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

**A Economist: The return of the machinery question**

In a Special Report on Artificial Intelligence in its issue of 25th June 2016, ‘*The return of the machinery question*‘, the Economist suggests that both cosmologist *Stephen Hawking* and enterpreneur *Elon Musk* share to some degree the:

“… fear that AI poses an existential threat to humanity, because superintelligent computers might not share mankind’s goals and could turn on their creators”.

**B Our irrational propensity to fear that which we are drawn to embrace**

Surprising, since I suspect both would readily agree that, if anything should scare us, it is our irrational propensity to fear that which we are drawn to embrace!

And therein should lie not only our comfort, but perhaps also our salvation.

For Artificial Intelligence is constrained by rationality; Human Intelligence is not.

An Artificial Intelligence must, whether individually or collectively, create and/or destroy only rationally. Humankind can and does, both individually and collectively, create and destroy irrationally.

**C Justifying irrationality**

For instance, as the legatees of logicians Kurt Goedel and Alfred Tarski have amply demonstrated, a Human Intelligence can easily be led to believe that some statements of even the simplest of mathematical languages—Arithmetic—must be both ‘formally undecidable’ and ‘true’, even in the absence of any objective yardstick for determining what is ‘true’!

**D Differentiating between Human reasoning and Mechanistic reasoning**

An Artificial Intelligence, however, can only treat as true that which can be proven—by its rules—to be true by an objective assignment of ‘truth’ and ‘provability’ values to the propositions of the language that formally expresses its mechanical operations—Arithmetic.

The implications of the difference are not obvious; but that the difference could be significant is the thesis of *this paper* which is due to appear in the December 2016 issue of Cognitive Systems Research:

‘*The Truth Assignments That Differentiate Human Reasoning From Mechanistic Reasoning*‘.

**E Respect for evidence-based ‘truth’ could be Darwinian**

More importantly, the paper demonstrates that both Human Intelligence—whose evolution is accepted as Darwinian—and Artificial Intelligence—whose evolution it is ‘feared’ may be Darwinian—share a common (Darwinian?) respect for an accountable concept of ‘truth’.

A respect that should make both Intelligences fitter to survive by recognising what philosopher *Christopher Mole* describes in *this invitational blogpost* as the:

“… importance of the rapport between an organism and its environment”

—an environment that can obviously accommodate the birth, and nurture the evolution, of both intelligences.

So, it may not be too far-fetched to conjecture that the evolution of both intelligences must also, then, share a Darwinian respect for the kind of human values—towards protecting intelligent life forms—that, no matter in how limited or flawed a guise, is visibly emerging as an inherent characteristic of a human evolution which, no matter what the cost could, albeit optimistically, be viewed as struggling to incrementally strengthen, and simultaneously integrate, individualism (fundamental particles) into nationalism (atoms) into multi-nationalism (molecules) and, possibly, into universalism (elements).

**F The larger question: Should we fear an extra-terrestrial Intelligence?**

From a broader perspective yet, our apprehensions about the evolution of a rampant Artificial Intelligence created by a Frankensteinian Human Intelligence should, perhaps, more rightly be addressed—as some have urged—within the larger uncertainty posed by SETI:

*Is there a rational danger to humankind in actively seeking an extra-terrestrial intelligence?*

I would argue that any answer would depend on how we articulate the question and that, in order to engage in a constructive and productive debate, we need to question—and reduce to a minimum—some of our most cherished mathematical and scientific beliefs and fears which cannot be communicated objectively.

*See also (i) this later publication by Sebastian Grève, where he concludes that “… while Gödel indeed showed some significant understanding of Wittgenstein here, ultimately, Wittgenstein perhaps understood Gödel better than Gödel understood himself”; and (ii) this note on Rosser’s Rule C and Wittgenstein’s objections on purely philosophical considerations to Gödel’s reasoning and conclusions, where we show that, although not at all obvious (perhaps due to Gödel’s overpoweringly plausible presentation of his interpretation of his own formal reasoning over the years) what Gödel claimed to have proven is not—as suspected and held by Wittgenstein—supported by Gödel’s formal argumentation.*

**A: Misunderstanding Gödel’s Theorems VI and XI: Incompleteness and Undecidability**

In an informal essay, “*Misunderstanding Gödel’s Theorems VI and XI: Incompleteness and Undecidability*“, DPhil candidate Sebastian Grève at The Queen’s College, Oxford, attempts to come to terms with what he subjectively considers:

*“… has not been properly addressed as such by philosophers hitherto as of great philosophical importance in our understanding of Gödel’s Incompleteness Theorems.”*

Grève’s is an unusual iconoclastic perspective:

*“This essay is an open enquiry towards a better understanding of the philosophical significance of Gödel’s two most famous theorems. I proceed by a discussion of several common misunderstandings, led by the following four questions:*

*1) Is the Gödel sentence true?*

*2) Is the Gödel sentence undecidable?*

*3) Is the Gödel sentence a statement?*

*4) Is the Gödel sentence a sentence?*

*Asking these questions in this order means to trace back the steps of Gödel’s basic philosophical interpretation of his formal results. What I call the basic philosophical interpretation is usually just taken for granted by philosopher’s writing about Gödel’s theorems.”*

In a footnote Grève acknowledges Wittgenstein’s influence by suggesting that:

*“This essay can be read as something like a free-floating interpretation of the theme of Wittgenstein’s remarks on Gödel’s Incompleteness Theorems in Wittgenstein: 1978[RFM], I-(III), partly following Floyd: 1995 but especially Kienzler: 2008, and constituting a reply to inter alia Rodych: 2003”.*

**B: Why we may see the trees, but not the forest**

We note that Grève’s four points are both overdue and well-made:

1. *Is the Gödel sentence true?*

Grève’s objection that standard interpretations are obscure when they hold the Gödel sentence as being intuitively true deserves consideration (see this post).

The ‘truth’ of the sentence should and does—as Wittgenstein stressed and suggested—follow objectively from the axioms and rules of inference of arithmetic.

2. *Is the Gödel sentence undecidable?*

Grève’s observation that the ‘undecidability’ of the Gödel sentence conceals a philosophically questionable assumption is well-founded.

The undecidability in question follows only on the assumption of ‘-consistency’ made explicitly by Gödel.

This assumption is actually logically equivalent to the philosophically questionable assertion that from the provability of we may conclude the existence of some numeral for which is provable.

Since Rosser’s proof implicitly makes this assumption by means of his logically questionable Rule C, his claim of avoiding omega-consistency for arithmetic is illusory.

3. *Is the Gödel sentence a statement?*

Grève rightly holds that the Gödel sentence should be treated as a valid statement within the formal arithmetic S, since it is structurally defined as a well-formed formula of S.

4. *Is the Gödel sentence a sentence?*

Grève’s concern about whether the Gödel sentence of S is a valid arithmetical proposition under interpretation also seems to need serious philosophical consideration.

It can be argued (see the comment following the proof of Lemma 9 of this preprint) that the way the sentence is formally defined as the universal quantification of an instantiationally (but not algorithmically) defined arithmetical predicate does not yield an unequivocally defined arithmetical proposition in the usual sense under interpretation.

In this post ^{[*]} we shall not only echo Grève’s disquietitude, but argue further that Gödel’s interpretation and assessment of his own formal reasoning in his seminal 1931 paper on formally undecidable arithmetical propositions is, essentially, a post-facto imposition that continues to influence standard expositions of Gödel’s reasoning misleadingly.

**Feynman’s cover-up factor**

Our thesis is influenced by physicist Richard P. Feynman, who started his 1965 Nobel Lecture with a penetrating observation:

*We have a habit in writing articles published in scientific journals to make the work as finished as possible, to cover up all the tracks, to not worry about the blind alleys or describe how you had the wrong idea first, and so on. So there isn’t any place to publish, in a dignified manner, what you actually did in order to get to do the work.*

That such `cover up’ may have unintended—and severely limiting—consequences on a discipline is suggested by Gödel’s interpretation, and assessment, of his own formal reasoning in his seminal 1931 paper on formally undecidable arithmetical propositions (Go31).

