0809.2135v1[1] Cantor Paradox

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EXTENDING CANTOR’S PARADOX

A CRITIQUE OF INFINITY AND SELFREFERENCE

Antonio Leon SanchezI.E.S Francisco Salinas, Salamanca, Spain

http://[email protected]

Abstract. This paper examines infinity and self-reference froma critique perspective. Starting from an extension of Cantor Para-dox that suggests the inconsistency of the actual infinite, the papermakes a short review of the controversial history of infinity and sug-gests several indicators of its inconsistency. Semantic self-referenceis also examined from the same critique perspective by comparingit with self-referent sets. The platonic scenario of infinity and self-reference is finally criticized from a biological and neurobiologicalperspective.

1. Introduction

The most relevant problems of contemporary philosophy were alreadyposed by Presocratic thinkers as early as the VII century b.C. (some ofthem perhaps suggested or directly taken from the cultural precedentsdeveloped in the Neolithic Fluvial Cultures [16], [147], [123], [161]).Three among those problems deserves special consideration: the prob-lem of change; infinity; and self-reference. The first one is surely themost difficult problem ever posed by man. For that reason, it surprisesthe little attention we currently pay to such a challenging question,specially when compared to the attention we pay to the other two. Af-ter more than twenty seven centuries, the problem of change remainsunsolved (for a general background see [117]). In spite of its apparentobviousness, no one has been capable of explaining how one leaves aplace to occupy another. The science of change -physics- seems to haveforgotten its most fundamental problem. In their turn, some philoso-phers from Hegel declared it as an inconsistent process! [78], [80], [110],[125], [136], [169], while others, as McTaggart, came to the same con-clusion as Parmenides on the impossibility of change [109].

Contrarily to the problem of change, which has an immediate re-flection in the agitated evolution of the Universe, both infinity andself-reference are pure theoretical devices without reflection in the nat-ural world. Cantor and Godel (the princes of infinity and self-referencerespectively) were two platonic fundamentalists of scarce devotion tonatural sciences and enormous influence in contemporary mathematics(for the case of Cantor see [44], [112], [33, pag. 141]; for that of Godel

1

http://arXiv.org/abs/0809.2135v1

linkto:http://www.interciencia.es

mailto:[email protected]

2 Extending Cantor’s paradox

[66, pags. 235-236], [67, pag. 359], [46] [118], [68]). Twenty seven cen-turies of discussions were not sufficient to prove the consistency of theactual infinity, which finally had to be legitimated by the expeditiousway of axioms. Although the consistency of self-reference has been lessdiscussed, the paradoxes it originates have been, and continue to be,the source of interminable discussions. One of them, the Liar Paradox,became (via Richard Paradox) the first of the two celebrated Godel’sincompleteness theorems1 [64].

Infinity and self-reference may be two of the most important ob-structions in the history of human thought. Obstructions that mayhave been driving our attention, effort and resources in the wrong wayif what we pretend is to explain the physical world our lives evolve in.The scarce (frequently null) naturalist education of mathematicians aswell as the scarce mathematical education of naturalists perhaps ex-plain because we have come so far in both affairs.

2. Paradoxes and inconsistencies

It is paradoxical that formal literature be not more exigent in the useof the term ’paradox,’ frequently used in the place of ’contradiction’ or’inconsistency.’ When, for instance, we write:

H ⇒ (c ∧ ¬c) (1)

we say H is inconsistent because it is the immediate cause of con-tradiction (c ∧ ¬c). But when the cause of the contradiction is notso immediate and well known it is relatively frequent to use the termparadox instead of contradiction. When, finally, we write a self-referentproposition as:

p = p is false (2)

we have on the one hand, and according to the Principle of Identity:

p ⇒ p (3)

and on the other, and according to the own proposition p:

p ⇒ ¬p (4)

Whence:p ⇒ (p ∧ ¬p) (5)

In these cases we invariably use the term ’paradox’ instead of ’contra-diction’ (in this case the celebrated ’if p then not p’ and ’if not p thenp’).

1In secondary literature, the first Godel’s incompleteness theorem is invariablypresented in a dishonest way. The reader get the idea that in every formal systemthere are true propositions which are undemonstrable. Godel actually proved theexistence of one proposition which is both true and undemonstrable; but thatproposition is a very very special one: it is a self-referent proposition. One that,conveniently interpreted, says of itself that it is undemonstrable in a certain formalcalculus. Something like: this proposition is undemonstrable

Extending Cantor’s paradox 3

The paradoxes of Set Theory we will examine here are surely in-consistencies derived from both the actual infinity and self-reference.As is well known, Burali-Forti made use of his famous paradox as aninconsistency to prove that the law of trichotomy does not hold for alltransfinite ordinals [25]. In his turn, Cantor did not hesitate in usingthe expression ’inconsistent totalities’ (letter to Dedekind quoted in[44, pag. 245]) to refer to certain sets we will immediately deal with.In any case, it is not the purpose of this work to state when and howwe should use the terms paradox and inconsistency or contradiction.In this sense we will be so informal as the formal literature related tomathematical formalism.

3. Extending Cantor Paradox

Although Burali-Forti was the first in publishing an inconsistency re-lated to transfinite sets [24], [62], Cantor was the first to discover aparadox in the nascent set theory: I am referring to the maximumcardinal paradox [62], [44]. There is no agreement regarding the dateCantor discovered his paradox [62] (the proposed dates range from 1883[135] to 1896 [70]). Burali-Forti Paradox on the set of all ordinals andCantor Paradox on the set of all cardinals are both related to the sizeof the considered totalities2, perhaps too big as to be consistent. Froman infinitist perspective it results ironic that a set may be inconsistentjust because of its excessive size. One can be infinite but only withincertain limits.

A common way of presenting Cantor Paradox is as follows: Let U bethe set of all sets, the so called universal set3 and P (U) its power set,

the set of all its subsets. As Cantor did, let us denote by U and P (U)their respective cardinals. Being U the set of all sets, we can write:

U ≥ P (U) (6)

On the other hand, and according to Cantor theorem on the power set[29], it holds:

U < P (U) (7)

which contradicts (6). This is the famous Cantor inconsistency or para-dox. As is well known, Cantor gave no importance to it [59] and solvedthe question by assuming the existence of two types of infinite totali-ties, the consistent and the inconsistent ones [28]. In Cantor opinion,the inconsistency of the lasts would surely due to their excessive size.

