When is the concept of a completed infinity consistent with a formal language?

The second issue is when, and whether, the concept of a completed infinity is consistent with the interpretation of a formal language.

Clearly, the consistency of the concept would follow immediately in any sound interpretation [1] of the axioms (and rules of inference) of a set theory such as ZF, but whether such an interpretation exists at all is, of course, another question.

In view of the perceived power of ZF [2] as an unsurpassed language of rich and adequate expression of mathematically expressible abstract concepts precisely, it is not surprising that many of the semantic and logical paradoxes depend on the implicit assumption that:

(i) such an interpretation may be assumed to exist, and that

(ii) the domain over which the paradox quantifies can always be assumed to be a well-defined mathematical object that can be formalised in ZF, even if this domain is not explicitly defined (or definable) set-theoretically.

This assumption is rooted in the questionable belief that ZF can express all mathematical `truths’ [3].

From this it is but a short step to the non-constructive argument—rooted in Gödel’s Platonic interpretation of his own formal reasoning in his seminal 1931 paper on `undecidable’ arithmetical propositions [4]—that PA must have non-standard models.

However, it is our contention that both of the above foundational issues need to be reviewed carefully, and that we need to recognize explicitly:

(a) the limitations on the ability of highly expressive mathematical languages such as ZF to communicate effectively; and

(b) the limitations on the ability of effectively communicating mathematical languages such as PA to adequately express abstract concepts—such as those involving Cantor’s first limit ordinal \omega [5].

Prima facie, the semantic and logical paradoxes—as also the seeming paradoxes associated with fractal constructions such as the Cantor ternary set and the constructions described below—seem to arise out of a blurring of this distinction, and an attempt to ask of a language more than it is designed to deliver.

A paradoxical fractal `construction’

For instance, consider the claim [6] that fractal `constructions’—such as the `Cantor ternary set’ [7]—yield (presumably in some interpretable sense) valid mathematical objects (sets) in the `limit’.

We consider an equilateral triangle BAC of height h and side s.

Divide the base BC in half and construct two isosceles triangles of height h.d and base s/2 on BC, where 1 \geq d>0.

Iterate the construction on each constructed triangle ad infinitum (see Figs 1-3 below).

Thus, the height of each of the 2^{n} triangles on the base BC at the n‘th construction is h.d^n, and the base of each triangle s/2^{n}.

Hence, the total area of all these triangles subtended by the base BC is s.h.d^{n}/2.

Now, if d=1, the total area of all the constructed triangles after each iteration remains constant at s.h/2, although the total length of all the sides opposing the base BC increases monotonically.

Moreover, if 1>d>0, it would appear that the base BC of the original equilateral triangle will always be the `limiting’ configuration of the sides opposing the base BC.

Fig 1

Fig 1

This is indeed so if 0<d<1/2, since (see fig. 1 above) the total length of all the sides opposing the base BC at the n‘th iteration—say l_{n}—yields a Cauchy sequence whose limiting value is, indeed, the length s of the base BC.

Fig 2

Fig 2

However, if d=1/2, the total length (see fig. 2 above) of all the sides opposing their base on BC is always 2s!

Fig. 3

Fig. 3

Finally, if 1>d>1/2, the total length (see fig. 3 above) of all the sides opposing their base on BC is a monotonically increasing value.

Consider now:

Case 1 Treating the above iteration as the fragmentation of a land-holding, how would one interpret the `limit’ of such an interpretation (which is postulated as existing in a putative `completion’ of Euclidean space)?

Case 2 Let s be one light-year and consider how long it would take a light signal to travel from B to C along the sides opposing the base in each of the above cases.

Case 3 Let the area BAC denote the population size of a virus cluster, where each virus cell has a `virulence’ measure h/s. Let each triangle at the n‘th iteration denote a virus cluster—with a virulence factor h.d^n/(s/2^{n})—that reacts to the next generation anti-virus by splitting into two smaller clusters with inherited virulence h.d^{n+1}/(s/2^{n+1}).

If d<1/2, the effects of the virus can—in a sense—be contained and eventually `eliminated’. If d=1/2, the effects of the virus can be `contained’, but never `eliminated’. However, if d>1/2, the effects of the virus can neither be `contained’ nor `eliminated’.

Case 4 Let the base BC denote an elastic string, stretched iteratively into the above configurations. If d<1/2, the elastic will, in principle, eventually return to its original state. If d>1/2, then the elastic must break at some point. However, what if d=1/2?