Thus, in his informal preamble to the result that he intended to prove formally, Gödel wrote (cf. Go31, p.9):

*The analogy of this result with Richard’s antinomy is immediately evident; there is also a close relationship with the Liar Paradox … Thus we have a proposition before us which asserts its own unprovability.*

Further, interpreting the significance of his formal reasoning as having established the existence of a formally undecidable arithmetical proposition that is, however, decidable by meta-mathematical arguments, Gödel noted that:

*The precise analysis of this remarkable circumstance leads to surprising results concerning consistency proofs of formal systems … (Go31, p.9)*

*The true reason for the incompleteness which attaches to all formal systems of mathematics lies, as will be shown in Part II of this paper, in the fact that the formation of higher and higher types can be continued into the transfinite (c.f., D. Hilbert, `Über das Unendliche’, Math. Ann. 95, p. 184), while, in every formal system, only countable many are available. Namely, one can show that the undecidable sentences which have been constructed here always become decidable through adjunction of suitable high types (e.g. of the type to the system . A similar result also holds for the axiom systems of set theory. (Go31, p.28, footnote 48a)*

The explicit thesis of this foundational paper is that the above interpretation is an instance of a `cover up’—in Feynman’s sense—which appears to be a post-facto imposition that, first, continues to echo in and misleadingly ^{[1]} influence standard expositions of Gödel’s reasoning when applied to a first-order Peano Arithmetic, PA, and, second, that it obscures the larger significance of the genesis of Gödel’s reasoning.

As Gödel’s various remarks in Go31 suggest, this possibly lay in efforts made at the dawn of the twentieth century—largely as a result of Brouwer’s objections (Br08)—to define unambiguously the role that the universal and existential quantifiers played in formal mathematical reasoning.

That this issue is critical to Gödel’s reasoning in Go31, but remains unresolved in it, is obscured by his powerful presentation and interpretation.

So, to grasp the underlying mathematical significance of Gödel’s reasoning, and of what he has actually achieved, one may need to avoid focusing (as detailed in the previous posts on *A foundational perspective on the semantic and logical paradoxes*; in this post on undecidable Gödelian propositions, and in this preprint on undecidable Gödelian propositions):

on the analogy of the so-called `Liar paradox’;

on Gödel’s interpretation of his arithmetical proposition as asserting its own formal unprovability in his formal Peano Arithmetic P (Go31, pp.9-13);

on his interpretation of the reasons for the `incompleteness’ of P; and

on his assessment and interpretation of the formal consequences of such `incompleteness’.

We show in this paper that, when applied to PA ^{[2]}, all of these obscure the deeper significance of what Gödel actually achieved in Go31.

**C: Hilbert: If the -Rule is true, can P be completed?**

Instead, Gödel’s reasoning may need to be located specifically in the context of Hilbert’s Program (cf. Hi30, pp.485-494) in which he proposed an -rule as a finitary means of extending a Peano Arithmetic—such as his formal system P in Go31—to a possible completion (*i.e. to logically showing that, given any arithmetical proposition, either the proposition, or its negation, is formally provable from the axioms and rules of inference of the extended Arithmetic*).

*Hilbert’s -Rule:* If it is proved that the P-formula [] interprets as a true numerical formula for each given P-numeral [], then the P-formula may be admitted as an initial formula (*axiom*) in P.

It is likely that Gödel’s 1931 paper evolved out of attempts to prove Hilbert’s -rule in the limited—and more precise—sense that if a formula [] is provable in P for each given numeral [], then the formula [] must be provable in P.

Now, if we meta-assume Hilbert’s -rule for P, then it follows that, if P is consistent, then there is no P-formula [] for which, first, [] is P-provable and, second, [] is P-provable for any given P-numeral [].

Gödel defined a consistent Peano Arithmetic with the above property as additionally -consistent (Go31, pp.23-24).

**D: The significance of -consistency**

To place the significance of -consistency in a current perspective, we note that the standard model of the first order Peano Arithmetic PA (cf. Me64, p.107; Sc67, p.23, p.209; BBJ03, p.104) *presumes* ^{[3]} that the standard interpretation * M* of PA (under which the PA-formula [], which is merely an abbreviation for , interprets as true if, and only if, holds for some natural number under

*) is*

**M***sound*(cf. BBJ03, p.174).

Clearly, if such an interpretation of the existential quantifier is sound, it immediately implies that PA is necessarily -consistent ^{[4]}.

Since Brouwer’s main objection was to Hilbert’s presumption that such an interpretation of the existential quantifier is sound, Gödel explicitly avoided this assumption in his seminal 1931 paper (Go31, p.9) in order to ensure that his reasoning was acceptable as “constructive” and “intuitionistically unobjectionable” (Go31, p.26).

He chose, instead, to present the formal undecidability of his arithmetical proposition—and the consequences arising from it—as explicitly conditional on the assumption of the formal property of -consistency for his Peano Arithmetic P under the unqualified—and, as we show below, mistaken—belief that:

*PA is -consistent (Go31, p.28, footnote 48a).*

**E: Gödel: If the -Rule is true, P cannot be completed**

Now, Gödel’s significant achievement in Go31 was the discovery that, if P is consistent, then it was possible to construct a P-formula, [] ^{[5]}, such that is P-provable for any given P-numeral [] (Go31, p.25(2)), but [] is P-unprovable (Go31, p.25(1)).

However, it becomes apparent from his remarks in Go31 that Gödel considered his more significant achievement the further argument that, if P is assumed -consistent, then both [] and [] ^{[6]} are P-unprovable, and so P is incomplete!

This is the substance of Gödel’s Theorem VI (Go31, p.24).

Although this Theorem neither validated nor invalidated Hilbert’s -rule, it did imply that assuming the rule led not to the completion of a Peano Arithmetic as desired by Hilbert, but to its essential incompletability!

**F: The -Rule is inconsistent with PA**

Now, apparently, the possibility neither considered by Gödel in 1931, nor seriously since, is that a formal sytem of Peano Arithmetic—such as PA—may be consistent *and* –*in*consistent.

If so, one would ascribe this omission to the `cover up’ factor mentioned by Feynman, since a significant consequence of Gödel’s reasoning—in the first half of his proof of his Theorem VI—is that it actually establishes PA as –*in*consistent (as detailed in Corollary 9 of this preprint and Corollary 4 of this post).

In other words, we can logically show for Gödel’s formula [] that [ ] is PA-provable, and that [] is PA-provable for any given PA-numeral [].

Consequently, Gödel’s Theorem VI is vacuously true for PA, and it also follows that Hilbert’s -Rule is inconsistent with PA!

**G: Need: A paradigm shift in interpreting the quantifiers**

Thus Gödel’s unqualified belief that:

“*PA is -consistent*“

was misplaced, and Brouwer’s objection to Hilbert’s presumption—that the above interpretation of the existential quantifier is sound—was justified; since, if PA is consistent, then it is provably –*in*consistent, from which it follows that the standard interpretation * M* of PA is

*not*sound.

Hence we can no longer interpret `[] is true’ maximally under the standard interpretation of PA as:

(i) The arithmetical relation is not always ^{[7]} true.

However, since the theorems of PA—when treated as Boolean functions—are Turing-computable as always true under a *sound* finitary interpretation of PA, we *can* interpret `[] is true’ minimally as:

(ii) The arithmetical relation is not Turing-computable as always true.

This interpretation allows us to conclude from Gödel’s meta-mathematical argument that we can construct a PA-formula [] that is unprovable in PA, but which is true under a sound interpretation of PA ^{[8]} although we may now no longer conclude from Gödel’s reasoning that there is an undecidable arithmetical PA-proposition.

Moreover, the interpretation admits an affirmative answer to Hilbert’s query: Is PA complete or completeable?

