has essentially shown in his famous 1931 paper on formally undecidable arithmetical propositions, which is that (Lemma 8.1, p.16, of the Epsilon 2015 paper):

Gödel: There is a PA formula [R(x, p)]—which Gödel refers to by its Gödel number r—which is not provable in PA, even though [R(x, p)] interprets as an arithmetical relation that is algorithmically verifiable as always true (see Definition 4, p.7, of the Epsilon 2015 paper) over the structure \mathbb{N} of the natural numbers.

C: If you by-pass the intuitionist objections, would all logicist and post-formalist theories hold?

If I have understood correctly, you claim that the PA system is \omega-inconsistent from an intuitionistic point of view? If you by-pass the intuitionist objections, would all logicist and post-formalist theories hold?

There is nothing to bypass—the first-order Peano Arithmetic PA is a formal axiomatic system which is \omega-inconsistent as much for an intuitionist, as it is for a realist, a finitist, a formalist, a logicist or a nominalist.

Philosophers may differ about beliefs that are essentially unverifiable; but the \omega-incompleteness of PA is a verifiable logical meta-theorem that none of them would dispute.

D: Isn’t Gödel’s undecidable formula [(\forall x)R(x, p)]—which Gödel refers to by its Gödel number 17Gen\ r—self-referential?

Isn’t Gödel’s undecidable formula [(\forall x)R(x, p)]—which Gödel refers to by its Gödel number 17Gen\ r—self-referential and covertly paradoxical?

According to Wittgenstein it interprets in any model as a sentence that is devoid of sense, or even meaning. I think a good reason for this is that the formula is simply syntactically wrongly formed: the provability of provability is not defined and can not be consistently defined.

What you propose may be correct, but for automation systems of deduction wouldn’t \omega-inconsistency be much more problematic than undecidability?

How would you feel if a syntax rule is proposed, that formulas containing numerals are instantiations of open formulas that may not be part of the canonical language? Too daring, may be?

Let me briefly respond to the interesting points that you have raised.

1. The \omega-inconsistency of PA is a meta-theorem; it is a Corollary of the Provability Theorem of PA (Theorem 7.1, p.15, of the Epsilon 2015 paper).

2. Gödel’s PA-formula [(\forall x)R(x, p)] is not an undecidable formula of PA. It is merely unprovable in PA.

3. Moreover, Gödel’s PA-formula [\neg(\forall x)R(x, p)] is provable in PA, which is why the PA formula [(\forall x)R(x, p)] is not an undecidable formula of PA.

4. Gödel’s PA-formula [(\forall x)R(x, p)] is not self-referential.

5. Wittgenstein correctly believed—albeit purely on the basis of philosophical considerations unrelated to whether or not Gödel’s formal reasoning was correct—that Gödel was wrong in stating that the PA formula [(\forall x)R(x, p)] asserts its own unprovability in PA.

Reason: We have for Gödel’s primitive recursive relation Q(x, y) that:

Q(x, p) is true if, and only if, the PA formula [R(x, p)] is provable in PA.

However, in order to conclude that the PA formula [(\forall x)R(x, p)] asserts its own unprovability in PA, Gödel’s argument must further imply—which it does not—that:

(\forall x)Q(x, p) is true (and so, by Gödel’s definition of Q(x, y), the PA formula [(\forall x)R(x, p)] is not provable in PA) if, and only if, the PA formula [(\forall x)R(x, p)] is provable in PA.

In other words, for the PA formula [(\forall x)R(x, p)] to assert its own unprovability in PA, Gödel must show—which his own argument shows is impossible, since the PA formula [(\forall x)R(x, p)] is not provable in PA—that:

The primitive recursive relation Q(x, p) is algorithmically computable as always true if, and only if, the arithmetical relation R'(x, p) is algorithmically computable as always true (where R'(x, p) is the arithmetical interpretation of the PA formula [R(x, p)] over the structure \mathbb{N} of the natural numbers).

