New Zeno and Actual Infinity

doi:10.4236/ojpp.2011.12010

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Open Journal of Philosophy

2011. Vol.1, No.2, 57-60

New Zeno and Actual Infinity

Casper Storm Hansen

Northern Institute of Philosophy, University of Aberdeen, Aberdeen, UK.

Email: casper_storm_hansen@hotmail.com

Received July 9th, 2011; revised August 15th, 2011; accepted August 25th, 2011.

In 1964 José Benardete invented the “New Zeno Paradox” about an infinity of gods trying to prevent a traveller

from reaching his destination. In this paper it is argued, contra Priest and Yablo, that the paradox must be re-

solved by rejecting the possibility of actual infinity. Further, it is shown that this paradox has the same logical

form as Yablo’s Paradox. It is suggested that constructivism can serve as the basis of a common solution to New

Zeno and the paradoxes of truth, and a constructivist interpretation of Kripke’s theory of truth is given.

Keywords: New Zeno, Yablo’s Paradox, Infinity, Constructivism

The New Zeno Paradox

I wish to discuss the “New Zeno Paradox” invented by José

Benardete, criticise the proposed solutions of Graham Priest

and Stephen Yablo, and argue that the paradox should be re-

solved by rejecting the possibility of actual infinity. The para-

dox goes as follows:

“A man decides to walk one mile from A to B. A god waits

in readiness to throw up a wall blocking the man’s further ad-

vance when the man has travelled 1/2 mile. A second god (un-

known to the first) waits in readiness to throw up a wall of his

own blocking the man’s further advance when the man has

travelled 1/4 mile. A third god...&c. ad infinitum. It is clear that

this infinite sequence of mere intentions (assuming the contrary

to fact conditional that each god would succeed in executing his

intentions if given the opportunity) logically entails the conse-

quence that the man will be arrested at point A; he will not be

able to pass beyond it, even though not a single wall will in fact

be thrown down in his path.” (Benardete, 1964)

Following Yablo (2000), we will assume that the man does

not stop before B unless a barrier is raised to prevent him from

proceeding. From this, together with the conclusion that no

walls are raised, follows that the man does move beyond A—a

contradiction.

According to Yablo, it is the combination of the god’s inten-

tions that is impossible. When the man tries to move away from

A it will turn out that some of the gods won’t be able to fulfil

their intentions. Either some of the gods will raise their wall,

even though walls before theirs have been raised, or some of

the gods will refrain from raising their wall even though none

of the prior walls were raised. Yablo writes:

“If there’s a paradox here, it lies in the difficulty of combin-

ing individually operational subsystems into an operational

system. But is this any more puzzling than the fact that al-

though I can pick a number larger than whatever number you

pick, and vice versa, we can’t be combined into a system pro-

ducing two numbers each larger than the other?” (Yablo, 2000)

The answer to that question is: Yes, it is much more surpris-

ing! For the two situations are not alike. If Yablo and I both try

to pick the higher number, he who gets to choose last will suc-

ceed. If I choose after Yablo I will be able to fulfil my intention

no matter what Yablo does (I can simply choose the number

that is one higher than Yablo’s). This resembles the situation

described in the paradox, as each god has a course of action

(raising the wall or refraining from doing so) available that is

consistent with his intentions, no matter what the gods before

him have done. But I can be prevented from fulfiling my inten-

tion of picking the higher number by someone (Yablo) acting

after me. That is not the case for the gods. For each given god,

it is irrelevant to his purposes what the succeeding gods do.

Actually, the intentions of a given god (let us call him Zeus

and let him be placed 1/2n of the way from A to B) are consis-

tent with any combination of actions that the other gods can

perform (that is, any combination of raisings and non-raisings

of walls by the other gods irrespective of their intentions), and

at the time when Zeus must decide what to do, he will have all

the information needed to ensure that his intention is fulfiled.

For the intention of Zeus is given by the biconditional

(In) a wall shall be raised at the 1/2n point, if and only if for

all m > n no wall has been raised at the 1/2m point,

which makes no reference to the actions of the gods after Zeus,

and is such that Zeus has full power to determine the truth value

of the left hand side at a time when the truth value of the right

hand side is determined. The contradictory combined intention

of the gods

(I) for all n ≥ 1, a wall shall be raised at the 1/2n point, if and

only if for all m > n no wall has been raised at the 1/2m point,

is not Zeus’ intention.