Zeno’s paradox in 2-dimensions!

If d=1/2, we then arrive at a two-dimensional version of Zeno’s paradox; one way of resolving which is by admitting the possibility that such an elastic `length’ undergoes a phase change in the limit that need not correspond to the limit of its associated Cauchy sequence!

The question arises: Since the raison d’etre of a mathematical language is—or should be [8]—to express our abstractions of natural phenomena precisely and communicate them unequivocally, in what sense can we sensibly admit an interpretation of a mathematical language that asserts all the above cases as having `limiting’ configurations in a putative `completion’ of Euclidean Space?


Ba88 Michael Barnsley. 1988. Fractals Everywhere. Academic Press, Inc., San Diego.

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.

Fe85 Richard P. Feynman. 1985. Judging Books by Their Covers in Surely You’re Joking, Mr. Feynman! (Adventures of a curious character). Norton, New York. (Extracts: Judging Books by Their Covers.)

Ff02 Solomon Feferman. 2002. Predicativity. Source: http://math.stanford.edu/~feferman/papers/predicativity.pdf.

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.

He04 Catherine Christer-Hennix. 2004. Some remarks on Finitistic Model Theory, Ultra-Intuitionism and the main problem of the Foundation of Mathematics. ILLC Seminar, 2nd April 2004, Amsterdam.

Hi27 David Hilbert. 1927. The Foundations of Mathematics. Text of an address delivered in July 1927 at the Hamburg Mathematical Seminar. In Jean van Heijenoort. 1967. Ed. From Frege to Gödel: A source book in Mathematical Logic, 1878 – 1931. Harvard University Press, Cambridge, Massachusetts.

Me64 Elliott Mendelson. 1964. Introduction to Mathematical Logic. Van Norstrand.

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.

Na08 Melvyn B. Nathanson. 2008. Desperately Seeking Mathematical Truth. Opinion in the August 2008 Notices of the American Mathematical Society, Vol. 55, Issue 7.

Pa95 Charles Parsons. 1995. Platonism and Mathematical Intuition in Kurt Gödel’s Thought. The Bulletin of Symbolic Logic, Volume 1, Number 1, March 1995, pp. 44-74.

Ru53 Walter Rudin. 1953. Principles of Mathematical Analysis. McGraw Hill, New York.

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

Ta33 Alfred Tarski. 1933. The concept of truth in the languages of the deductive sciences. In Logic, Semantics, Metamathematics, papers from 1923 to 1938 (p152-278). ed. John Corcoran. 1983. Hackett Publishing Company, Indianapolis.

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.

Tu39 Alan Turing. 1939. Systems of logic based on ordinals. In M. Davis (ed.). 1965. The Undecidable. Raven Press, New York. Reprinted from the Proceedings of the London Mathematical Society, ser. 2. vol. 45 (1939), pp.161-228.

Ya93 Stephen Yablo. 1993. Paradox without self-reference. Analysis, 53(4), pp. 251-252.

An07 Bhupinder Singh Anand. 2007. Why we shouldn’t fault Lucas and Penrose for continuing to believe in the Gödelian argument against computationalism – I. The Reasoner, Vol(1)6 p3-4.

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.


Return to 1: We define an interpretation I_{S} of a formal system S as sound if, and only if, every provable formula [F] of S translates as a true proposition F under the interpretation (cf. BBJ03, p.174).

Return to 2: More accurately, ZFC.

Return to 3: In a series of nine previous posts starting here we have shown how—in the case of Goodstein’s Theorem—such a belief leads to a curious argument.

Return to 4: Go31.

Return to 5: In this previous post we show why we cannot add a symbol corresponding formally to the concept of an `infinite’ mathematical entity (such as is referred to symbolically in the literature by `\aleph‘ or `\omega‘) to the first-order Peano Arithmetic PA without inviting inconsistency.

Return to 6: See Ba88, p.35, §2.7, The completeness of the space of fractals.

Return to 7: Defined as a set-theoretical limit of an iterative process in the completion of a metric space; see, for instance, Ru53, p.54; Ba88, p.45.

Return to 8: See, for instance, Richard P. Feynman’s delightful commentary (Fe85) on his horrifying experiences of how the teaching of mathematical languages was actually introduced into the schools curricula in the State of California in 1964.

Bhupinder Singh Anand