2Later we will deal with Russell Paradox, although this paradox is related toself-reference rather than to cardinality. As we will see, Burali-Forti and CantorParadoxes are also related to self-reference.

3Cantor Paradox arose in what we now call naive set theory, which admits setsas the universal set.


We would be in the face of the mother of all infinities, the absoluteinfinity which directly leads to God [28].

As we will immediately see, it is possible to extend Cantor Paradoxto other sets much more modest than the universal set. But neitherCantor nor his successors considered such a possibility. We will doit here. This is just the objective of the discussion that follows. Adiscussion that will take place within the context of Cantor set theory(naive set theory), although with some side effects on modern axiomaticset theories.

We begin by considering a set X such that X = C > 1. From X wedefine set TX as:

TX : P (X) ⊂ TX ∧ ∀Y (Y ⊂ TX ⇒ P (Y ) ⊂ TX) (8)

Id est, TX contains as elements all subsets of set X; in addition, if Y isa subset of TX , its power set P (Y ) is also a subset of TX . We will seenow that TX is an inconsistent totality. For this, let A be any propernonempty subset of X. From A we define set TA in accordance with:

TA = {Y |Y ∈ TX ∧ Y ∩ A 6= A} (9)

TA is, therefore, the set of all elements of set TX that do not contain assubset the subset A of X. TA is then a subset of TX . Consequently, andaccording to (8), its power set P (TA) is also a subset of TX . Thereforethe elements of set P (TA) are also elements of TX . Moreover, it is clairthat:

∀Y ∈ P (TA) : Y ∩ A 6= A (10)

In consequence all elements of set P (TA) satisfy definition (9). So,them all belong to TA. We have therefore a similar situation to thatof Cantor Paradox. On the one hand and taking into account that TA

contains all elements of P (TA) we will have

TA ≥ P (TA) (11)

On the other and according to Cantor theorem it must hold:

TA < P (TA) (12)

Again a contradiction. And then the totalities TA and TX from whichit arises must be inconsistent. But X is any set of any cardinal C > 1and A any one of its nonempty proper subsets. We can therefore statethat every set of cardinal C gives rise to at least 2C − 2 inconsistenttotalities.

The above argument not only proves the number of inconsistent to-talities is inconceivable greater than the number of consistent ones, italso suggests the excessive size of the sets could not be the cause ofthe inconsistency. In fact, consider, for instance, the set X = {1, 2};according to the above argument, the totality T{1,2} defined in accor-dance with (8) is also inconsistent. But compared to the universal set


it is an insignificant totality: it lacks of practically all sets of numericalcontent4 or related to numbers (integers, rationals, reals, complexes,hiperreals, etc.); and all of non numerical content. T{1,2} is in fact aninsignificant subset of the universal set, it lacks of almost all its ele-ments. But it is also inconsistent. What both sets have in common,from the infinitist perspective, is that they are complete infinite total-ities; i.e. totalities whose infinitely many elements exist all at once.

Had we know the existence of so many inconsistent infinite totali-ties, and not necessarily so greater as the absolute infinity, and perhapsCantor transfinite set theory would have been received in a differentway. Perhaps the very notion of actual infinity would have been putinto question; and perhaps we would have discovered the way of prov-ing its inconsistency: just the ω-order (see below). But this was notthe case. The history of the reception of set theory and the way ofdealing with its inconsistencies -all of them promoted by the actualinfinity and self-referenceis well known. From the beginnings of theXX century a great effort have been carried out to found set theory ona consistent background free of inconsistencies. Although the objectivecould only be reached with the aid of the appropriate axiomatic parch-ing. At least half a dozen axiomatic set theories have been developedever since5 [84]. Some hundred pages are needed to explain all axiomsof contemporary axiomatic set theories. Just the contrary one expectson the foundations of a formal science as mathematics.

4. Criticism of the actual infinity

The history of infinity is one of a long and interminable controversybetween its supporters and opponents. A controversy in which one ofthe alternatives always dominated on the other, although the domi-nant alternative was not always the same. Fortunately we dispose ofan excellent and abundant literature on the history of infinity and itscontroversies (see for instance: [178], [101], [149], [18], [141], [39], [94],[113], [116], [89], [90], [1], [114], [38], [170] etc.). Here we will limit our-selves to recall the most significant details of that history, from Zenoof Elea to the actual paradise. Apart from the absolute hegemony ofinfinitism (supporters of the actual infinity) in contemporary math-ematics, it surprises the lack of critical attitudes towards the actualinfinity, even among its opponents (finitists). And that in spite of theparadoxes and extravagances its assumption gives rise, not to say itscognitive sterility in experimental sciences or the tremendous difficul-ties it poses in certain areas of physics (as renormalization in physicsof elementary particles).

4Recall, for instance, that between any to real numbers an uncountable infinity(2ℵ0) of other different reals numbers do exist. What, as Wittgenstein would surelysay, makes one feel dizzy [174].

5There are also some contemporary attempts to recover naive set theory.


As is well known, the history of infinity begins with Zeno of Elea andhis famous paradoxes. Although we know nothing on the true Zeno’sintentions, surely more related to the problem of Change6 than to themathematical infinity. Scarcely 300 words from the original Zeno’stexts have survive until the present [16]. It is by his doxographers7

that we know Zeno’s original works. Thus it seems proved he written abook with more than forty arguments in defense of Parmenides’ thesis.We also know the book was available in Plato Academy and that itwas read by the young Aristotle [40]. By way of example consider thereader to following original Zeno’s argument ([16, pag. 177]):

What moves, don’t move neither in the place it is norin that it is not.

