Paradoxes of Infinity

Infinity

I Few concepts lend themselves to as many paradoxes asthat of infinity

I Most of the paradoxes will of the form “if there couldbe infinitely many things, then there is a bizarreconsequence”

I One important point to note is what counts as thethings in the various paradoxes, because we may wantto accept some infinites and not others

I Examples of things include:I Past eventsI Past and future eventsI Steps in a taskI Physical objects (e.g. people, rooms)I Causal steps

Past years and months

(1) If the past is infinite, then there have been infinitelymany years in the past.

(2) If the past is infinite, then there have been infinitelymany months in the past.

(C) Therefore, if the past is infinite, the numbe or years inthe past is equal to the number of months in the past.

Stacchato runSuppose you thought that the correct answer to Zeno’s Dichotomy paradoxwas that we can in fact run infinitely many mid-points since we have in-finitely many blocks of time in which to run them. That is, infinite tasksare possible as long as they are bounded and well-defined. It them seemslike the following would be possible:

Achilles runs for 1/2 a minute, then stops for half a minute towrite the first decimal of a decimal expansion of π, then runs for1/4 of a minute, then stops for 1/4 of a minute to write the seconddecimal of a decimal expansion of π, and so on. After two minutes,he will have not only run the distances we supposed he ran inthe dichotomy paradox, he will also have written out a completedecimal expansion of π.

While this is not physically possible (given acceleration and speed limits,limits on the width of a pencil, etc.) it seems like if one accepts infinites,then this should be possible, at least conceptually, which seems absurd.

Stacchato run

(1) If it is possible for an infinite number of tasks to becompleted, then one could write out a completedecimal expansion of π.

(2) There cannot be a completed decimal expansion of π.

(C) Therefore, it is not possible for an infinite number oftasks to be completed.

Thomson’s lamp

Consider a lamp that has one button which when pushedturns the light from on to off or off to on. It is pushed, andthen pushed again 1/2 a minute later, and then pushed again1/4 of a minute later, then pushed again 1/8 of a minutelater, and so on. Will the lamp be on or off after a minute?

Thomson’s lamp

(1) After 1 minute, the lamp is either off or on.

(2) After 1 minute, the state of the lamp is determined bythe button pushings of the previous minute.

(3) An infinite number of button pushings could not resultin the lamp being on.

(4) An infinite number of button pushings could not resultin the lamp being off.

(5) Therefore, the lamp’s button could not be pushedinfinitely many times prior to 1 minute. (1, 2, 3, 4)

(6) If an infinite number of tasks is completable in a finiteamount of time, then the lamps button could bepushed infinitely many times prior to 1 minute.

(C) Therefore, an infinite number of tasks is notcompletable in a finite amount of time. (5,6)

Hilbert’s hotelHilbert owns a Grand Hotel with infinitely many rooms,each of which is filled. A new guest shows up. Hilbert, notwanting to miss out on business, decides to accommodatethe new guest by moving the guest in room 1 to room two,room 2 to room 3, room 3 to room 4, and so on, leavingroom 1 open for the new guest. Then, infinitely many guestsshow up. Hilbert really does not want to miss out on all thisbusiness, so he moves the guest in room 1 to room 2, room 2to room 4, room 3 to room 6, room 4 to room 8, and so on,leaving an infinite number of odd numbered rooms availablefor the new guests (so the sign on the front of his hotel reads“No Vacancy, Guests Welcome”). The next day, all theguests in the even numbered rooms check out, leaving aninfinite number of rooms filled. The day after, all the guestsexcept the guests in rooms 1 and 3 check out, leaving onlytwo rooms filled. Thus, it seems that infinity minus infinityis infinity, and that infinity minus infinity is 2 (or any othernumber one likes)

Hilbert’s hotel

(1) If transfinite numbers can count things, then weshould be able to perform all arithmetical functionswith them.

(2) Subtraction is not well-defined for transfinite numbers.

(3) Subtraction is an arithmetical function.

(C) Therefore the transfinite numbers cannot count things.

Ross’s urn

Consider an urn and a series of balls numbered 1 to infinity.At step 1 we put in balls 1-10 and take out ball 1. At step 2(after 1/2 minute), we put in balls 11-20 and take out ball 2.At step 3 (after another 1/4 of a minute), we put in balls21-30 and take out ball 3, and so on. After 1 minute, we willhave completed the process infinitely many times. Howmany balls are in the urn?

Someone might complain (as they do with Thompson’slamp) that we did not specify what happens to the balls at 1minute, but it seems like we can add in the followingassumption: “if at some moment prior to 1 minute, a ballcomes to rest at a particular position, and it does not moveagain prior to 1 minute, then it is still at that position atone minute.” If so, then after a minute there will be no ballsin the urn.

Ross’s urn

(1) If a process increases the the number of balls in theurn at each step, then completing the process infinitelymany times will not result in no balls in the urn.

(2) If Ross’s urn is possible, then it is possible for aprocess which increases the number of balls in the urnat each step to result in no balls in the urn afterinfinitely many steps.

(3) Therefore Ross’s urn is impossible. (1,2)

(4) If it is possible to complete an infinite number oftasks, then Ross’s urn is possible.

(C) It is not possible to complete an infinite number oftasks.(4,5)

Paradox of the gods

Achilles intends to run a mile. However, unbeknownst tohim, a powerful god has resolutely determined toinstantaneously put a wall in his path to stop him as soon ashe completes 1/2 the run. However, unbeknownst to Achillesand the first god, a second god waits resolutely determinedto instantaneously put a wall in his path to stop him as soonas he complete 1/4 the run. There are in fact infinitelymany such gods intending to do so after 1/8 of the run,1/16th of the run, and so on. Thus, Achilles cannot move,because for any finite distance he moves, he would havealready been blocked by a wall. However, no wall is thrownup, since he does not move. Thus, he cannot move, andnothing stops him from moving.