6. Hence, Gödel’s PA-formula [(\forall x)R(x, p)] is not covertly paradoxical.

7. IF Wittgenstein believed that the PA formula [(\forall x)R(x, p)] is empty of meaning and has no valid interpretation, then he was wrong, and—as Gödel justifiably believed—he could not have properly grasped Gödel’s formal reasoning that:

(i) ‘17Gen\ r is not \kappa-provable’ is a valid meta-theorem if PA is consistent, which means that:

‘If PA is consistent and we assume that the PA formula [(\forall x)R(x, p)] is provable in PA, then the PA formula [\neg(\forall x)R(x, p)] must also be provable in PA; from which we may conclude that the PA formula [(\forall x)R(x, p)] is not provable in PA’

(ii) ‘Neg(17Gen\ r) is not \kappa-provable’ is a valid meta-theorem ONLY if PA is \omega-consistent, which means that:

‘If PA is \omega-consistent and we assume that the PA formula [\neg(\forall x)R(x, p)] is provable in PA, then the PA formula [(\forall x)R(x, p)] must also be provable in PA; from which we may conclude that the PA formula [\neg(\forall x)R(x, p)] is not provable in PA’.

8. In fact the PA formula [(\forall x)R(x, p)] has the following TWO meaningful interpretations (the first of which is a true arithmetical meta-statement—since the PA formula [R(n)] is provable in PA for any PA-numeral [n]—but the second is not—since the PA formula [(\forall x)R(x, p)] is not provable in PA):

(i) For any given natural number n, there is an algorithm which will verify that each of the arithmetical meta-statements ‘R'(1, p) is true’, ‘R'(2, p) is true’, …, ‘R'(n, p) is true’ holds under the standard, algorithmically verifiable, interpretation \mathbb{M} of PA (see \S 5, p.11 of the Epsilon 2015 paper);

(ii) There is an algorithm which will verify that, for any given natural number n, the arithmetical statement ‘R'(n, p) is true’ holds under the finitary, algorithmically computable, interpretation \mathbb{B} of PA (see \S 6, p.13 of the Epsilon 2015 paper).

9. IF Wittgenstein believed that the PA formula [(\forall x)R(x, p)] is not a well-defined PA formula, then he was wrong.

Gödel’s definition of the PA formula [(\forall x)R(x, p)] yields a well-formed formula in PA, and cannot be treated as ‘syntactically wrongly formed’.

10. The Provability Theorem for PA shows that both ‘proving something in PA’ and ‘proving that something is provable in PA’ are finitarily well-defined meta-mathematical concepts.

11. The Provability Theorem for PA implies that PA is complete with respect to the concepts of satisfaction, truth and provability definable in automated deduction systems, which can only define algorithmically computable truth.

12. The Provability Theorem for PA implies that PA is categorical, so you can introduce your proposed syntax rule ONLY if it leads to a conservative extension of PA.

13. Whether ‘daring’ or not, why would you want to introduce such a rule?

E: Consider these two statements of yours …

Consider these two statements of yours:

“(iv): p is the Gödel-number of the formula [(\forall x)][R(x, y)] of PA” and

“D(4): Gödel’s PA-formula [(\forall x)R(x, p)] is not self-referential.”

If ‘p‘ is the Gödel-number of the open formula in para (iv), and the second argument of the closed formula R in para D(4) is ‘p‘, then the second formula is obtained by instantiating the variable ‘y‘ in the first with its own Gödel-number.

So how would you call, in one word, the relation between the entire formula (in D(4)) and its second argument?

Para D(4) is an attempt to clarify precisely this point.

1. Apropos the first statement ‘(iv)’ cited by you:

From a pedantic perspective, the “relation between the entire formula (in D(4)) and its second argument” cannot be termed self-referential because the “second argument”, i.e., p, is the Gödel-number of the PA formula [(\forall x)R(x, y)], and not that of “the entire formula (in 4)”, i.e., of the formula [(\forall x)R(x, p)] itself (whose Gödel number is 17Gen\ r).

Putting it crudely, 17Gen\ r is neither self-referential—nor circularly defined—because it is not defined in terms of 17Gen\ r, but in terms of p.

2. Apropos the second statement ‘D(4)’ cited by you:

I would interpret:

Gödel’s PA-formula [(\forall x)R(x, p)] is self-referential

to mean, in this particular context, that—as Gödel wrongly claimed:

[(\forall x)R(x, p)] asserts its own unprovability in PA.

Now, if we were to accept the claim that [(\forall x)R(x, p)] is self-referential in the above sense, then (as various critics of Gödel’s reasoning have pointed out) we would have to conclude further that Gödel’s argument leads to the contradiction:

(\forall x)Q(x, p) is true—and so, by Gödel’s definition of Q(x, y)—the PA formula [(\forall x)R(x, p)] is not provable in PA—if, and only if, the PA formula [(\forall x)R(x, p)] is provable in PA.