It seems impossible to say more with so few words. The famous race be-tween Achilles and the tortoise and Achilles passing over the so called[168] Z-points8 (Dichotomy I) and Z∗-points9 (Dichotomy II) posedvery difficult problems in which the actual infinity was already involved[144], [114], [106]. Aristotle suggested a solution to both dichotomies:the one to one correspondence between the successive Z-points (or Z∗-points) and the successive instants at which Achilles passes over them[9]. But Aristotle was not convinced of his own solution and suggestedanother radically different one: the consideration of two types of in-finities, namely the actual and the potential infinity. According to thefirst, infinite totalities would be complete totalities. In accordance withthe second, infinite totalities would always be incomplete so that theonly complete totalities would be the finite totalities. As is well known,Aristotle preferred the second type of infinitude, the potential infinity.And until the ends of the XIX century that was the dominant alter-native on the infinity, although the other always maintained an activeaffiliation. In its turn, Zeno’s Paradoxes still remains unsolved, in spiteof certain proposed solutions and pseudo-solutions10

6Both Parmenides and his pupil Zeno defended the impossibility of change withmany convincing arguments.

7Among them Plato [127], Aristotle [9] and Simplicius [152].8The ω-ordered sequence of points 1/2, 3/4, 7/8, 15/16, . . . Achilles traverses in

his race from point 0 to 1.9The ω

∗-ordered sequence of points . . . , 1/16, 1/8, 1/4, 1/2 Achilles traverses inhis race from point 0 to 1.

10Most of those solutions needed the development of new areas of mathematics,as Cantor’s transfinite arithmetics, topology, or measure theory [73], [74], [176], [75],[77], [76], and more recently internal set theory [108], [107]. It is also remarkablethe solutions posed by P. Lynds in the framework of quantum mechanics [97], [98].Some of those solutions, however, have been seriously contested [124], [3], and inmost cases the proposed solutions do not explain where Zeno arguments fail [124],[133]. In addition, the proposed solution have given rise to new problems so excitingas Zeno’s original ones [144], [86] [150].


Galileo famous paradox11 on the natural numbers and their squares[61] introduced a significant novelty in the controversy on the mathe-matical infinity: the properties of infinite numbers could be differentfrom those of the finite numbers. That is at least the only explanationGalileo found to the paradoxical fact that existing more natural num-bers than perfect squares both sets can be put into a one to one corre-spondence (f(n) = n2, ∀n ∈ N). In fact, Galileo paradox only arises ifboth sets are considered as complete totalities (hypothesis of the actualinfinity). From the perspective of the potential infinite Galileo para-dox don’t arise because from that perspective both sets are incompletetotalities so that we can only pair finite totalities of each one of them,finite totalities with the same number of elements. As large as we wishbut always finite and incomplete. If in the place of the perfect squaresGalileo would had considered the expofactorials perhaps he would nothave come to the same conclusion. The expofactorial of a number n,written as n!, is the factorial n! raised n! times to the power of n!. So,while the expofactorial of 2 is 16, the expofactorial of 3 is:

3! = 6666666

= 6666646656

= 666626591197721532267796824894043879...

(13)

where the incomplete exponent of the last expression has nothing lessthan 36306 digits (roughly ten pages of standard text). Modest 3! isa number so large that no modern (nor presumably future) computercan calculate it. Imagine, for instance 100!. The set E of expofactori-als lacks of practically all natural numbers and, however, can also beput into a one to one correspondence with the set of natural numbers(through the bijection f(n) = n!). Naturally, there is no problem inpairing off the firsts n natural numbers with the firsts n expofactorials,being the n-th expofactorial inconceivably greater than the n-th natu-ral. Other thing is to claim that both sets exist as complete totalitieswith the same number of elements. And things may get worse withhyperfactorials, being the hyperfactorial of n, denoted by n!!, the expo-factorial n! raised n! times to the power of n!. Or ultrafactorials, beingthe ultrafactorial of n, denoted by n!!! the hyperfactorial n!! raised n!!

times to the power of n!! . . .The Euclidean Axiom of the Whole and the Part12 controlled the in-

finitist impulses of authors as al-Harrani [96], Leibniz13 [91] or Bolzano

11Un example of the so called paradoxes of reflexivity, in which a whole is putinto a one to one correspondence with one of its proper parts [149], [50]. Thistype of paradoxes had already posed by many other authors as Proclus, J. Filopon,Thabit ibn Qurra al-Harani, R. Grosseteste, G. de Rimini, G. de Ockham etc. [149].

12”The whole is greater than the part”. Common Notion 5. Book 1 of Euclid’sElements [57]

13Leibniz position on the infinity is more ambiguous [114], [10] than could beexpected from his famous declaration of supporter of the actual infinity [91, pag.416]


[22]. Among the most famous bijections (one to one correspondences)between a whole and one of its parts are the classical ones defined byBolzano between a real interval and one of its proper subintervals. Inspite of which, Bolzano did not believe they were a sufficient conditionto prove the equipotence of the paired intervals. [21, §22, 33, translatedfrom [149]]. Dedekind took the next step by assuming that exhaustiveinjections (bijections or one to one correspondences) were in fact a suf-ficient condition to prove equipotence. This implies to assume thattwo sets have the same number of elements if they can be put into aone to one correspondence. Or in other words, if after pairing everyelement of a set B with a different element of a set A, no element ofA results unpaired (exhaustive injection), then sets A and B have thesame number of elements. This is a reasonable assumption. But it isalso reasonable to assume that if after pairing every element of a set B

with a different element of a set A, one or more elements of A remainsunpaired (non-exhaustive injection), then A and B have not the samenumber of elements. We are then faced with a dilemma: if exhaustiveinjections and non-exhaustive injections have or not the same level ofconclusiveness when used as instruments to compare the number ofelements of two infinite sets. If they have, then the actual infinity isinconsistent because in these conditions it can be proved that an infi-nite set have (exhaustive injection) and does not have (non-exhaustiveinjection) the same number of elements as one of its proper subsets14.