Paradox of the gods

(1) If there can be infinitely many objects, then Achillescan be stopped from running without anythingstopping him.

(2) Achilles cannot be stopped from running withoutanything stopping him.

(C) Therefore, there cannot be infinitely many objects.

Infinity

I There are worse problems still for infinity (such as theproblems of reverse hypertasks), but this is a samplingof the bizarre things that can arise from there beinginfinitely many things or from completing infinitelymany tasks (and it is generally assumed that infinitetime will allow infinite tasks)

I However, many do not want to say that that there arelimits as to how many tasks can be completed, howmany years there can be, or how small something canbe divided.

I One might then try to say that there can be potentiallyinfinite things but no actually infinite things.

Distinguishing InfinitiesI One way to distinguish good and bad infinite is as

follows:I A collection of objects is potentially infinite if it is finite

but can be increased without limit.I A collection of objects is actually infinite if it contains

an infinite number of things.I There are several other rival theses though:

I There can be infinites across time but never anythingthat is infinite at a single time (Aristotle)

I The word “infinite” can modify certain other words, butthere is no number “infinity” (Scholastic/medievalphilosophers)

I There can be infinite sets, but not infinite causal chains(Pruss)

I Things can be modeled using infinite numbers but thosenumbers are mere useful fictions (Craig)

I Note: if one rejects “actual” infinites, then it is falsethat space is either discrete or continuous; it is alwaysfinitely divided but can be divided further.

Given that both Zeno’s paradoxes and the paradoxes ofinfinite can be addressed by claiming that things can onlyever be potentially infinite, why might someone want toaccept actual infinites?

I Arguably, the potential infinite can only be defined interms of the actual infinite, so you can’t have potentialinfinites without actual infinites.I How do we know that counting 1, 2, 3, 4, won’t

eventually reach an endpoint?It seems like this requires knowing that these numbersare actually a small subset of the natural numbers, andthat because the natural numbers are actually infinite,we know this process will not run into an endpoint.

I Something is only divisible if there are places for it to bedivided. But then it seems like the stick is potentiallyinfinitely divisible in virtue of the actually infinitenumber of places along the stick that it can be divided.

Despite the success of Aristotle’s distinction at addressingZeno, a lot of people still want to endorse that there are (orat least can be) actual infinites. Why might someone wantto accept actual infinites?

I Arguably, the potential infinite can only be defined interms of the actual infinite, so you can’t have potentialinfinites without actual infinites.

I It is difficult to understand mathematical objects (likethe natural numbers) without actual infinites. Are moreof them coming into being? How many currently exist?

I Actual infinites seem to be required for some scientifictheories.

I Axiomatizations of math (and in particular set theory)almost always imply the existence of an actually infinitenumber of numbers.

I It is well-defined, useful, and non-paradoxical within settheory.

I The main reason most mathematicians and philosopherstoday accept actual infinites is because Georg Cantor(and some others) showed in the late 19th century howto define them without contradiction.

I The theory is in terms of actually infinite sets

I The key to mathematically defining an infinite set is inthe definition of what it takes for two sets to have thesame number of members (same cardinality)

I Specifically, two sets have the same number of membersif and only if there is a 1-1 correspondence (or bijectivefunction) between them.

I We can then define an actually infinite set as follows:A set is actually infinite if there exists a 1-1correspondence between it and one of its proper subsets.

I With this definition in hand, Cantor showed how to order, add, andmultiply transfinite numbers without contradiction.

I This, combined with the fact that set theory (which at the time wasbecoming the foundations of math) implied the existence of actuallyinfinite sets, led to actually infinite sets being accepted.

I We can understand Cantor’s response to the problems of infinity asthis: Infinite sets are quite unlike the finite sets we are used to;however, the mere fact that they are different does not mean thatthey are contradictory.

The whole difficulty of the subject lies in the necessity of thinkingin an unfamiliar way, and in realising that many properties whichwe have thought inherent in number are in fact peculiar to finitenumbers. If this is remembered, the positive theory of infinity...willnot be found so difficult as it is to those who cling obstinately to theprejudices instilled by the arithmetic which is learnt in childhood.–Bertrand Russell

Consider the inductive case:

I We thought all numbers counted objects, but then wediscovered negative numbers

I We thought all numbers could be put in ratio to eachother, but then we discovered irrational numbers

I We thought all numbers were real, but then wediscovered imaginary numbers

I We thought that a subset was always smaller than theset it was a subset of, but then we discovered transfinitenumbers

Conclusions

I There thus seems to be a strong case against infinity,but also a strong case for it

I One could accept actual infinites, in which case onegenerally says that the paradoxes were under-defined(e.g. Thompson’s lamp) or one accepts theconsequences of the paradoxes as surprising but notimpossible things (e.g. Hilbert’s hotel)

I One could deny actual infinite,s in which case one willneed to say how we manage to use infinites and in whatsense (if any) are infinites possible

An ApplicationThe Kalaam Cosmological argument is:

(1) Everything that begins to exist has a cause

(2) The universe began to exist

(C) Therefore the universe has a cause

I One can make the case that the cause of the beginningof the universe must be something with free will, inwhich case we have reason to believe that there is acause of the universe and that this cause is a mind

I While there are many reasons one might accept premise2 (e.g. scientific evidence for the big bang), theparadoxes of infinity are seen as one of the classic proofsfor the premise.

I Most of the distinctions of infinity will say that aninfinite past is impossible, in which case we have strongevidence for God’s existence that does not depend onan knowledge from experience.