So if Yablo is right that the solution is that it will turn out

that some of the gods fail in their attempts to fulfil their inten-

tions, we lack an explanation why. Yablo’s “explanation” is that

“logic stops them”. But that does not explain why the individ-

ual gods fail.

If an individual fails in achieving some goal, “being stopped

by logic” will only be a sufficient explanation if that goal is

self-contradictory. And if a group of individuals have goals that

are self-consistent separately and contradictory combined, so

that there is at least one of these individuals who will not

achieve his goal, his failure will have a more concrete explana-

tion, i.e. an explanation that makes reference to contingent

states of affairs that contradict the goal of this individual. If, for

example, Achilles and the Tortoise race each other and both

intend to be the first to reach the finishing line, their goals are

contradictory combined, and so “being stopped by logic” can

explain their failure in reaching both their goals. But if the

Tortoise is the one to fail his goal, there is also a more concrete

explanation for this, namely that Achilles got to the finishing

line first (or at the exact same time as the Tortoise).

C. S. HANSEN

Assume that Zeus is one of the gods who fail their goal. The

goal of Zeus is not self-contradictory, and so his failure will not

be sufficiently explained by saying that he was “stopped by

logic”. The goals of all the gods are contradictory combined,

but there can be given no concrete explanation for the failure of

Zeus. No combination of raised and non-raised walls of the

other gods will serve to explain why Zeus couldn’t make the

truth value of the left hand side of (In) equal to the right hand

side.

This of course generalises to all the gods. So if the man

moves away from A, something inexplicable will happen (i.e.

an event without a cause will happen). And that is also the case

if the man can’t move away from A: Either there will be no

raised walls and so the man’s failure will be inexplicable, or

some walls will be raised and then the actions of the associated

gods will be inexplicable.

So given that Yablo is right, the situation described in the

paradox will necessarily result in an inexplicable state of affairs.

That is unacceptable, and so his solution must be rejected.

So where does that leave us? Let us examine the premises.

Letting “Rx” mean that the man reaches point x and “Bx” mean

that a barrier is raised at point x, where x ranges over the real

numbers, and A is placed at x = 0 and B at x = 1, the premises

are stated thus by Yablo (I have made some inessential

changes):

(A1)









,0,1:

yRxyx Ry

(A2)







,0,1:

yByyx Rx

(A3)













0,1 :0,1 :yxx yBxRy 

(A4)













0,1 ::1 2&2

xBxnx Rx

Priest (1999) suggests that the paradox could be resolved by

denying the possibility of motion, i.e. rejecting premises (A1),

(A2) or (A3) or some combination thereof. Yablo shows that

this won’t work. He does so with the example of an infinite

series of demons calling off YES’s and NO’s in reverse order,

with demon n calling after demon n + 1. The nth demon calls at

the time t = 1/2n. The intention of each demon is to call YES iff

all the earlier-calling demons have called NO. This amounts to

using the same premises with “Rx” and “Bx” reinterpreted to

mean “up to (and including) the time t = x no demon has called

YES” and “at t = x a demon calls YES” respectively. In this

version of the paradox motion plays no role, but the contradic-

tion still ensues. And the first three premises have been reduced

to truisms that can’t be rejected with any degree of reasonabili-

ty. The interpretations are as follows:

(A1) If no demon has called YES up to t = x then no demon

has called YES up to any earlier time.

(A2) If at t = y a demon calls YES then there is no later time

up to which no demon has called YES.

(A3) If no demon has called YES up to t = y then no demon

has called YES up to t = y.

So premise (A4) must be rejected. I agree with Yablo that far.

In order to analyze the situation in more detail, I will “split

up” premise (A4), i.e. replace it with two premises whose con-

junction implies (A4). Let g be a function from the set of natu-

ral numbers to the set of gods. Then the two new premises are

(A4')

 

,:nmnmgngm 

(A4")













::nmn mgngm 









12 12





Premise (A4') says that there exists infinitely many gods.