There is no reason to assume that non-exhaustive injections are lessrelevant than exhaustive ones to state if two sets have, or not, thesame number of elements. Thus modern infinitism should explicitlydeclare, by an appropriate ad hoc axiom, a new singularity of infinitesets, namely that the number of their elements cannot be comparedby means of non-exhaustive injections. But, as is well known, no stephas been taken in this direction. On the contrary, infinitists defineinfinite set just as those that can be put into an exhaustive and injectivecorrespondence with one of its proper subsets, ignoring that this subsetcan also be put into a non-exhaustive and injective correspondence withthe infinite set. As Dedekind proposed, infinitist definition of infinitesets is entirely based on the violation of the old Euclidean Axiom of theWhole and the Part. And committed the violation, the infinitist orgydon’t make us wait: Cantor inaugurated the actual infinitist paradise

just at the end of the XIX century. Paradise that for others authors asBrouwer, Poincare o Wittgenstein, was rather a nightmare, a maladyor even a joke. Thus, from the beginning of the XX century, infinitismbecome absolutely dominant in mathematics. Although finitist hasnot disappeared its presence in contemporary mathematics is almostirrelevant. In addition, the critique of infinity is practically non-existentand the scarce efforts carried out have been systematically rejected in

14See ’Reintrepreting the Paradoxes of Reflexivity’ at http://www.interciencia.es



behalf of the supposed differences between the infinite and the finitenumbers (see for instance [142]).

At the beginning of the XXI century a new way of criticism has beenopened. It is based on the notion of ω-order, a formal consequence ofthe actual infinity as Cantor himself proved [32, pags. 110-118]. Al-though the new way of criticism has its roots in certain discussions thattook place at the beginning of the second half of the XX century. Infact, in the year 1954 J. F. Thomson introduced the term supertask15

in a famous argument regarding a lamp that is turned on and off infin-itely many times. Thomson defended the impossibility of performingsuch a supertask [166]. His argument was motivated for other similararguments defended by M. Black with its infinite machines [19], andcriticized by R. Taylor [165] and J. Watling [171]. The argument ofThomson lamp was in turn severally criticized by P. Benacerraf [15],and that criticism gave rise to a new infinitist theory at the end of theXX century: supertask theory (see, for instance, [23], [37], [133]). Inthe last years supertask theory has extended its theoretical scenarioto the physical world. The possibility of actually performing certainsupertask have been discussed ([126], [129], [133], [144], [75], [77], [76]([129], [130], [131], [55], [132], [122], [4], [5], [134] [172], [83], [53], [54],[122], [52], [150]. As could be expected the performance of such su-pertasks would imply the most extravagant pathologies in the physicalworld, both newtonian and relativist. But what really surprises is thatin the place of considering such unbelievable pathologies as clair indica-tives of the inconsistency of supertasks (and then of the actual infinity),its defenders prefer to accept those pathologies before questioning theconsistency of the actual infinity.

As is well known, a set is ω-ordered if it has a first element andevery element has an immediate successor and an immediate prede-cessor (except the first one). The set N of natural number with itsnatural order of precedence is a clair example of ω-ordered set. Fromthe point of view of the actual infinity, an ω-ordered set is a completetotality, although no last element completes it. Thus an ω-ordered listis one that is simultaneously complete (as the actual infinity requires)and uncompletable (because no last element completes it). The exis-tence of lists that are simultaneously complete and uncompletable isanother inevitable extravagance of the actual infinity, although in thiscase it is less known than others (perhaps because being simultaneouslycomplete and uncompletable seems contradictory even for infinitists).Still less known is the following asymmetry also derived from ω-order.Imagine a straight line segment of 30000 millions light-years (withinthe scale of the supposed size of the Universe) and denote its two ex-tremes by A and B. Assume AB is divided into an ω-ordered sequence

15The notion of supertask is much more older. J. Gregory used it in the XVIIcentury [114, pag. 53].


of adjacent intervals (an ω-ordered partition as is usually called) whosefirst interval begins just at point A. Let C be a point on AB placed at adistance of B far less than Planck distance (10−33 cm). ω-Order makesit inevitable that between A and C only a finite number of intervalslie, while an infinite number of them (i.e. practically all of them) mustnecessarily lie between C and B. No matter how close is C from B,between C and B will always be an infinite number of intervals whilebetween A and C only a finite number of them will exist. The sameapplies to supertasks: if tb if the first instant at which a supertaskhas been accomplished, then at any instant prior to tb, whatsoever itbe, only a finite number of tasks will have been carried out and in-finitely many of them must still be performed. There is no way thatonly a finite number n of tasks remain to be performed simply becausethey would be the impossible last n tasks of an ω-ordered sequenceof tasks16. Thus while a supertask be performing only a finite num-ber of tasks will have been performed and an infinite number of themwill always remain to be performed. By way of example, imagine thatGod decides to successively count all natural numbers in its naturalorder of precedence (which, as St. Augustine believed [2], He could).Although He has been counting during an eternity and only a timeless than Planck time (10−43 s) remains to finish, He will have countedonly a finite number of numbers and practically all of them -infinitelymany of them- have still to be counted. Or in other words, while He becounting He will have counted only a finite number of numbers and aninfinitude of them remain still to be counted. It is impossible to avoidthis huge and unaesthetic asymmetry. As Aristotle claimed, perhapsit is impossible to traverse the untraversable [8], or to complete theuncompletable.

The criticism of ω-order I have just referred to, analyzes the as-sumption that it is possible to complete an uncompletable sequence ofactions, be them the tasks of a supertask or the definitions of a recur-sive ω-ordered sequence of definitions. Benacerraf was right in that wecannot infer consequences on the final state of a supermachine from thesuccessive performed actions with that supermachine. But Benacerrafdid not consider the consequences of having completed the supertaskon the supermachine that performs it. The formal development of thisline of argument leads to many contradictions involving ω-order andthen the actual infinity. It can be proved, for instance, that, as a conse-quence of a supertask, the magnetic beads of Hilbert machine are andare not placed in the left side of the machine17. None of those argu-ments have been refuted. Although some recalcitrant infinitists defendarguments so picturesque as for instance that the magnetic beads -inthe above example- disappear for reasons unknown; others pretend to

16See ’The Aleph-Zero or Zero Dichotomy’ at http://www.interciencia.es17A collection of these arguments are available at http://www.interciencia.es




invalidate an argument by confronting its conclusions with the conclu-sions of other arguments with have nothing to do with the original one;or defend the idea that the inconsistencies of ω-order found in super-task theory do not apply to ω-ordered sets. But ω-order is ω-order inall circumstances, and consists of being a complete totality that has afirst element (be it the element of a set or the task of a supertask) andsuch that every element has an immediate successor and an immediatepredecessor (except the first one); i.e. a totality that is both completeand uncompletable. To be complete and uncompletable not only seemscontradictory, the criticism of ω-order I am referring to, actually provesthat is in fact the case.