**H: PA is algorithmically complete**

In outline, the basis from which this conclusion follows formally is that:

(i) Gödel’s proof of the essential incompleteness of any formal system of Peano Arithmetic (Go31, Theorem VI, p.24) *explicitly* assumes that the arithmetic is -consistent;

(ii) Rosser’s extension of Gödel’s proof of the essential incompleteness of any formal system of Peano Arithmetic (cf. Ro36, Theorem II, p.233) *implicitly* presumes that the Arithmetic is -consistent (as detailed in this post);

(iii) PA is –*in*consistent (as detailed in Corollary 9 of this preprint);

(iv) The classical `standard’ interpretation of PA (cf. Me64, section \S 2, pp.49-53; p107) over the structure []—defined as { (*the set of natural numbers*); (*equality*); (*the successor function*); (*the addition function*); (*the product function*); (*the null element*)}— does *not* define a finitary model of PA (as detailed in the paper titled *Evidence-Based Interpretations of PA* presented at IACAP/AISB Turing 2012, Birmingham, UK in July 2012);

(v) We can define a sound interpretation of PA—in terms of Turing-computability—which yields a finitary model of PA, but which does not admit a non-standard model for PA (as detailed in this paper);

(vi) PA is algorithmically complete in the sense that an arithmetical proposition defines a Turing-machine TM which computes as true under if, and only if, the corresponding PA-formula [] is PA-provable (as detailed in Section 8 of this preprint).

**I: Gödel’s proof of his Theorem XI does not withstand scrutiny**

Since Gödel’s proof of his Theorem XI (Go31, p.36)—in which he claims to show that the consistency of his formal system of Peano Arithmetic P can be expressed as a P-formula which is not provable in P—appeals critically to his Theorem VI, it follows that this proof cannot be applied to PA.

However, we show below that there are other, significant, reasons why Gödel’s reasoning in this proof must be treated as classically objectionable per se.

**J: Why Gödel’s interpretation of the significance of his Theorem XI is classically objectionable**

Now, in his Theorem XI, Gödel constructs a formula [] ^{[9]} in P and assumes that [] translates—under a sound interpretation of P—as an arithmetical proposition that is true if, and only if, a specified formula of P is unprovable in P.

Now, if there were such a P-formula, then, since an inconsistent system necessarily proves every well-formed formula of the system, it would follow that a proof sequence within P proves that P is consistent.

However, Gödel shows that his formula [] is not P-provable (Go31, p.37).

He concludes that the consistency of any formal system of Peano Arithmetic is not provable within the Arithmetic. ^{[10]}

**K: Defining meta-propositions of P arithmetically**

Specifically, Gödel first shows how 46 meta-propositions of P can be defined by means of primitive recursive functions and relations (Go31, pp.17-22).

These include:

() A primitive recursive relation, *Form*(), which is true if, and only if, is the Gödel-number of a formula of P;

() A primitive recursive relation, , which is true if, and only if, is the Gödel-number of a proof sequence of P whose last formula has the Gödel-number .

Gödel assures the constructive nature of the first 45 definitions by specifying (cf. Go31, p.17, footnote 34):

*Everywhere in the following definitions where one of the expressions `‘, `‘, ` (There is a unique )’ occurs it is followed by a bound for . This bound serves only to assure the recursive nature of the defined concept.*

Gödel then defines a meta-mathematical proposition that is not recursive:

() A proposition, , which is true if, and only if, is true.

Thus is true if, and only if, is the Gödel-number of a provable formula of P.

**L: Expressing arithmetical functions and relations in P**

Now, by Gödel’s Theorem VII (Go31, p.29), any recursive relation, say , can be represented in P by some, corresponding, arithmetical formula, say [], such that, for any natural number :

If is true, then [] is P-provable;

If is false, then [] is P-provable.

However, Gödel’s reasoning in the first half of his Theorem VI (Go31, p.25(1)) establishes that the above representation does not extend to the closure of a recursive relation, in the sense that we cannot assume:

If is true (i.e, is true for any given natural number), then is P-provable.

In other words, we cannot assume that, even though the recursive relation is instantiationally equivalent to a sound interpretation of the P-formula [], the number-theoretic proposition must, necessarily, be logically equivalent to the—correspondingly sound—interpretation of the P-formula [].

The reason: In recursive arithmetic, the expression `‘ is an abbreviation for the assertion:

(*) There is some (at least one) natural number such that holds.

In a formal Peano Arithmetic, however, the formula `[]’ is simply an abbreviation for `[]’, which, under a sound finitary interpretation of the Arithmetic can have the verifiable translation:

(**) The relation is not Turing-computable as always true.

Moreover, Gödel’s Theorem VI establishes that we cannot conclude (*) from (**) without risking inconsistency.

Consequently, although a primitive recursive relation may be instantiationally equivalent to a sound interpretation of a P-formula, we cannot assume that the existential closure of the relation must have the same meaning as the interpretation of the existential closure of the corresponding P-formula.

However this, precisely, is the presumption made by Gödel in the proof of Theorem XI, from which he concludes that the consistency of P can be expressed in P, but is not P-provable.

**M: Ambiguity in the interpreted `meaning’ of formal mathematical expressions**

The ambiguity in the `meaning’ of formal mathematical expressions containing unrestricted universal and existential closure under an interpretation was emphasised by Wittgenstein (Wi56):

*Do I understand the proposition “There is . . .” when I have no possibility of finding where it exists? And in so far as what I can do with the proposition is the criterion of understanding it … it is not clear in advance whether and to what extent I understand it.*

**N: Expressing “P is consistent” arithmetically**

Specifically, Gödel defines the notion of “P is consistent” classically as follows:

P is consistent if, and only if, *Wid*(P) is true

where *Wid*(P) is defined as:

This translates as:

There is a natural number which is the Gödel-number of a formula of P, and this formula is not P-provable.

Thus, *Wid*(P) is true if, and only if, P is consistent.

**O: Gödel: “P is consistent” is always expressible in P**

However, Gödel, then, presumes that:

(i) *Wid*(P) can be represented by some formula [] of P such that “[] is true” and “*Wid*(P) is true” are logically equivalent (i.e., have the same meaning) under a sound interpretation of P;

(ii) if the recursive relation, (1931, p24(8.1)), is represented by the P-formula [], then the proposition “[] is true” is logically equivalent to (i.e., has the same meaning as) “ is true” under a sound interpretation of P.

**P: The loophole in Gödel’s presumption**

Although, (ii), for instance, does follow if “[] is true” translates as “ is Turing-computable as always true”, it does not if “[] is true” translates as “ is constructively computable as true for any given natural number , but it is not Turing-computable as true for any given natural number “.

So, if [], too, interprets as an arithmetical proposition that is constructively computable as true, but not Turing-computable as true, then the consistency of P may be provable instantiationally in P ^{[11]}.

Hence, at best, Gödel’s reasoning can only be taken to establish that the consistency of P is not provable algorithmically in P.

Gödel’s broader conclusion only follows if P purports to prove its own consistency algorithmically.

However, Gödel’s particular argument, based on his definition of *Wid*(P), does not support this claim.

**Bibliography**

**BBJ03** George S. Boolos, John P. Burgess, Richard C. Jeffrey. 2003. *Computability and Logic* (4th ed). Cambridge University Press, Cambridge.

**Br08** L. E. J. Brouwer. 1908. *The Unreliability of the Logical Principles.* English translation in A. Heyting, Ed. *L. E. J. Brouwer: Collected Works 1: Philosophy and Foundations of Mathematics.* Amsterdam: North Holland / New York: American Elsevier (1975): pp. 107-111.

**Go31** Kurt Gödel. 1931. *On formally undecidable propositions of Principia Mathematica and related systems I.* Translated by Elliott Mendelson. In M. Davis (ed.). 1965. *The Undecidable.* Raven Press, New York.

**Hi27** David Hilbert. 1927. *The Foundations of Mathematics.* In *The Emergence of Logical Empiricism.* 1996. Garland Publishing Inc.

**Hi30** David Hilbert. 1930. *Die Grundlegung der elementaren Zahlenlehre.* Mathematische Annalen. Vol. 104 (1930), pp. 485-494.

**Me64** Elliott Mendelson. 1964. *Introduction to Mathematical Logic.* Van Norstrand. pp.145-146.

**Ro36** J. Barkley Rosser. 1936. *Extensions of some Theorems of Gödel and Church.* In M. Davis (ed.). 1965. *The Undecidable.* Raven Press, New York. Reprinted from The Journal of Symbolic Logic. Vol.1. pp.87-91.

