12 Some Paradoxes Mostly Resolved HL

12 Some Paradoxes Mostly Resolved
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12.1   The arithmetic of infinity
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from ‘∞+1 = ∞’, one can seemingly derive that 1=0;
Note:PUZZLE

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The expression ‘∞+1’ makes no sense because addition is an operation on numbers,
Note:SOLUZIONE FACILE

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The Cantorian approach, by contrast, holds that there are infinite numbers, but they simply obey different rules from all the other numbers.
Note:ALTERNATIVA CLASSICA

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12.2   The paradox of geometric points
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an extended region of space with some positive volume is supposed to be composed entirely out of parts (points) each of which has zero volume.
Note:PUZZLE 2...ASSOMIGLIA UN PO A ZENONE

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reject additivity of sizes for uncountable collections.
Note:SOLUZIONE TRADIZIONALE

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If a region is really built up from points, how is it that its magnitude is not explained by their magnitudes and number?

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NN C È UNA VERA SPIEGAZ

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this is what we must say to avoid paradox.
Note:L UNICA SPIEGAZ

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The simplest way to avoid these paradoxes is to reject the existence of size-zero parts of an object.
Note:COME EVITARE IL PARADOSSO

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there in fact are no such things as points; there are only positive-sized regions.
Note:ABOLIRE ABOLIRLA PUNTI

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12.3   Infinite sums
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the same (infinite collection of) numbers can seemingly have different sums, depending on the order in which the numbers are added.
Note:IL PARADOSSO

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we can obtain 0 if we add them like this: (1−1) + (2−2) + ... . But we can obtain ∞ if we add them like this: 1 + (−1+2) + (−2+3) + ... .
Note:ES PRATICO

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the correct sum sometimes depends upon the order in which the numbers are added.
Note:STANDARD VERSION

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The sum of an infinite series of numbers is defined to be the limit (if there is one) of the sequence
Note:NIENTE LIMITE NIENTE SOMMA

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it has the consequence that ‘sum’ has a different meaning
Note:CONSEGUENZE

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I agree in essence with the standard view of infinite sums,
Note:CONCLUS

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one cannot add up infinitely many numbers.
Note:IN SINTESI...LA SOMMA ESISTE MA NN RIUSCIAMO CALCOLARLA

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The real thesis is this: as a conceptual matter, addition is an operation defined on pairs of numbers.
Note:MA LA VERA TESI

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One can define an ‘infinite sum’ using limits, but then one is changing the meaning of ‘sum’.
Note:SOMME E LIMITI

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This explains why ‘infinite sums’ may violate the rules of ordinary addition, specifically, the commutative and associative laws
Note:ESSENDO COSE DIVERSE

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‘Infinite sums’ violate the rules for addition because they are not actually sums.
Note:CONCLUSIONE

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12.4   Galileo’s paradox
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in answer to the question ‘Which are more numerous: the perfect squares or the natural numbers?’, we can only answer that both are beyond number;
Note:RICRDA...L INFINITO NN È UN NUMERO...TUTTI I PARADOSSI CHE LO CONSIDERANO TALE SONO SMASCHERATI

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This thesis implies a rejection of the conclusion that is usually said to be established by Cantor’s famous Diagonalization Argument
Note:LA TESI DEL NON NUMERO

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the set of real numbers is larger than the set of natural numbers.
Note:LA TESI DELLA DIAGONALE....ASSURDA SE GLI NFINITI NN SONO NUMERI

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there is no one-to-one function from the natural numbers onto the real numbers,
Note:CIÒ CHE SI LIMITA A DIRE LA DIAGONALE

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it is not the same as the conclusion that the real numbers are more numerous than
Note:Ccccccvccc ALTRO ASSUNTO NECESSARIO > = ONE-TO-ONE

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there is a one-to-one function from a proper subset of the former onto the latter.
Note:RICORDIAMOCI PERÓ CHE ESISTE UNA ONE TO ONE TRA I MATURA I E UN SOTTOINSIEME DEI REALI

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the proper subset criterion (every set is greater than any proper subset of itself) – which is equally important.
Note:UN ALTRO CRITERIO PER IL GREATER THAN

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the best explanation for the fact that the two ‘criteria’ for greaterness often diverge in the case of infinite collections is that infinity is not a kind of number.
Note:IN QS CASO NN DIVERGONO MA SPESSO SÌ IL CHE RENDE PRUDENTE ASSUMERE CHE LA RELAZIONE <> NN ESISTA TRA GLI INSIEMI