Premise (A4") expresses the individual god’s ability to raise a

barrier at his unique point iff the man gets half the way to his. If

we assume the logical possibility of the existence of gods with

the ability to raise arbitrarily thin walls arbitrarily fast (or just

demons with the ability to call YES or NO arbitrarily fast) and

base their decision of whether to do so on previous events, then

(A4") can’t be rejected without accepting the possibility of

inexplicable states of affairs as argued above. So (A4') must be

rejected instead. That amounts to rejecting the possibility of

actual infinity.

That solves the paradox because if only potential infinity and

not actual infinity can exist, the “closest” situations to the one

described in the paradox are these:

 One god intends to stop the man the first time he arrives

at a point in the set . {1/ 2|}

nn

 A potential infinity of gods are created one after the other

and when each god is created he is assigned to a point on

the route.

And they do not give rise to a paradox. The god in the first

situation simply has an inconsistent intention, and so his failure

to fulfil it can be sufficiently explained by saying that “logic

stops him”. In the second situation there will only exist a finite

number of gods at the time when the man begins his journey.

So one of these gods will be the first on the route, and he will

raise his wall while none of the others will.

The Logical Essence of New Zeno

I will use the rest of this paper to provide further support for

this conclusion; that New Zeno should be solved by rejecting

actual infinity. I will do this by first carrying out a deeper logi-

cal analysis of the paradox than the one above. This analysis

will reveal a close affinity to the semantic paradoxes, in par-

ticular the one named after Yablo. And then (in the next section)

I will reach the conclusion through an appeal to Priest’s Princi-

ple of Uniform Solution.

One step towards identifying the “logical essence” of New

Zeno has been taken with Yablo’s modification into a simpler

form not involving movement and the demonstration that this

modified paradox has the same logical structure as the original.

Another step can be taken by making an alternative and simpler

formalization of the modified paradox, where only the predicate

B and not R is used. Still using “Bx” to mean that a demon calls

YES at t = x, all the premises (A1) - (A4) can be replaced with

just this one:







12 ::

nBxyxB  y

But instead of using the reals and the natural numbers in a

naive way, where it is not clear what properties of the metric

and order relations on these sets are necessary to achieve the

contradiction, the premises of the paradox can be given as a set

of formulas from which the contradiction can be deduced using

only standard first order predicate logic. Let B and



unary predicates, written prefix and postfix respectively, and <

a binary predicate, written infix. Then the set of formulas con-

sists of these four:

(B1) :





(B2) ,, :

yzA xyyzxz





(B3) :

Ay Ay x





(B4)





ABxy xBy 

C. S. HANSEN 59

The contradiction is derived as follows. From (B1) by exis-

tential specification we have aA



, and then from (B4)

follows by universal specification. As-

sume



:Bay aBy 



:yAya 

ba

:ya By

and then deduce. From (B3) follows

and hence by existential specification again we

have . From this follows in conjunction with (B2) and

that  and

:ya

By

:yb

b



hold. As we also

get from (B4) this implies the contra-

diction. Discarding the assumption, we have



:Bbyb By

a



:aBy

, from

which it follows together with that

holds, and then the use of existential specification

yet again produces



Ba y

:yaBy

b. From this a contradiction can be de-

rived analogously to how it was derived from

What we see is that in addition to (B4) the only thing that is

required is a non-empty set equipped with a transitive and

non-wellfounded order relation. The metric on the reals and the

linearity of the standard order relation on the same do not play

any role in the derivation of the contradiction; their purpose is

only to make the premises plausible. The metric serves to make

it plausible that the traveller can cover the distance from point

A to point B in a finite amount of time. The linearity makes it

plausible that the right hand side of (In) is a given at the time

where god number n must decide whether or not to make the

left hand side true.

What is interesting about the fact that (B1) - (B4) is the “lo-

gical essence” of the paradox is that it reveals that New Zeno

has the same logical structure as Yablo’s Paradox (Yablo, 1993).