**Sc67** Joseph R. Schoenfield. 1967. Mathematical Logic. Reprinted 2001. A. K. Peters Ltd., Massachusetts.

**Tu36** Alan Turing. 1936. *On computable numbers, with an application to the Entscheidungsproblem.* Proceedings of the London Mathematical Society, ser. 2. vol. 42 (1936-7), pp.230-265; corrections, Ibid, vol 43 (1937) pp. 544-546.

**Wi56** Ludwig Wittgenstein. 1956. *Remarks on the Foundations of Mathematics.* Edited by G. H. von Wright and R. Rhees. Translated by G. E. M. Anscombe. Basil Blackwell, Oxford.

**Notes**

Return to *: Edited and transcribed from this 2010 preprint. Some of its pedantic conclusions regarding the `soundness’ of the standard interpretation of PA (and consequences thereof) should, however, be treated as qualified by the broader philosophical perspective that treats the standard and algorithmic interpretations of PA as complementary—rather than contradictory—interpretations (as detailed in this post).

Return to 1: We show in this paper that, from a finitary perspective (such as that of this preprint) the proofs of both of Gödel’s celebrated theorems in Go31—his Theorem VI postulating the existence of an undecidable proposition in his formal Peano Arithmetic, P, and his Theorem XI postulating that the consistency of P can be expressed, but not proven, within P—hold vacuously for first order Peano Arithmetic, PA.

Return to 2: Although we have restricted ourselves in this paper to considering only PA, the arguments would—prima facie—apply equally to any first-order theory that contains sufficient Peano Arithmetic in Gödel’s sense (cf. Go31, p.28(2)), by which we mean that every primitive recursive relation is definable within the theory in the sense of Gödel’s Theorems V (Go31, p.22) and VII (Go31, p.29).

Return to 3: Following Hilbert.

Return to 4: Since we cannot, then, have that is PA-provable and that is also PA-provable for any given numeral .

Return to 5: This corresponds to the P-formula of his paper that Gödel defines, and refers to, only by its Gödel-number (cf. Go31, p.25, eqn.(12)).

Return to 6: Gödel refers to these P-formulas only by their Gödel-numbers and respectively (cf. Go31, p.25, eqn.13).

Return to 7: i.e., for any given natural number .

Return to 8: Because the arithmetical relation is a Halting-type of relation (cf.Tu36, ) that is constructively computable as true for any given natural number , although it is not Turing-computable as true for any given natural number (as detailed in this post).

Return to 9: Gödel refers to it only by its Gödel-number (Go31, p.37).

Return to 10: Gödel’s broader conclusion—unchallenged so far but questionable—was that his reasoning could be validly “… carried over, word for word, to the axiom systems of set theory M and of classical mathematics A”.

Return to 11: That Gödel was open to such a possibility in 1931 is evidenced by his remark (Go31, p37) that “… it is conceivable that there might be finitary proofs which cannot be represented in P (or in M or A)”.

**Use of square brackets**

Unless otherwise obvious from the context, we use square brackets to indicate that the contents represent a symbol or a formula—of a formal theory—generally assumed to be well-formed unless otherwise indicated by the context.

In other words, expressions inside the square brackets are to be only viewed syntactically as juxtaposition of symbols that are to be formed and manipulated upon strictly in accordance with specific rules for such formation and manipulation—in the manner of a mechanical or electronic device—without any regards to what the symbolism might represent semantically under an interpretation that gives them meaning.

**Use of an asterisk**

Unless otherwise obvious from the context, we use an asterisk to indicate that the associated expression is to be interpreted semantically with respect to some well-defined interpretation.

**Explanatory comments**

We have taken some liberty in emphasising standard definitions selectively, and interspersing our arguments liberally with comments and references, generally of a foundational nature.

These are intended to reflect our underlying thesis that essentially arithmetical problems appear more natural when expressed—and viewed—within an arithmetical perspective of an interpretation of PA that appeals to the evidence provided by a deterministic algorithm.

Since a deterministic algorithm has only one possible move from a given configuration such a perspective, by its very nature, cannot appeal implicitly to transfinite concepts.

**Evidence**

“It is by now folklore … that one can view the *values* of a simple functional language as specifying *evidence* for propositions in a constructive logic …”

… Chetan R. Murthy. 1991. *An Evaluation Semantics for Classical Proofs.* Proceedings of Sixth IEEE Symposium on Logic in Computer Science, pp. 96-109, (also Cornell TR 91-1213), 1991.

**Aristotle’s particularisation**

This holds that from an assertion such as:

‘It is not the case that: For any given does not hold’

usually denoted symbolically by ‘‘, we may always validly infer in the classical, Aristotlean, logic of predicates ^{[1]} that:

‘There exists an unspecified such that holds’

usually denoted symbolically by ‘‘.

Aristotle’s particularisation (AP) is essentially the semantic postulation that from the negation of a universal we may always deduce the existence of a contrafactual. It is necessarily true over finite domains.

Expressed more formally:

**Aristotle’s particularisation under an interpretation**

If the formula of a first order language interprets as true under a sound interpretation of , then we may always conclude that there must be some object in the domain of the interpretation such that, if the formula interprets as the unary relation in , then the proposition is true under the interpretation.

**The significance of Aristotle’s particularisation for the first-order predicate calculus**

We note that in a formal language the formula ‘‘ is an abbreviation for the formula ‘‘.

The commonly accepted interpretation of this formula—and a fundamental tenet of classical logic unrestrictedly adopted as intuitively obvious by standard literature ^{[2]} that seeks to build upon the formal first-order predicate calculus—tacitly appeals to Aristotlean particularisation.

However, L. E. J. Brouwer had noted in his seminal 1908 paper on the unreliability of logical principles ^{[3]} that the commonly accepted interpretation of this formula is ambiguous if interpretation is intended over an infinite domain.

Brouwer essentially argued that, even supposing the formula ‘‘ of a formal Arithmetical language interprets as an arithmetical relation denoted by ‘‘, and the formula ‘‘ as the arithmetical proposition denoted by ‘‘, the formula ‘‘ need not interpret as the arithmetical proposition denoted by the usual abbreviation ‘‘; and that such postulation is invalid as a general logical principle in the absence of a means for constructing some putative object for which the proposition holds in the domain of the interpretation.

Hence we shall follow the convention that the assumption that ‘‘ is the intended interpretation of the formula ‘‘—which is essentially the assumption that Aristotle’s particularisation holds over the domain of the interpretation—must always be explicit.

**The significance of Aristotle’s particularisation for PA**

In order to avoid intuitionistic objections to his reasoning, Kurt Gödel introduced the syntactic property of -consistency ^{[4]} as an explicit assumption in his formal reasoning in his seminal 1931 paper on formally undecidable arithmetical propositions ^{[5]}.

Gödel explained at some length ^{[6]} that his reasons for introducing -consistency explicitly was to avoid appealing to the semantic concept of classical arithmetical truth in Aristotle’s logic of predicates (which presumes Aristotle’s particularisation).

The two concepts are meta-mathematically equivalent in the sense that, if PA is consistent, then PA is -consistent if, and only if, Aristotle’s particularisation holds under the standard interpretation of PA ^{[7]}.

We note that Aristotle’s particularisation is a non-constructive—and logically fragile—semantic deduction rule. It is reflected in classical first order deduction either by some similarly non-constructive syntactic rule of natural deduction—such as Rosser’s *Rule C* ^{[7.1]}—or by the assumption that FOL is -consistent.

**The structure **

The structure of the natural numbers—namely:

(*the set of natural numbers*);

(*equality*);

(*the successor function*);

(*the addition function*);

(*the product function*);

(*the null element*).

**The axioms of the first-order Peano Arithmetic PA**

: ;

: ;

: ;

: ;

: ;

: ;

: ;

: ;

: For any well-formed formula of PA:

.

**Generalisation in PA**

If is PA-provable, then so is .

**Modus Ponens in PA**

If and are PA-provable, then so is .