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12.5   Hilbert’s hotel

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the completely occupied hotel with infinitely many rooms is able to accommodate one new guest by moving everyone to the next room down.
Note:PARADOSSO

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there will always be one guest temporarily out of a room.
Note:NN CI STANNO TUTTI

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‘Why suppose that the hotel guests relocate in sequence,
Note:OBIEZIONE

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Why not suppose that the hotel manager notifies all the guests of the new arrangement, using his PA system
Note:SINCRONIA

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The answer is that there cannot be such a PA system,
Note:RISPOSTA

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would require a signal that travels at infinite velocity.
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Velocity is a natural intensive magnitude,
Note:UN DATO CHE RENDE TUTTO IMPOSSIBILE

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now suppose that no communication was sent out. Instead, each of the guests, completely on his own, just decides
Note:SUPERIAMO L OSTACOLO

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This is metaphysically possible,
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the probability of such a coincidence happening is zero.
Note:PROBLEMA PLAUSIBILITÀ

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12.6   Gabriel’s horn
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in the real world, there is a limit to how thin a coat of paint can be (for example, it cannot be less than one molecule thick).
Note:NEL MONDO REALE UN SIMILE OGGETTO NN PUÓ ESISTERE...È DI X SÈ METAFISICAMENTE IMPOSSIBILE

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then the Horn could not in fact be painted with a finite amount of paint.
Note:QUINDI

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12.7   Smullyan’s infinite rod
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12.8   Zeno’s paradox
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1.To reach the ground, the ball must complete the series, ½, ¾, ⅞, ... . 2.This series is endless. 3.It is impossible to complete an endless series. 4.Therefore, it is impossible for the ball to reach the ground.
Note:RICORDIAMO L ARGOMENTO

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confusion between two meanings of ‘endless’ and two meanings of ‘complete’.
Note:CONFUSIONE

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no last member.
Note:INFINITO 1

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There is no time at which every member of S has    occurred.
Note:INFINITO2

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The last member of S has occurred.
Note:COMPLETO 1

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Every member of S has occurred.
Note:COMPLETO 2

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cannot be completed1

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INFINITO 1

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it is never completed2
Note:INFINITO 2

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Suppose I am going out of town for the week, and I get you to promise to feed all of my pets while I’m gone. If I have a turtle, then to keep your promise, you would have to feed the turtle. On the other hand, if I don’t have a turtle, then feeding my nonexistent turtle is not required. Similarly, if series S has a last member, then going through the last member is required
Note:ANALOGIA

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if S does not have a last member, then going through this nonexistent member is not required. The Zeno series has no last member, so reaching this nonexistent member is not required
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the series is endless1 but not endless2; it is never completed1 but it is completed2
Note:LA SERIE DI ZENONE

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12.8.2   The staccato run
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a person goes through an endless Zeno series, but the person pauses after each part of the journey.
Note:VARIAZIONE CON LO STACCATO

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this infinite series is impossible
Note:TESI

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this variation of the Zeno series, unlike the original version, requires infinite natural intensive magnitudes.
Note:METAFISICAMENTE IMPOSSIBILE

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average velocity remains constant
Note:PRIMA NOTAZIONE

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in each stage, you have to accelerate from speed zero up to your peak velocity, then decelerate back to speed zero
Note:ESIGENZA

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This means that your (average) acceleration must double with each succeeding stage.
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The runner’s body will thus need infinite material strength in order to avoid disintegration;
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12.8.3   The short staccato run
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12.9   The divided stick
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12.18   Comment: shallow and deep impossibilities 12.18.1   Solving paradoxes by appeal to impossibility
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One way of solving such a paradox is to explain how the scenario is impossible.
Note:LA SOLUZIONE PIÙ COMUNE DEI PARADOSSI

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there are stronger and weaker kinds of impossibility.
Note:MA......

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some reasons for counting a scenario impossible would fail to resolve a paradox.
Note:PURTROPPO

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Suppose I could argue, completely convincingly, that (i) a budget of at least $200 trillion would be required to construct a lamp such as the one Thomson describes, and (ii) no one would ever be able to marshal so many resources for such a frivolous purpose.