It is even possible to develop arguments of this type in set theoreticalterms. For instance, in each step of the recursive definition of an strictlyincreasing ω-ordered sequence of finite nested sets, a real variable x isdefined as the cardinal of the set just defined at each step. It canthen be proved that x is and is not defined as a finite cardinal18. Wecould also consider the set Q+ of positive rational numbers and provea consistency derived from the fact that Q+ may be densely ordered(between any two rationals infinitely many other rationals do exist)and ω-ordered (between any two successive rationals no other rationalexists). In effect, being Q+ denumerable there exists a bijection f

between Q+ and the set N of natural numbers so that Q+ can be writtenas {f(1), f(2), f(3), . . .}. If x is a rational variable whose initial valueis 1, and 〈di〉i∈N a sequence of positive rationals, both defined accordingto:

i = 1, 2, 3, . . .

{

di =| f(i + 1) − f(1) |

di < x ⇒ x = di

(14)

where | f(i+1)−f(1) | is the absolute value of f(i+1)−f(1); and ’<’represents the natural order of Q, then it can be proved that f(1) + x

is and is not the less rational greater than f(1)19.

5. Criticism of self-reference

The other great obstruction in the history of human thought is self-reference. Also of Presocratic origins (paradoxes of Epimenides andEubulides) it has been, and continues to be, an inexhaustible sourceof discussions (see, for instance, [104], [13], [81], [151], [160], etc.). Tobegin, the very term ’self-reference’ is equivocal: in all cases there isa sentence or symbol chain referring to itself. But no object has theability to refer to itself or to other objects, only humans beings havethat ability. No sentence nor symbol chain has autonomous existence.Even if they are automatically generated, the engine that generated

18See ’On the consistency of ω-order’ at http://www.interciencia.es19See section ’An inconsistency in the field of rational numbers’



them is always conceived of and turn on by a human being. In addi-tion, they have no meaning if no human interpret them. They do notsymbolize for themselves, they exclusively symbolize for us. So, beyondany sentence or symbol chain there always is a man or a woman tryingto say something. Thus, beyond the celebrated liar sentence:

This sentence is false (15)

there is a human trying to say something. He exactly says:

This I say is false (16)

If in the place of saying (16) our human says

This I tread is yellow (17)

we would look at his foots and could identify the object he is referringto. But what object is being referred to when he says (16)? As far asI’m concerned, I must recognize I don’t know of what I’m predicatingits falseness when I say (16). I even have the impression of be sayingnothing when I say (16). In any case let us consider the following twoalternatives regarding the possibilities of the subject whose falseness Ipredicate when I say (16). It may be either:

This I say is false (18)

or

This I say (19)

In the first alternative the words ’is’ and ’false’ have a confusing doublesyntactic function, the one as a part of the subject of the sentence; thesecond as the verb (in the case of ’is’) or the predicate (in the case of’false’) of the same sentence. But no language exists whose grammarconsider the possibility for a term to have two syntactic functions atthe same time in the same sentence. This should suffice to rule out thefirst alternative. In the second alternative the true subject is:

This (20)

because the words ’I’ and ’say’ are to determine a particular (20), theone I utter when I say (19). But to say this or that and nothing moreis to say nothing. I must therefore conclude that, in fact, I don’t knowof what I am stating its falseness when I say (16). And I suspect noone knows it20. If that were the case, and taking into account that we,the humans, are its only interprets, to say (16) is to say nothing. Andthe same applies when I say things as:

This I say is P (21)

20And that assuming we have previously selected the appropriate theory of trueto deals with those declarative sentences, because some dozens of them are available[121]


where P is any predicate, for instance:

This I say is not C-demonstrable (22)

where C is a certain formal calculus. Furthermore, since the subjectof a self-referent sentence is the whole sentence, we will not know thesubject of the sentence until the whole sentence has been uttered; butat this moment we have already predicated the subject. This is asto count one’s chickens before they are hatched. That is in fact thecase of the liar paradox and the like21, we are saying that somethingis false before to know of what are we predicating its falseness. Thissituation resembles the one originated by Russell’s Paradox as we willimmediately see. Probably our mind processes two times self-referentsentences, the first to state the subject -the referent- and the secondto predicate it. If that were the case, self-reference would be only asemantic illusion.

Russell Paradox of the set R of all sets that do not belong to them-selves22 is a set theoretical version of the Liar Paradox. In fact, accord-ing to its definition R belongs to itself whenever it does not belong toitself. We are again in the same above situation of trying to count one’schickens before the are hatched. In effect, all elements of a set shouldhave been defined before grouping them as a set. That is the only wayto know which elements are we grouping (in the same way we shouldknow the subject of a predicate before we can predicate it). But theset R as element of the set R is not defined until the set R be definedas a set, which in turn is impossible until R be defined as an element.Shortly before Russell discover his famous paradox, Charles Dogson,better known as Lewis Carroll, proposed the following definition ofClass: ([34], p. 31) :

Classification or the formation of Classes is a MentalProcess, in which we imagine that we have put together,in a group, certain things. Such a group is called a”Class”.

Carroll’s proposal is clearly non platonic. Classes are mental construc-tions, theoretical objects resulting from our mental activity. Carroll’sdefinition could be rewritten as:

Definition 1. A set is the theoretical object that results from a mental

process of grouping arbitrary objects previously defined.

Constructive Definition 1 is incompatible with self-referent sets becausebefore defining the set, all its elements have to be previously defined-either by enumeration or comprehension. And this excludes the set

21Self-reference can be ingenuously hidden through a group of sentences thatrefer to each other in a circular way.