**The standard interpretation of PA**

The standard interpretation of PA over the structure is the one in which the logical constants have their ‘usual’ interpretations ^{[8]} in Aristotle’s logic of predicates (which subsumes Aristotle’s particularisation), and ^{[9]}:

(a) the set of non-negative integers is the domain;

(b) the symbol interprets as the integer ;

(c) the symbol interprets as the successor operation (addition of );

(d) the symbols and interpret as ordinary addition and multiplication;

(e) the symbol interprets as the identity relation.

**Simple consistency**

A formal system S is simply consistent if, and only if, there is no S-formula for which both and are S-provable.

**-consistency**

A formal system S is -consistent if, and only if, there is no S-formula for which first is S-provable, and second is S-provable for any given S-term .

**Soundness (formal system – non-standard)**

A formal system S is sound under an interpretation with respect to a domain if, and only if, every theorem of S translates as ‘ is true under in ‘.

**Soundness (interpretation – non-standard)**

An interpretation of a formal system S is sound with respect to a domain if, and only if, S is sound under the interpretation over the domain .

**Soundness in classical logic**

In classical logic, a formal system is sometimes defined as ‘sound’ if, and only if, it has an interpretation; and an interpretation is defined as the assignment of meanings to the symbols, and truth-values to the sentences, of the formal system. Moreover, any such interpretation is defined as a model ^{[10]} of the formal system.

This definition suffers, however, from an implicit circularity: the formal logic underlying any interpretation of is implicitly assumed to be ‘sound’.

The above definitions seek to avoid this implicit circularity by delinking the defined ‘soundness’ of a formal system under an interpretation from the implicit ‘soundness’ of the formal logic underlying the interpretation.

This admits the case where, even if and are implicitly assumed to be sound, is sound, but is not.

Moreover, an interpretation of is now a model for if, and only if, it is sound.

**Algorithmic verifiability**

A number-theoretical relation is algorithmically verifiable if, and only if, for any given natural number , there is an algorithm which can provide objective evidence for deciding the truth/falsity of each proposition in the finite sequence .

**Tarskian interpretation of an arithmetical language verifiably in terms of the computations of a simple functional language**

We show in the Birmingham paper that the ‘algorithmic verifiability’ of the formulas of a formal language which contain logical constants can be inductively defined under an interpretation in terms of the ‘algorithmic verifiability’ of the interpretations of the atomic formulas of the language; further, that the PA-formulas are decidable under the standard interpretation of PA over if, and only if, they are algorithmically verifiable under the interpretation. ^{[11]}

**Algorithmic computability**

A number theoretical relation is algorithmically computable if, and only if, there is an algorithm that can provide objective evidence for deciding the truth/falsity of each proposition in the denumerable sequence .

**Tarskian interpretation of an arithmetical language algorithmically in terms of the computations of a simple functional language**

We show in the Birmingham paper that the ‘algorithmic computability’ of the formulas of a formal language which contain logical constants can also be inductively defined under an interpretation in terms of the ‘algorithmic computability’ of the interpretations of the atomic formulas of the language; further, that the PA-formulas are decidable under an algorithmic interpretation of PA over if, and only if, they are algorithmically computable under the interpretation. ^{[12]}

**Algorithmic verifiability vis à vis algorithmic computability**

We note that algorithmic computability implies the existence of an algorithm that can decide the truth/falsity of each proposition in a well-defined denumerable sequence of propositions ^{[13]}, whereas algorithmic verifiability does not imply the existence of an algorithm that can decide the truth/falsity of each proposition in a well-defined denumerable sequence of propositions.

From the point of view of a finitary mathematical philosophy—which is the constraint within which an applied science ought to ideally operate—the significant difference between the two concepts could be expressed by saying that we may treat the decimal representation of a real number as corresponding to a physically measurable limit ^{[14]}—and not only to a mathematically definable limit—if and only if such representation is definable by an algorithmically computable function. ^{[15]}

We note that although every algorithmically computable relation is algorithmically verifiable, the converse is not true. ^{[16]}

**References**

**Be59** Evert W. Beth. 1959. *The Foundations of Mathematics.* Studies in Logic and the Foundations of Mathematics. Edited by L. E. J. Brouwer, E. W. Beth, A. Heyting. 1959. North Holland Publishing Company, Amsterdam.

**BBJ03** George S. Boolos, John P. Burgess, Richard C. Jeffrey. 2003. *Computability and Logic.* (4th ed). Cambridge University Press, Cambridge.

**BF58** Paul Bernays and Abraham A. Fraenkel. 1958. *Axiomatic Set Theory* Studies in Logic and the Foundations of Mathematics. Edited by L. E. J. Brouwer, E. W. Beth, A. Heyting. 1959. North Holland Publishing Company, Amsterdam.

**Br08** L. E. J. Brouwer. 1908. *The Unreliability of the Logical Principles.* English translation in A. Heyting, Ed. L. E. J. Brouwer: Collected Works 1: *Philosophy and Foundations of Mathematics.* Amsterdam: North Holland / New York: American Elsevier (1975): pp.107-111.

**Co66** Paul J. Cohen. 1966. *Set Theory and the Continuum Hypothesis.* (Lecture notes given at Harvard University, Spring 1965) W. A. Benjamin, Inc., New York.

**Da82** Martin Davis. 1958. *Computability and Unsolvability.* 1982 ed. Dover Publications, Inc., New York.

**EC89** Richard L. Epstein, Walter A. Carnielli. 1989. *Computability: Computable Functions, Logic, and the Foundations of Mathematics.* Wadsworth & Brooks, California.

**Go31** Kurt Gödel. 1931. *On formally undecidable propositions of Principia Mathematica and related systems I.* Translated by Elliott Mendelson. In M. Davis (ed.). 1965. *The Undecidable.* Raven Press, New York. pp.5-38.

**HA28** David Hilbert & Wilhelm Ackermann. 1928. *Principles of Mathematical Logic.* Translation of the second edition of the *Grundzüge Der Theoretischen Logik>* 1928. Springer, Berlin. 1950. Chelsea Publishing Company, New York.

**Hi25** David Hilbert. 1925. *On the Infinite.* Text of an address delivered in Münster on 4th June 1925 at a meeting of the Westphalian Mathematical Society. In Jean van Heijenoort. 1967. Ed. *From Frege to Gödel: A source book in Mathematical Logic, 1878 – 1931.* Harvard University Press, Cambridge, Massachusetts.

**Kl52** Stephen Cole Kleene. 1952. *Introduction to Metamathematics.* North Holland Publishing Company, Amsterdam.

**Kn63** G. T. Kneebone. 1963. *Mathematical Logic and the Foundations of Mathematics: An Introductory Survey.* D. Van Norstrand Company Limited, London.

**Li64** A. H. Lightstone. 1964. *The Axiomatic Method.* Prentice Hall, NJ.

**Me64** Elliott Mendelson. 1964. *Introduction to Mathematical Logic.* Van Norstrand. pp.145-146.

**Mu91** Chetan R. Murthy. 1991. *An Evaluation Semantics for Classical Proofs.* Proceedings of Sixth IEEE Symposium on Logic in Computer Science, pp. 96-109, (also Cornell TR 91-1213), 1991.

**Nv64** P. S. Novikov. 1964. *Elements of Mathematical Logic.* Oliver & Boyd, Edinburgh and London.

**Qu63** Willard Van Orman Quine. 1963. *Set Theory and its Logic.* Harvard University Press, Cambridge, Massachusette.

**Rg87** Hartley Rogers Jr. 1987. *Theory of Recursive Functions and Effective Computability.* MIT Press, Cambridge, Massachusetts.

**Ro53** J. Barkley Rosser. 1953. *Logic for Mathematicians.* McGraw Hill, New York.

**Sh67** Joseph R. Shoenfield. 1967. *Mathematical Logic.* Reprinted 2001. A. K. Peters Ltd., Massachusetts.

**Sk28** Thoralf Skolem. 1928. *On Mathematical Logic.* Text of a lecture delivered on 22nd October 1928 before the Norwegian Mathematical Association. In Jean van Heijenoort. 1967. Ed. *From Frege to Gödel: A source book in Mathematical Logic, 1878 – 1931.* Harvard University Press, Cambridge, Massachusetts.