However, in view of the Provability Theorem of PA (Theorem 7.1, p.15, of the Epsilon 2015 paper), this contradiction would only follow if Gödel’s argument were to establish (which it does not) that:

The primitive recursive relation Q(x, p) is algorithmically computable as always true if, and only if, the arithmetical interpretation R'(x, p) of the PA formula [R(x, p)] is algorithmically computable as always true over the structure \mathbb{N} of the natural numbers.

The reason Gödel cannot claim to have established the above is that his argument only proves the much weaker meta-statement:

The arithmetical interpretation R'(x, p) of the PA formula [R(x, p)] is algorithmically verifiable as always true over the structure \mathbb{N} of the natural numbers.

Ergo—contrary to Gödel’s claim— Gödel’s PA-formula [(\forall x)R(x, p)] is not self-referential (and so, even though Gödel’s claimed interpretation of what his own reasoning proves is wrong, there is no paradox in Gödel’s reasoning per se)!

F: Is the PA system \omega-inconsistent without remedy?

Is the PA system \omega-inconsistent without remedy? Is it possible to introduce a new axiom or new rule which by-passes the problematic unprovable statements of the Gödel-Rosser Theorems?

1. Please note that the first-order Peano Arithmetic PA is:

(i) consistent (Theorem 7.3, p.15, of the Epsilon 2015 paper); which means that for any PA-formula [A], we cannot have that both [A] and [\neg A] are Theorems of PA;

(ii) complete (Theorem 7.1, p.15, of the Epsilon 2015 paper); which means that we cannot add an axiom to PA which is not a Theorem of PA without inviting inconsistency;

(iii) categorical (Theorem 7.2, p.15, of the Epsilon 2015 paper); which means that if \mathbb{M} is an interpretation of PA over a structure \mathbb{S}, and \mathbb{B} is an interpretation of PA over a structure \mathbb{T}, then \mathbb{S} and \mathbb{T} are identical and denote the structure \mathbb{N} of the natural numbers defined by Dedekind’s axioms; and so PA has no model which contains an element that is not a natural number (see Footnote 54, p.16, of the Epsilon 2015 paper).

2. What this means with respect to Gödel’s reasoning is that:

(i) PA has no undecidable propositions, which is why it is not \omega-consistent (Corollary 8.4, p.16, of the Epsilon 2015 paper);

(ii) The Gödel formula [(\forall x)R(x, p)] is not provable in PA; but it is algorithmically verifiable as true (Corollary 8.3, p.16, of the Epsilon 2015 paper) under the algorithmically verifiable standard interpretation \mathbb{M} of PA (see Section 5, p.11, of the Epsilon 2015 paper) over the structure \mathbb{N} of the natural numbers;

(iii) The Gödel formula [(\forall x)R(x, p)] is not provable in PA; and it is algorithmically computable as false (Corollary 8.3, p.16, of the Epsilon 2015 paper) under the algorithmically computable finitary interpretation \mathbb{B} of PA (see Section 6, p.13, of the Epsilon 2015 paper) over the structure \mathbb{N} of the natural numbers;

(iv) The Gödel formula [\neg(\forall x)R(x, p)] is provable in PA; and it is therefore also algorithmically verifiable as true under the algorithmically verifiable standard interpretation \mathbb{M} of PA over the structure \mathbb{N} of the natural numbers—which means that the logic by which the standard interpretation of PA assigns values of ‘satisfaction’ and ‘truth’ to the formulas of PA (under Tarski’s definitions) may be paraconsistent (see http://plato.stanford.edu/entries/logic-paraconsistent) since PA is consistent;

(v) The Gödel formula [\neg(\forall x)R(x, p)] is provable in PA; and it is therefore algorithmically computable as true (Corollary 8.2, p.16, of the Epsilon 2015 paper) under the algorithmically computable finitary interpretation \mathbb{B} of PA over the structure \mathbb{N} of the natural numbers.

3. It also means that:

(a) The “Gödel-Rosser Theorem” is not a Theorem of PA;

(b) The “unprovable Gödel sentence” is not a “problematic statement”;

(c) The “PA system” does not require a “remedy” just because it is “\omega-inconsistent”;

(d) No “new axiom or new rule” can “by-pass the unprovable sentence”.

4. Which raises the question:

Why do you see the “unprovable Gödel sentence” as a “problematic statement” that requires a “remedy” which must “by-pass the unprovable sentence”?

Author’s working archives & abstracts of investigations

Bhupinder Singh Anand

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