22Russell discovered his famous paradox when he was analyzing Cantor’s paradoxon the set of all cardinals [24], which is also a self-referent set.


being defined, simply because the set cannot be defined as element be-fore it be defined as set; and if it is not previously defined as elementit cannot be grouped to form the set. It is remarkable, on the otherhand, that constructive Definition 1 is not circular, a mental processof grouping is not a group but a mental process. In addition, it seemsreasonable to require that all elements to be grouped be previously de-fined, particularly if we pretend to know what are we grouping. Beingself-referent, Russell’s set R is not a set according to Definition 1. Andfor the same reasons of self-reference neither the set of all cardinalsnor the set of all ordinals could be sets according to the same defini-tion. Thus, to assume the above non platonic definition of set sufficesto solve Cantor, Burali-Forti and Russell paradoxes. But modern settheories have never considered this non platonic option. Self-referencehad finally to be removed from set theory, but by less elegant meansas the Axiom of Regularity [120], [179], the Theory of Types [143] orthe Theory of Classes [17], [65].

The removal of self-reference from set theory should serve us to reflexon the future of semantic self-reference. Sets and monadic sentencesare more related to each other than it may seem. When we say:

S is P (23)

we assert the subject S has the predicate or property P. The elementsof a set are as the subject of the above monadic sentence, all themshare the same property of membership. A set that belong to itself isas a self-referent sentence: it also states a predicate of itself, in thiscase its membership to the set, i.e. the membership to itself. Nowthen, if self-reference is not appropriate for set theory and we had toremove it in order to avoid inconsistencies, why not to remove semanticself-reference from languages?

6. Platonism and biology

Dobzhansky written a famous paper entitled Nothing in Biology Makes

Sense Except in the Light of Evolution [51]. I think it would have beenmore appropriate to write reproduction in the place of evolution23. Andnot only because evolution is powered by reproduction. Mainly it isbecause only reproduction explains the extravagances of living being.Living organisms are, in effect, extravagant objects, i.e. objects withproperties that cannot be derived from physical laws. To have redfeathers, to move by jumping, or to be devored by the female in ex-change for copulating with it, are examples (and the list would be

23Of course, evolution is a natural process and denying it is so stupid as denyingphotosynthesis or glycolysis. Other different affair is the scientific theory availableto explain it. As any other scientific theory, the theory of organic evolution remainsunfinished and currently opened to numerous discussions (see for instance [157],[20], [162], [138], [146], [102], [56], [137], [36], [69], [145], [35] etc.)


interminable) of properties that cannot be derived from physical laws.Living beings are subjected to a biological law that dominates overall physical laws, the Law of Reproduction (reproduce as you might).This law is the only responsible for all past, present and future extrav-agances of living beings. The success in reproduction depends uponcertain characteristics of living beings that frequently have nothing todo with the efficient accomplishment of physical laws but with arbitrarypreferences as singing, or dancing, or having brilliant colors. Although,on the other hand, to achieve reproduction is previously necessary tobe alive, which in turn involves a lot of abilities related to the partic-ular environment of each living being. But this is in fact secondary:adapted and efficient as an organism may be, if it don’t reproduce, itsexcellence will have no future in the Biosphere.

Living beings are topically viewed as systems efficiently adapted totheir environment. No attention is usually payed to its extravagantnature, although extravagance -in the above sense- is undoubtedly itsmore relevant characteristic. We are the only (known) extravagant sys-tems in the Universe. By the way, these extravagances could only be theresult of capricious evolution rather than of intelligent design. Capri-cious evolution restricted by the physical laws governing the world. Oneof the latest extravagances appeared on Earth is the consciousness ex-hibited by some primates and notably by most of humans. Surely, thesensation of individual subjectivity is the responsible for our peculiarway of interpreting our internal symbolic reflex of the world (see be-low). I refer to Platonism, the belief that ideas do exist independentlyof the mind that elaborate them and use them in its continuous inter-action with the physical world. It is surely a side effect of consciousnessthat ideas become alive.

Animals must have a symbolic and abstract representation of theirnatural environments (surely inscribed in some neural networks), par-ticularly of all those objects and processes that may have consequenceson their own survival and reproduction. It is a symbolic, abstractknowledge: a leopard has in its mind the (abstract) idea of gazelle, itknows what to do with a gazelle, whatsoever be the particular gazelleit encounters with. The abstract idea of gazelle, and of any other,is elaborated in the brain by means of different components -atomsof knowledge- that not only serve to form the idea of gazelle but ofmany other abstract ideas. And not only ideas, sensorial perceptionsare also elaborated by similar process in atomic and abstract terms,which surely serves to filter the irrelevant details from the highly vari-able information coming from the physical world, and thus to identifywith sufficient security the many objects forming part of the naturalenvironments [177], [115]. It is, on the other hand, much more efficient


and plausible this way of functioning. To have a symbolic represen-tation of every particular object of a natural environment would beinconceivable more complex and expensive from all points of view.

To have an abstract representations of the world is indispensable foranimals in order to find the appropriate response to each of the nu-merous stimuli coming from the highly dynamic world they evolves in.And an error in this affair may cost the higher price. A ball rollingdown towards a precipice will not stop to avoid falling down; but thedog running behind it, will do; the dog knows gravity and its conse-quences. Organisms interact with their surroundings and must knowits singularities, its peculiar ways of being and evolving, i.e. its itsphysical logic, and even its mathematical logic24. Organisms need anabstract and symbolic representation of the physical world, and that isnot a minor detail. It must therefore be an efficient and precise repre-sentation, otherwise life would be impossible. It is through their ownactions and experiences, including imitation and innovation [87], [63],[139], [173], that living beings develop their neurobiological represen-tations (symbolic reflections) of the world. Although genetics is alsoinvolved in those cognitive abilities [128].

Perception and cognition are constructive process in which take partdifferent brain areas, as we are now knowing with certain details [140],[42], [156], [43]. Doubtless, concepts and ideas are mental elaborationsrather then transcendent entities we have the ability to contact with.Evolutionary biology and neurobiology are then incompatible with pla-tonism. Our consciousness of ideas together with our recursive thought(an exclusive ability of humans [41], [79]) could have promoted theraising and persistence of platonism. But the evolutive and neurobio-logical fundaments of human cognition suggest that platonic idealismis a wrong way of thinking. If there is a word expressing the essence ofthe Universe, that word is evolution. And as E. Mayr said Plato is theantihero of evolution [105]. It seems reasonable that Plato were pla-tonic in Plato times, but is certainly surprising the persistence of thatprimitive way of thinking in the community of contemporary mathe-maticians, though as could be expected, a certain level of disagreementon this affair also exists [99], [92], [100], [12]. It is remarkable the factthat many non platonic authors, as Wittgenstein, were against boththe actual infinity and self-reference [103].