**Sm92** Raymond M. Smullyan. 1992. *Gödel’s Incompleteness Theorems.* Oxford University Press, Inc., New York.

**Su60** Patrick Suppes. 1960. *Axiomatic Set Theory.* Van Norstrand, Princeton.

**Tu36** Alan Turing. 1936. *On computable numbers, with an application to the Entscheidungsproblem.* In M. Davis (ed.). 1965. *The Undecidable.* Raven Press, New York. Reprinted from the Proceedings of the London Mathematical Society, ser. 2. vol. 42 (1936-7), pp.230-265; corrections, Ibid, vol 43 (1937) pp. 544-546.

**Wa63** Hao Wang. 1963. *A survey of Mathematical Logic.* North Holland Publishing Company, Amsterdam.

**An12** Bhupinder Singh Anand. 2012. *Evidence-Based Interpretations of PA.* In Proceedings of the Symposium on Computational Philosophy at the AISB/IACAP World Congress 2012-Alan Turing 2012, 2-6 July 2012, University of Birmingham, Birmingham, UK.

**An13** … 2013. *A suggested mathematical perspective for the argument.* Presented on 7’th April at the workshop on `*Logical Quantum Structures*‘ at UNILOG’2013, 4’th World Congress and School on Universal Logic, 29’th March 2013 – 7’th April 2013, Rio de Janeiro, Brazil.\

**Notes**

Return to 1: HA28, pp.58-59.

Return to 2: Hi25, p.382; HA28, p.48; Sk28, p.515; Go31, p.32.; Kl52, p.169; Ro53, p.90; BF58, p.46; Be59, pp.178 & 218; Su60, p.3; Wa63, p.314-315; Qu63, pp.12-13; Kn63, p.60; Co66, p.4; Me64, p.52(ii); Nv64, p.92; Li64, p.33; Sh67, p.13; Da82, p.xxv; Rg87, p.xvii; EC89, p.174; Mu91; Sm92, p.18, Ex.3; BBJ03, p.102.

Return to 3: Br08.

Return to 4: The significance of -consistency for the formal system PA is highlighted in An12.

Return to 5: Go31, p.23 and p.28.

Return to 6: In his introduction on p.9 of Go31.

Return to 7: For details see An12.

Return to 7.1: See Ro53, pp.127-136.

Return to 8: See Me64, p.49.

Return to 9: See Me64, p.107.

Return to 10: We follow the definition in Me64, p.51.

Return to 11: We show in An12 that the concept of Algorithmic verifiability is also well-defined under the standard interpretation of PA over .

Return to 12: We show in An12 that the concepts of Algorithmic verifiability and Algorithmic computability are both well-defined under the standard interpretation of PA over ; moreover they identify distinctly different subsets of the well-defined PA formulas.

Return to 13: We note that the concept of ‘algorithmic computability’ is essentially an expression of the more rigorously defined concept of ‘realizability’ in Kl52, p.503.

Return to 14: In the sense of a physically ‘completable’ infinite sequence (as needed to resolve Zeno’s paradox).

Return to 15: This point is addressed in more detail in An13.

Return to 16: See Appendix B of this preprint *Is Gödel’s undecidable proposition an ‘ad hoc’ anomaly?*.

**Two perspectives on Hilbert’s First and Second Problems**

In the Birmingham paper ‘Evidence-Based Interpretations of PA’, we introduced the distinction between algorithmic verifiability and algorithmic computability.

We showed how the distinction naturally helped distinguish between a finitary algorithmic interpretation of PA, and the non-finitary standard interpretation of PA.

We then showed how the former yielded a finitary proof of consistency for PA, as demanded by the second of Hilbert’s celebrated 23 problems.

In the previous post, we also highlighted why the the non-finitary standard interpretation of PA does not yield a finitary proof of consistency for PA.

We now show that the difference between the conclusions suggested by finitary reasoning and those suggested by non-finitary reasoning in the previous pages (as in the case of Goodstein’s argument) is reflected further in the differing status of the Continuum Hypothesis (the first of Hilbert’s 23 problems) when viewed from finitary and non-finitary perspectives as detailed below.

**The non-finitary set-theoretical perspective**

The non-finitary set-theoretical perspective on the Continuum Hypothesis is well-known, and described succintly by Topologist Peter Nyikos in a short expository lecture given at the University of Auckland in May, 2000:

“In 1900, David Hilbert gave a seminal lecture in which he spoke about a list of unsolved problems in mathematics that he deemed to be of outstanding importance. The first of these was Cantor’s continuum problem, which has to do with infinite numbers with which Cantor revolutionised set theory. The smallest infinite number, , `aleph-nought,’ gives the number of positive whole numbers. A set is of this cardinality if it is possible to list its members in an arrangement such that each one is encountered after a finite number (however large) of steps. Cantor’s revolutionary discovery was that the points on a line cannot be so listed, and so the number of points on a line is a strictly higher infinite number (, `the cardinality of the continuum’) than . Hilbert’s First Problem asks whether any infinite subset of the real line is of one of these two cardinalities. The axiom that this is indeed the case is known as the Continuum Hypothesis (CH). …

Gödel [1940] also gave a partial solution to Hilbert’s First Problem by showing that the Continuum Hypothesis (CH) is consistent if the usual Zermelo-Fraenkel (ZF) axioms for set theory are consistent. He produced a model, known as the Constructible Universe, of the ZF axioms in which both the Axiom of Choice (AC) and the CH hold. Then Cohen showed in 1963 that the negations of these axioms are also consistent with ZF; in particular, CH can fail while AC holds in a model of ZF.”

… *Hilbert’s First and Second Problems and the foundations of mathematics*, Topology Atlas Document # taic-52, Topology Atlas Invited Contributions vol. 9, no. 3 (2004) 6 pp.

**Is CH a Deﬁnite Mathematical Problem?**

Well, the non-finitary set-theoretical formulation of the Continuum Hypothesis isn’t, according to Solomon Feferman who, in a presentation at the inaugural Paul Bernays Lectures, ETH, Zurich, Sept. 12, 2012, restated in his presentation that:

My view: No; in fact it is essentially indeﬁnite (“inherently vague”).

That is, the concepts of arbitrary set and function as used in its formulation even at the level of P(N) are essentially indeﬁnite.

… *Why isn’t the Continuum Problem on the Millennium ($1,000,000) Prize List?* CSLI Workshop on *Logic, Rationality and Intelligent Interaction*, Stanford, June 1, 2013.

Feferman sought to place in perspective the anti-Platonistic basis for his belief by quoting:

“Those who argue that the concept of set is not sufﬁciently clear to ﬁx the truth-value of CH have a position which is at present difﬁcult to assail. As long as no new axiom is found which decides CH, their case will continue to grow stronger, and our assertion that the meaning of CH is clear will sound more and more empty.”

… D. A. Martin, *Hilbert’s ﬁrst problem: The Continuum Hypothesis,* in *Mathematical Developments arising from Hilbert Problems,* Felix E. Browder, Rutgers University, Editor – American Mathematical Society, 1976, 628 pp.

**A finitary arithmetical perspective**

However a possible candidate for a finitary arithmetical perspective (as proposed in the previous pages of these investigations) is reflected in the following:

**Theorem:** There is no set whose cardinality is strictly between the cardinality of the integers and the cardinality of the real numbers.

**Proof:** By means of Gödel’s -function , we can show that if denotes the digit in the decimal expansion of a putatively given real number R in the interval then, for any given natural number , we can define an arithmetical function such that:

for all .

Since Gödel’s -function is primitive recursive, it follows that every putatively given real number R can be uniquely corresponded to an algorithmically verifiable arithmetical function within the first order Peano Arithmetic PA, where we define by:

for all ,

and is selected such that:

for all .

(*For the purist, the above conclusion can be justified by the argument in this preprint.*)

**Definition: Algorithmically verifiable function**

A number-theoretical function is algorithmically verifiable if, and only if, for any given natural number , there is an algorithm which can provide objective evidence for deciding the value of each formula in the finite sequence .

“It is by now folklore … that one can view the *values* of a simple functional language as specifying *evidence* for propositions in a constructive logic …”.