This paradox results from this infinite list of sentences:

Y1: For all n > 1 the sentence Yn is not true

Y2: For all n > 2 the sentence Yn is not true

Y3: For all n > 3 the sentence Yn is not true



This set of sentences satisfies (B1) - (B4) when the predi-

cates are interpreted as follows: “Bx” means that x is true,

“

A” means that x is a sentence on the list and “x < y”

means that x is further down on the list than y.

The (semi)formal proof of contradiction above then corre-

sponds to this informal proof: There is some sentence on the list,

call it “a”. a is true iff all sentences further down on the list are

false. Assume that a is true. Then all sentences further down on

the list are false. There is a sentence further down on the list,

call it “b”. All sentences further down than b are false. b is false.

b is true iff all sentences further down on the list than b are

false. Contradiction! So a is false. There is a true sentence fur-

ther down than a, call it “b”. b is true. Contradiction follows

analogously.

The Principle of Uniform Solution

According to Priest’s so-called Principle of Uniform Solution,

paradoxes with the same logical structure should be solved in a

similar way. When it has now been demonstrated that New

Zeno and Yablo’s Paradox have the same logical structure, this

puts a significant restriction on what can be accepted as possi-

ble solutions to these two paradoxes. On the one hand, the de-

nial-of-the-possibility-of-motion solution and the logic-stops-

them solution to New Zeno can not be transfered to Yablo’s

Paradox, and on the other hand, the denial of either semantical

closure, the validity of the Tarskian T-schema or classical logic

—the conjunction of which are normally considered the source

of the semantic paradoxes—will not solve New Zeno.

I claim that constructivism can serve as such a unifying solu-

tion. Three aspects of constructivism, relevant to this paper, can

be distinguished. The first is the thesis that any infinity is

merely a potiential such, in the sense that its elements are cre-

ated in time and at any given time is finite in number. The sec-

ond is mentalism with regard to certain abstract entities, e.g.

mathematical objects. And the third is the rejection of tertium

non datur. It is the first of these three aspects which can unify

the solutions. The other two will become relevant in the fol-

lowing, but do not apply to New Zeno and are therefore not part

of what unifies. The second is a way to explain why the first

applies to abstract entities. And the third, even though often

considered the central thesis of constructivism, is but a side-

effect of the two others.

Constructivism qua the first aspect solves New Zeno as ex-

plained in the first section. Yablo’s Paradox can be solved with

Kripke’s theory of truth (Kripke, 1975)1. And that theory can, I

will argue, be interpreted as a constructivistic theory.

In (Beall, 2007, chapter 1) Beall presents Kripke’s theory

through a metaphor about books. It goes as follows. Imagine a

world initially consisting only of non-semantic facts. In this

world, there is a writer with two very large books. They carry

the titles The True and The False. In the beginning they are

empty, but the writer sets out to fill them so that they accurately

reflect their titles. In the first book, he records every fact of the

world, and in the second, he records every state of affairs that

fails to obtain in the world. For instance, he writes “Snow is

white” in the first book and “Snow is green” in the second.

After having done so, he realises that his work is not complete.

For now there are more facts than when he started. By writing

in the books, he has added facts to the world, namely facts

about what is written in the books, and he did not include these

facts in The True, nor did he include non-obtaining facts about

the books in The False. So in each book, he puts the heading

“Chapter 1” over what he has written so far and starts writing

the more comprehensive chapters 2 of each book. Chapter 2 of

The True is a complete record of all facts about the world out-

side the books as well as about chapter 1 of each of the books.

He uses the predicate “is true” in the meaning “is a sentence

written in The True” and similarly the predicate “is false” in the

meaning “is a sentence written in The False”. So “‘Snow is

white’ is true” and “‘Snow is green’ is false” both appear in this

chapter. Because “Snow is green” is in The False, it is deter-

mined that this sentence will never be in The True, no matter

how many chapters are written, so the writer can put “‘Snow is

green’ is true” in chapter 2 of The False.