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ES THOMPSON

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this obviously would not count as a solution to the paradox.
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Even physical impossibility may not be enough.
Note:ANCORA

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my argument turned on precise estimates of the value of Coulomb’s constant (the proportionality constant that relates electrostatic force to charge and distance).
Note:ES SEMPRE SULLA LAMPADA

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Suppose that if Coulomb’s constant were 10% larger than it actually is, then my argument would fail and the Thomson Lamp would be constructible; yet given the constant’s actual value, the Lamp is unconstructible.
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‘If Coulomb’s constant had been 10% larger, and the Thomson Lamp had been constructed, then what state would the lamp have been in at the end of the infinite series of switchings?’
Note:RISPOSTA OVVIA DELL OBBIETTORE

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if a certain device fails to exist due to some deep conceptual, logical, or metaphysical impossibility,
Note:DI COSA ABBIAMO BOSOGNO

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a metaphysically impossible scenario would have metaphysically impossible consequences.
Note:IN PARTICOLARE

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if the Thomson Lamp is impossible in a deep sense, we may think that there is no point to asking what would happen if the lamp were to exist, and that there need be no determinate answer to such a question.
Note:ESEMPIO

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12.18.2   An objection
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complain that my solutions to the paradoxes of infinity appeal to mere physical impossibilities, not metaphysical impossibilities,
Note:L OBIEZIONE

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have been at pains to emphasize that infinite natural intensive magnitudes are not ‘merely physically impossible’; they are metaphysically impossible, because they require there to exist a kind of number that does not exist.
Note:RISPOSTA.....INFINITI CHE NN SI COMPLETANO MA SI ESAURISCONO...VEDI ZENONE

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it is impossible to have more than four cats while also having fewer than three – this is deeply impossible because there is no number greater than four and less than three.
Note:ANALOGIA

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1.Scenario S requires magnitude M to take on an infinite value. 2.M is a natural intensive magnitude. 3.No natural intensive magnitude can assume an infinite value. 4.So S is (metaphysically) impossible. Having found S metaphysically impossible, I conclude that there is no need to concern ourselves with what would happen if S occurred.
Note:RIPETIAMO LA CLASSICA SOLUZIONE

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challenge whether (1) is metaphysically necessary.
Note:IL CRITICO POTREBBE OBIETTARE

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impossibility of infinite material strength
Note:ES...XCHÈ LA LAMPADA È IMPOSSIBILE

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One might object that it is only a contingent truth of our physics
Note:OBIEZIONE

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Perhaps there is some possible world with a different physics,
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12.18.3   Deep physical impossibilities
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a sharper division between physical and metaphysical impossibilities
Note:E X RISPONDERE ENTRA IN GIOCO...

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Some ‘physical impossibilities’ are sufficiently deep to resolve paradoxes.
Note:TESI

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One can easily imagine the constant having a different value,
Note:X LA COSTANTE DI COLOMBO

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the switch need not have infinite material strength. It really is extremely unclear how we are supposed to imagine this working.
Note:CONTROFATTUALE IMPOSSIBILE DA IMMAGINARE

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an object withstands ever-increasing forces despite having only limited material strength?

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IMPOSSIBILE DA IMMAGINARE...CONTRADDOZ IONE

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it isn’t clear that such alternative physics are genuinely metaphysically possible.
Note:IL CONTROFATTUALE SEMBREREBBE...

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such as that nothing can be both completely red and completely blue,
Note:ANALOGIA

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someone wants us to imagine a radically alien physics,
Note:CHI CI CHIEDE QS SFORZO

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this would simply be to describe a different problem from the original one.
Note:SE UNO CI CHIEDE DI IMMAGINARE UNA FISICA ALIENA

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a.If I were to add a teaspoon of salt to this recipe, how would it taste? b.If I were to add a teaspoon of salt to this recipe, in an alternate possible world in which salt is a compound of plutonium and mercury and we are sea creatures who evolved living on kelp and plankton, how would it taste? Whatever might be said about (b), it is surely a different question from (a),
Note:ANALOGIA

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It is hardly an indictment of a theory that it fails to provide answers to hypothetical problems that have yet to be clearly posed.
Note:L OBIEZIONE SI INDEBOLISCE

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12.18.4   Generalizing
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principles concerning binding energy are essential to the distinction between solid and non-solid materials
Note:ESEMPIO DI PRINCIPI TENUTI FERMI

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the relationship between resistivity and current flow is essential to the nature of electric circuits
Note:ALTRO ESEMPIO

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the Beer-Lambert Law is essential to the nature of transparency and opacity
Note:ESEMPIO

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To suppose these principles false is to suppose a fundamentally alien physics
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It is really not so clear that one could describe a coherent, metaphysically possible physics without these principles. Even if that would be possible, counterfactual questions about what would happen in an alternate world
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