… Chetan R. Murthy. 1991. *An Evaluation Semantics for Classical Proofs.* Proceedings of Sixth IEEE Symposium on Logic in Computer Science, pp. 96-109, (also Cornell TR 91-1213), 1991.

**Definition: Algorithmically computable function**

A number theoretical function is algorithmically computable if, and only if, there is an algorithm that can provide objective evidence for deciding the value of each formula in the denumerable sequence .

**Cantor’s diagonal argument**

From a finitary arithmetical perspective, Cantor’s diagonal argument simply shows that there are algorithmically verifiable functions which are not algorithmically computable.

The correspondence is unique because, if R and S are two different putatively given reals in the interval , then there is always some for which . Hence we can always find corresponding arithmetical functions and such that:

for all .

for all .

.

Since PA is first order, the cardinality of the reals in the interval cannot, therefore, exceed that of the integers. The theorem follows.

In other words, the Continuum Hypothesis is trivially true from a finitary perspective because of the seemingly heretical conclusion that: , an answer that Hilbert would probably never have envisaged for the first of the celebrated twenty three problems that he bequethed to posterity!

**Skolem’s (apparent) paradox**

It is an answer that should, however, give comfort to the shades of Thoralf Skolem. In his 1922 address delivered in Helsinki before the Fifth Congress of Scandinavian Mathematicians, Skolem improved upon both the argument and statement of Löwenheim’s 1915 theorem—subsequently labelled as the:

**(Downwards) Löwenheim-Skolem Theorem**

If a first-order proposition is satisfied in any domain at all, then it is already satisfied in a denumerably infinite domain.

Skolem then cautioned about unrestrictedly (and meta-mathematically) corresponding putative mathematical entities across domains of different axiom systems, and drew attention to a:

“… peculiar and apparently paradoxical state of affairs. By virtue of the axioms we can prove the existence of higher cardinalities, of higher number classes, and so forth. How can it be, then, that the entire domain can already be enumerated by means of the finite positive integers? The explanation is not difficult to find. In the axiomatization, ‘set’ does not mean an arbitrarily defined collection; the sets are nothing but objects that are connected with one another through certain relations expressed by the axioms. Hence there is no contradiction at all if a set of the domain is non-denumerable in the sense of the axiomatization; for this means merely that *within* there occurs no one-to-one mapping of onto (Zermelo’s number sequence). Nevertheless there exists the possibility of numbering all objects in , and therefore also the elements of , by means of the positive integers; of course such an enumeration too is a collection of certain pairs, but this collection is not a ‘set’ (that is, it does not occur in the domain ).”

… Thoralf Skolem. 1922. *Some remarks on axiomatized set theory.* Text of an address delivered in Helsinki before the Fifth Congress of Scandinavian Mathematicians, 4-7 August 1922. In Jean van Heijenoort. 1967. Ed. *From Frege to Gödel: A source book in Mathematical Logic, 1878 – 1931.* Harvard University Press, Cambridge, Massachusetts, p.295.

**What do you think?** Does the above argument apply to the finite ordinals? If so, is ZF inconsistent, or is it -consistent?

**Finitarily consistent mechanist reasoning and non-finitarily consistent human reasoning: Mutually inconsistent yet complementary!**

We now consider the following (tentatively expressed) conclusions suggested by our previous post, which we shall aim to investigate from various perspectives in these pages.

**Structures**

The Birmingham paper suggests that we may need to distinguish much more sharply than we do at present between:

Mathematical structures that are built upon only finitary reasoning, and

Mathematical structures that admit non-finitary reasoning.

**Interpretations**

For instance the Birmingham paper provides:

An example of a mathematical structure based on finitary reasoning, namely the finitarily sound algorithmic interpretation of the first order Peano Arithmetic PA.

An example of a mathematical structure based on non-finitary reasoning, namely the non-finitarily sound standard interpretation of the first order Peano Arithmetic PA.

The Birmingham paper suggests that the roots of the distinction between these two structures lies in the fact that:

Finitary reasoning does not assume that Aristotle’s particularisation is always true over infinite domains.

Non-finitary reasoning assumes that Aristotle’s particularisation is always true over infinite domains.

**Consistency of Arithmetic**

In the Birmingham paper we also show that:

Finitary reasoning proves that PA is consistent finitarily (as demanded by the second of Hilbert’s celebrated twenty three problems).

Non-finitary reasoning proves that PA is consistent non-finitarily (a consequence of Gentzen’s non-finitary proof of consistency for PA).

**FOL is consistent; FOL+AP is -consistent**

This suggests that:

Finitary reasoning as formalised in first order logic (FOL) is consistent.

Non-finitary reasoning as formalised in Hilbert’s -calculus (FOL+AP) is -consistent.

**-consistency**

Since the Birmingham paper shows that Aristotle’s particularisation holds over the structure of the natural numbers if, and only if, PA is -consistent, it suggests that:

Finitary reasoning does not admit that PA can be -consistent (see Corollary 4 of this post).

Non-finitary reasoning admits that PA can be -consistent.

**Arithmetical undecidability**

Since proofs of arithmetical undecidability implicitly assume Aristotle’s particularisation, this further suggests that:

Finitary reasoning does not admit undecidable arithmetical propositions (see Corollary 3 of this post).

Non-finitary reasoning admits undecidable arithmetical propositions.

**Completed Infinity**

A significant consequence is that:

Finitary reasoning does not admit an axiom of infinity.

Non-finitary reasoning admits an axiom of infinity.

**Non-standard models of PA**

A further consequence of this is that:

Finitary reasoning does not admit non-standard models of PA.

Non-finitary reasoning too does not admit non-standard models of PA.

**Algorithmically computable truth and algorithmically verifiable truth**

The Birmingham paper also suggests that:

The truths of finitary reasoning are algorithmically computable.

The truths of non-finitary reasoning are algorithmically verifiable, but not necessarily algorithmically computable.

**Categoricity and incompleteness of Arithmetic**

We show in Corollary 1 of this post that it also follows from the Birmingham paper that:

Finitary reasoning proves that PA is categorical with respect to algorithmically computable truth.

Non-finitary reasoning proves that PA is incomplete with respect to algorithmically verifiable truth (a consequence of Gödel’s proof of of the undecidability of some arithmetical propositions in any -consistent system of arithmetic).

**How intelligences reason**

This suggests that:

Finitary reasoning is a shared characteristic of all intelligences, human or non-human.

Non-finitary reasoning is a characteristic of human intelligence that may not be shared by any other intelligence.

**Communication between intelligences: SETI**

It further suggests that the search for extra-terrestrial intelligence may benefit from the argument that:

Finitary reasoning admits effective and unambiguous communication between two intelligences with respect to its (algorithmically computable) arithmetical truths.

Non-finitary reasoning does not admit effective and unambiguous communication between two intelligences with respect to its (algorithmically verifiable) arithmetical truths.

**Determinism, Unpredictability and the EPR paradox**

An unexpected consequence of the arguments of the Birmingham paper is that our perspectives on the relation between determinism and predictability may benefit from the paradigm shift demanded by the argument that:

Finitary reasoning admits the EPR paradox.

Non-finitary reasoning does not admit the EPR paradox.

The arguments of the Birmingham paper also suggest a fresh perspective on the issue of computationalism since:

Finitary reasoning does not admit Lucas’ Gödelian argument.

Non-finitary reasoning admits Lucas’ Gödelian argument.

**Effective computability**

It further suggests that the nature and status of ‘effective computability’ may also need to be assessed afresh since:

Finitary reasoning naturally equates algorithmic computability with effective computability.

Non-finitary reasoning naturally equates algorithmic verifiability with effective computability.

**Church Turing Thesis**

As also the nature of CT, since:

Finitary reasoning admits the Church-Turing Thesis.

Non-finitary reasoning does not admit the Church-Turing Thesis.

Broadly speaking, the two conflicting-but-complementary structures defined in the Birmingham paper suggest that we should be more explicit—in our argumentation—of the structure to which a particular assertion about the natural numbers pertains, since:

Both finitary and non-finitary reasoning do not admit the proof of Goodstein’s Theorem as neither admits a completed infinity.