The reader may come to his own conclusions on the effects the abovebiological criticism of platonic idealism could have on self-reference andthe actual infinity. Although, evidently, he can also maintain that hedoes not know through neural networks and then to persist in his pla-tonic habits. But for those of us that believe in the organic natureof our brains and in its abilities of perceiving and knowing modeled

24Primates and humans could dispose of neural networks to deal with numbers[48], [49], [79]


through more than 3600 millions years of organic evolution, platonismhas no longer sense. And neither self-reference nor the actual infinitymay survive away from the platonic scenario. On the other hand, itseems convenient to recall the long and conflictive history of both no-tions (would them have been so conflictive if they were consistent?); andabove all their absolute uselessness in order to know the natural world25.Physics [153], [155] and even mathematics [119], [154] could go withoutboth notions26. Experimental sciences as chemistry, biology and geol-ogy have never been related to them. The potential infinity probablysuffices27. Even the number of distinguishable sites in the universe isfinite [82]. Finite and discrete: not only matter and energy are discreteentities, space and time could also be of a discrete -quantum- nature asis being suggested from some areas of contemporary physics as super-string theory ([71], [72] [167], [58]), loop quantum gravity ([158], [11][159]), euclidean quantum gravity ([6]), quantum computation ([164],[14], [93]) or black holes thermodynamics [14], [164]. Beyond Planck’sdistance nature seems to loss its physical sense. The continuum space-time, as self-reference and the actual infinity, could only be a uselessnessrhetorical device. The reader can finally imagine the enormous simpli-fication of mathematics once liberated from infinity and self-reference(Ockham Razor agitates within its box).

25Not only uselessness, the actual infinity is also extremely annoying in certaindisciplines as physics of elementary particles (renormalization [60], [85], [94], [175],[148], [7])

26Except transfinite arithmetic and other related areas, most of contemporarymathematics are compatible with the potential infinity, including key concepts asthose of limit or integral.

27Some contemporary cosmological theories, as the theory of multiverse [45] orthe theory of cyclic universe [163], make use of the infinity in a rather impreciseway.


Appendix A. An inconsistency in the field of rational

numbers

As in modern physics with superstring theory [88], [95], the actualinfinity in contemporary mathematics plays such a dominant role thatit almost impossible to maintain a critical attitude (perhaps anothersociological version of the so called Principle of St. Matthew [111]).But the actual infinity is an inconsistent notion and soon or later thenightmare will finally end, as Poincare and Wittgenstein, among others,announced.

Along the last twelve years, and motivated by supertask theory, Icould develop several arguments against the actual infinity based onthe notion of ω-order, an inevitable consequence (as Cantor himselfproved [31]) of assuming the existence of actual infinite totalities. Tobe ω-ordered means to be complete (as the actual infinity requires)and uncompletable (because no last element completes ω-ordered se-quences). To be complete and uncompletable is not only apparentlycontradictory, it is an actual contradiction28. The following short ar-gument has nothing to do with this line of reasoning, although it isbased on another schizophrenic ability of the actual infinity; in thiscase the ability of the set of rational numbers of being both denselyordered (between any two rationals infinitely many other rationals doexist) and ω-ordered (between any two successive rationals no otherrational exists).

As Cantor did29 [30], we will also assume the existence of the set ofall finite cardinals as a complete totality (axiom of infinity in modernterms). The discussion that follows will be entirely based upon thatassumption. In those conditions, consider the field of rational numberswith its usual operations and ordering. As is well known, Cantor suc-cessfully proved the set Q of rationals is denumerable [26], and so it isthe set Q+ of positive rationals. In consequence, there exists a one toone correspondence f between the set N of natural numbers and Q+.We can then write:

Q+ = {f(1), f(2), f(3), . . .} (24)

In this way, bijection f induces a new ω-ordering in Q+. Consider nowa positive rational variable x whose initial value is 1, and a sequenceof positive rationals 〈di〉i∈N, both defined in accordance with:

i = 1, 2, 3, . . .

{

di =| f(i + 1) − f(1) |

di < x ⇒ x = di

(25)

28A collection of arguments proving it are available at http://www.interciencia.es29Although he never recognized the hypothetical nature of complete infinite

totalities.

http://www.interciencia.es


where | f(i+1)−f(1) | is the absolute value of f(i+1)−f(1); and ’<’stands for the usual ordering of Q; i.e d1 < x means x− di > 0. Infini-tist procedures as (25) are common in infinitist mathematics. Cantorhimself used them in the demonstrations of some of his most relevantinfinitist conclusions (for instance in his two celebrated proofs on theuncountable nature of the set R of real numbers [26], [27] [29]). In ourcase, and being Q closed with respect to addition and substraction, def-inition (25) defines the sequence 〈di〉i∈N as a complete infinite totalityof positive rationals. But it also redefines rational variable x wheneverthe term just defined be less than its current value. Notice we havemake used of both ω-order (<ω) and Q dense order (<). The first todefine the successive terms of 〈di〉i∈N; the second to redefine variablex. We are now in the appropriate position to prove the following twocontradictory results:

Proposition 1. f(1) + x is the less rational greater than f(1) in the

usual ordering of Q+.

Proof. Being Q+ closed with respect to addition, it is clear that f(1)+x

is a positive rational number. Assume that it is not the less rationalgreater than f(1). In this case there will be a positive rational f(n)such that f(n) − f(1) < x. But this is impossible because

x ≤| f(n) − f(1) | (26)

just from the definition of dn−1 on. f(1)+x is therefore the less rationalgreater than f(1) in the usual ordering of Q+.

�

Proposition 2. f(1) + x is not the less rational greater than f(1) in

the usual ordering of Q+.