There are sentences that the writer puts in neither of the two

chapters 2. One of them is “‘‘‘Snow is white’ is true’ is true’ is

true”. This is because the sentence “‘‘Snow is white’ is true’ is

true” is not in chapter 1 of either of the books, so whether it

will be written in The True is not yet determined. Another is

“This sentence is false”. And for the same reason. The sentence

referred to by “This sentence” namely “This sentence is false”.

itself, is not in chapter 1 of either of the books. After having

written the two new chapters, there are again new facts, so the

writer also compiles increasingly comprehensive chapters 3, 4,

5, etc. Of the two mentioned sentences, the first eventually gets

into one of the books (The True, in chapter 4), while the second

never does. It is “ungrounded”.

According to this metaphor, Kripke’s theory introduces an

element of temporality in semantics; the sentence “‘Grass is

green’ is true” is made true later than the sentence “Grass is

green”. I believe we should take this temporality seriously and

not just metaphorically. Imagine, in analogy to the ideal ma-

1Much criticism can be directed at Kripke’s theory, so whether the paradox

can really be solved by the theory is a controversial issue. I will not go into

that discussion here, but simply assume that Kripke’s theory is adequate.

C. S. HANSEN

thematician appealed to by writers in intuitionistic mathematics

(for example by Brouwer in his (1933)), an ideal linguist who

was to construct the set of truths and the set of falsities given

only non-semantical facts. We can then see Kripke’s theory as a

set of rules for the linguist to follow in this undertaking. Were

he to follow these rules, he would of course proceed just as the

writer in the metaphor except that we must abstract from the

concrete writing of books.

So a constructivist theory of truth that can be an interpreta-

tion and justification of Kripke’s theory is that the truth values

of sentences are the mental constructions of an ideal linguist.

Let me immediately prevent a possible misunderstanding: I am

not claiming some form of anti-realism or idealism in a general

sense. The existence of mind-independent facts is not rejected.

We can consistently believe that it is a mind-independent fact

that grass is green, while claiming that the truth of the sentence

“Grass is green” is mind-dependent. For the truth of this sen-

tence is not determined by the fact of the greenness of grass

alone; an equally important role is played by the rules of the

English language. These rules are a human construct, and the

act of applying them to sentences to assign a truth value is a

mental one.

Mental constructions happen in time, and when the language,

to whose sentences truth values are to be assigned, itself con-

tains predicates for the truth values, these mental constructions

cannot be carried out in any order. According to the correspon-

dence theory of truth, a sentence is true if it corresponds to or

represents a fact. For something to represent something else,

the represented must in some sense be logical prior to the rep-

resenting. So when not only the representing but also the repre-

sented is a sentence, i.e. when a sentence is about sentences,

temporality appears in semantics; the semantics of some sen-

tences must be prior to the semantics of other sentences. Only

after the sentence “Grass is green” is made true, is there a fact

to which the sentence “‘Grass is green’ is true” can correspond.

This I believe to be the lesson of Kripke’s theory (although

perhaps not of Kripke).

In this theory, Yablo’s Paradox is solved in exactly the same

way as the Liar Paradox. None of the sentences Y1, Y2, Y3, …

are ever put in either of the books and hence none of them are

true or false. This is because each of the sentences depend for

their truth value on an infinity of other sentences in the list, and

at no point in time do all these sentences and their truth values

exist so as to determine the truth value of any given sentence on

the list.

It is in other words premise (B3) that should be rejected, both

in the case of New Zeno and in the case of Yablo’s Paradox:

Rejecting the possibility of an actual infinity of gods and sen-

tences/truth values makes for a uniform solution to these para-

doxes.

Conclusion

The New Zeno Paradox and Yablo’s Paradox are of similar

logical form and hence, according to the Principle of Uniform

Solution, they should have similar solutions. This requirement

disqualifies many proposed solutions to each of the paradoxes.

On the other hand it serves to make the constructivist solution

to New Zeno proposed in this paper and Kripke’s solution to

the paradoxes of truth under a constructivist interpretation lend

reciprocal support to each other.

Of the three proposed solutions to New Zeno; the denial-

of-the-possibility-of-motion solution, the logic-stops-them so-

lution and the denial-of-actual-infinity solution, the first falls

for the Principle of Uniform Solution applied to New Zeno and

the paradox of the demons calling YES and NO, and the second

falls for the same principle applied to the latter paradox and

Yablo’s Paradox. Only constructivism can, it seems to me, be a

basis for a common solution to all these paradoxes.

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