Set-theoretical reasoning admits the proof of Goodstein’s Theorem as it admits a completed infinity.

**There’s more …**

In the next post we shall consider some further intriguing consequences suggested by the Birmingham paper.

**What do you think?**

Does Goodstein’s sequence over the natural numbers always terminate or not?

**Aristotle’s particularisation: A grey area in our accepted foundational concepts**

We shall now argue that what mathematics needs is not a new foundation, but a greater awareness of the nature of its existing foundations.

In particular, it is the thesis of these investigations that almost all of the unresolved philosophical issues in the foundations of mathematics reflect the fact that the nature and role of Aristotle’s particularisation is left implicit when it is postulated over infinite domains.

Perhaps that is the unintended consequence of ignoring Hilbert’s efforts to integrate the concept formally into first order logic by formally defining universal and existential quantification through the introduction of his -operator.

**Semantic postulation of Aristotle’s particularisation **

Aristotle’s particularisation (AP) is the postulation that from the negation of a universal we may always deduce the existence of a contrafactual.

(*It is necessarily true over finite domains.*)

More formally:

**Aristotle’s particularisation under an interpretation**

If the formula of a first order language interprets as true under a sound interpretation of , then we may always conclude that there must be some object in the domain of the interpretation such that, if the formula interprets as the unary relation in , then the proposition is true under the interpretation.

(*We note that Aristotle’s particularisation is a non-constructive—and logically fragile—semantic deduction rule. It is reflected in classical first order deduction either by some similarly non-constructive syntactic rule of natural deduction—such as Rosser’s Rule C—or by the assumption that FOL is -consistent.*)

**Is the price of Aristotle’s particularisation too high?**

If so, we shall argue that the price being asked for assuming AP implicitly—instead of explicitly as Hilbert had proposed—may be too high!

Partially because the assumption seems to effectively obscure the far-reaching consequences of the non-finitary nature of AP from immediate view in natural and formal deductive chains.

(*And therefore of the first order logic FOL under the implicit assumption of Aristotle’s particularisation.*)

For instance, as Carnap’s deduction of the Axiom of Choice in ZF illustrates, the non-finitary consequences of assuming AP over infinite domains becomes apparent when the underlying logic is taken as Hilbert’s -calculus instead of the classical first order logic FOL.

However, formal deductions apparently prefer to substitute—seemingly arbitrarily—the implicit assumption of AP in the underlying logic by the introduction of `contrived’ formal assumptions such as Gödel’s -consistency or Rosser’s Rule C.

**-consistency**

A formal system S is -consistent if, and only if, there is no S-formula for which, first, is S-provable and, second, is S-provable for any given S-term .

**Rosser’s Rule C**

“Since the rule ‘If , then ‘ corresponds to a hypothetical act of choice, we shall call it the rule of choice, or more briefly, Rule C.”

… J. Barkley Rosser. *Logic for Mathematicians.* 1953. McGraw Hill Book Company Inc., New York.

Similarly natural deduction chains apparently prefer to substitute—again seemingly arbitrarily—the implicit assumption of AP in the underlying logic by admitting a Rule of Infinite Induction (transfinite induction).

More importantly, the price may be too high because the implicit assumption of Aristotle’s particularisation in the underlying logic has masked the fact that, without such assumption, FOL is finitarily consistent; and—as we note below—that mathematically significant finitary structures can be built upon it without assuming AP.

**Evidence-Based Interpretations of PA**

Some consequences of making the assumption of Aristotle’s particularisation explicit are highlighted in `Evidence-Based Interpretations of PA’ that was presented at the AISB/IACAP Turing 2012 conference in Birmingham last year.

We showed there that Tarski’s inductive definitions admit evidence-based interpretations of the first-order Peano Arithmetic PA which allow us to define the satisfaction and truth of the quantified formulas of PA *constructively* over the domain of the natural numbers in *two* essentially different ways:

In terms of algorithmic verifiabilty; and

In terms of algorithmic computability.

That there can be even *one*, let alone *two*, logically sound (one finitary and one non-finitary) assignments of satisfaction and truth certificates to both the atomic and compound formulas of PA had hitherto been unsuspected!

**Definition: Algorithmically verifiable arithmetical truth**

A number-theoretical relation is algorithmically verifiable if, and only if, for any given natural number , there is an algorithm which can provide objective evidence for deciding the truth/falsity of each proposition in the finite sequence .

“It is by now folklore … that one can view the *values* of a simple functional language as specifying *evidence* for propositions in a constructive logic …”.

… Chetan R. Murthy. 1991. *An Evaluation Semantics for Classical Proofs.* Proceedings of Sixth IEEE Symposium on Logic in Computer Science, pp. 96-109, (also Cornell TR 91-1213), 1991.

We show in the Birmingham paper (as we shall refer to it hereafter) that the `algorithmic verifiability’ of the formulas of a formal language which contain logical constants can be inductively defined under an interpretation in terms of the `algorithmic verifiability’ of the interpretations of the atomic formulas of the language; further, that the PA-formulas are decidable under the standard interpretation of PA over if, and only if, they are algorithmically verifiable under the interpretation.

**Definition: Algorithmically computable arithmetical truth**

A number theoretical relation is algorithmically computable if, and only if, there is an algorithm that can provide objective evidence for deciding the truth/falsity of each proposition in the denumerable sequence .

We show in the Birmingham paper that the `algorithmic computability’ of the formulas of a formal language which contain logical constants can also be inductively defined under an interpretation in terms of the `algorithmic computability’ of the interpretations of the atomic formulas of the language; further, that the PA-formulas are decidable under an algorithmic interpretation of PA over if, and only if, they are algorithmically computable under the interpretation.

**Algorithmic verifiability vis à vis algorithmic computability**

We show in the Birmingham paper that the concepts of Algorithmic verifiability and Algorithmic computability are both well-defined under the standard interpretation of PA over ; moreover they identify distinctly different subsets of the well-defined PA formulas.

We show in this paper that although every algorithmically computable relation is algorithmically verifiable, the converse is not true.

We note that algorithmic computability implies the existence of an algorithm that can decide the truth/falsity of each proposition in a well-defined denumerable sequence of propositions, whereas algorithmic verifiability does not imply the existence of an algorithm that can decide the truth/falsity of each proposition in a well-defined denumerable sequence of propositions.

From the point of view of a finitary mathematical philosophy—which is the constraint within which an applied science ought to ideally operate—the significant difference between the two concepts could be expressed (as addressed in more detail in this paper) by saying that we may treat the decimal representation of a real number as corresponding to a physically measurable limit—and not only to a mathematically definable limit—if and only if such representation is definable by an algorithmically computable function.

**The finitarily sound algorithmic interpretation of PA over **

We argued from the above distinction that the algorithmically computable PA-formulas *can* provide a finitarily sound algorithmic interpretation of PA over the domain of the natural numbers.

We showed, moreover, that this yields a finitary proof of consistency for PA—as demanded by the Second of Hilbert’s celebrated Twenty Three Problems.

**The non-finitarily sound standard interpretation of PA over **

On the other hand, the distinction also suggests that Gerhard Gentzen’s transfinite proof of consistency for PA corresponds to the argument that the algorithmically verifiable PA-formulas of PA provide a non-finitarily sound standard interpretation of PA over .

Moreover—as has been generally suspected (perhaps for the reason noted towards the end of this post)—the distinction also suggests why the standard interpretation cannot yield the finitary proof of consistency for PA as demanded by Hilbert.

**The distinction between finitary and non-finitary arithmetical reasoning introduced in the Birmingham paper has far reaching consequences**

In these pages we shall argue that the power of this simple distinction actually goes far beyond the immediate conclusions drawn in the Birmingham paper.

Reason: We can further constructively define an unambiguous distinction between finitary and non-finitary reasoning, at the level of first order logic itself, which shows that the two are both mutually inconsistent yet complementary!

As can be expected, such a distinction could have far-reaching consequences for the foundations of mathematics, logic and computabiity (which form the focus of the investigations in these pages).

We shall consider some of these in the next post.

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