Proof. The rational f(1) + x × 10−1 is evidently grater than f(1) andless than f(1) + x. Therefore, the rational f(1) + x is not the lessrational greater than f(1) in the usual ordering of Q+.

�

We have therefore derived a contradiction from our initial assumption,the Axiom of Infinity, that legitimates the existence of complete infinitetotalities, the only ones that can be ω-ordered and densely ordered.From the perspective of the potential infinity, on the other hand, onlyfinite totalities can be considered, and finite totalities cannot be neitherω-ordered nor densely ordered. Thus no contradiction arises from thisperspective. It is significant that while the history of infinite sets ismarked out by a lot of contradictions that could only be avoided eitherby stating ad hoc differences between finite and infinite numbers or byaxiomatic parching set theories, neither paradoxes nor contradictionsare known in the case of finite sets.


Appendix B. Reinterpreting the paradoxes of reflexivity

The formal strategy of pairing off the elements of two sets is moreancient than it is usually recognized. In fact, Aristotle used it to rejectZeno’s Dichotomies [9]. And since then, the same strategy has beenused by many authors with different discursive purposes, almost alwaysrelated to infinity (Proclus, J. Filopon, Thabit ibn Qurra al-Harani,R. Grosseteste, G. de Rimini, Duns Scotus, G. de Ockham, Galileoetc. [149]). But none of those authors, including Bolzano, rejected theEuclidean Axiom of the Whole and the Part30. Things began to changewith Dedekind, who assumed the violation of the old Euclidean axiomand stated the definition of infinite sets based just on that violation:a set is infinite if it can be put into a one to one correspondence withone of its proper subsets [47]. Dedekind and Cantor inaugurated theso called paradise of the actual infinity, where exhaustive injections(bijections or one to one correspondences) play a capital role. Theabsolute dominance of infinitism in contemporary mathematics and thepractical absence of contestation perhaps explain some things otherwiseunexplainable. In fact, in the short discussion that follows we will havethe occasion to discover a really uncomfortable situation related to theviolation of the Axiom of the Whole and the Part.In the core of our discussion will be two functional definitions contend-ing with the number of elements (power or cardinality) of two differentsets. I refer to the two following ones:

Definition 2. Two sets have the same number of elements if there

exists an exhaustive injection from one of them to the other.

Definition 3. Two sets have not the same number of elements if there

exists a non-exhaustive injection from one of them to the other.

Or in other words, if after pairing every element of a set with a differentelement of other set all elements of this last set result paired (exhaustiveinjection) then both sets have the same number of elements. Otherwise,if at least one element of the last set results unpaired (non-exhaustiveinjection) then both sets have not the same number of elements. WhileDefinition 2 is fundamental in transfinite set theory, Definition 3 isusually ignored, although both definitions are complementary; and theway of comparing the number of elements in both definition is similar:if after pairing every element of a set B with a different element of setA, one or more elements of A remains unpaired then A and B havenot the same number of elements. This seems so reasonable as to saythat if after pairing every element of B with a different element ofA, no elements of A remains unpaired then A and B have the samenumber of elements. Thus, being both definitions of the same level of

30’The whole is greater than the part.’ Common Notion 5. Book 1 of Euclid’sElements [57]


basic reasoning31 and having the same reasonable credibility, we shouldassume that both exhaustive and non exhaustive injections have thesame level of conclusiveness when used as instruments to compare thenumber of elements of two finite or infinite sets (from now on thisassumption will be referred to as A1).

There is no reason (except reasons of convenience) to reject A1. Sup-pose then we assume it. In this case, consider any infinite set A. Bydefinition, at least a proper subset B of A exists such that both setscan be put into a one to one correspondence, and then both sets havethe same number of elements (Definition 2). But it also exists a non-exhaustive injection f from B to A defined as:

Bf→ A (27)

f(b) = b; ∀b ∈ B (28)

according to which A and B have not the same number of elements(Definition 3). Thus, A1 leads to a contradiction: A and B have anddo not have the same number of elements.

From the infinitist perspective the paradoxes of reflexivity (all thosein which a whole is put into a one to one correspondence with one ofits proper parts32) are usually interpreted as side effects resulting fromthe supposed singularities of the actual infinity. Infinity is different,is monotonously claimed33. The reasonability of A1 suggests, however,a new interpretation of those paradoxes: that rather than paradoxesthey are true contradictions. Contradictions derived from the incon-sistency of the actual infinity. It is the actual infinity, in fact, thatmakes it possible that two sets, as A and B, can be put into injectivecorrespondences that may be exhaustive and non exhaustive. This is aclear indicator of inconsistency. Unless we explain the reason for whichA and B have the same number of elements in spite of the fact thatafter having paired each element of B with a different element of A,at leats one element of A remains unpaired. And the reason cannot bethe exhaustive injection (bijection) between A and B just because itcould be the case that the non-exhaustive injection between both setsbe so legitimate and conclusive as the exhaustive one (A1). If we werefaced with a contradiction we could not use one of the contradictoryconclusions to arbitrarily deny the other.

There is a long tradition in the history of human thought, particu-larly in formal sciences, that consists in taking as paradoxes what really

31Transfinite arithmetic will be later derived from this basic initial assumptionsand then we will get results as ℵ0 + n = ℵ0; ℵ

n

0= ℵ0 and the like, where n is a

finite cardinal or natural number. Evidently those results cannot be used to statethe convenience or inconvenience of Definition 3, otherwise we will making use of acircular reasoning

32Galileo Paradox [61] is a well known example of this type of paradox.33So different that it could be inconsistent.


are contradictions. And this has surely contributed to increase the con-fusion on certain notions as the actual infinity. In our case, we mustdecide whether exhaustive and non-exhaustive injections have or notthe same level of conclusiveness when used as instruments to comparethe number of elements of two infinite sets. If we decide they have thesame level of conclusiveness, then the actual infinity is an inconsistentnotion (infinite sets have and do not have the same number of elementsof some of its proper subsets). Otherwise we should explicitly declare,by the appropriate ad hoc axiom, that the number of elements of infi-nite sets cannot be compared by means of non-exhaustive injections.


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0809.2135v1[1] Cantor Paradox