The four causes and supertasks

Suppose I make a geologist’s hammer out of a chunk of steel and break a rock with the hammer. Then the chunk of steel is the material cause of the hammer, and the hammer is the efficient cause of the rock breaking.

The hammer then is explanatorily prior to the rock breaking and the chunk of steel is explanatorily prior to the hammer.

Admittedly these are different kinds of explanatory priority. But they do nonetheless combine: it is clearly correct to say that the chunk of steel is explanatorily prior to the rock breaking. (I am not claiming that transitivity holds across all the kinds of explanatory priority, though I suspect it does, but only here.) But now notice that this instance of explanatory priority does not correspond to any of the four causes: in particular the chunk of steel is neither the material nor the efficient cause of the rock breaking (it is only insofar as the chunk was shaped into a geologist’s hammer that it broke the rock). Hence, the four causes do not exhaust all the types of explanatory priority.

Other examples are possible. I push a rock with my hand, and consider the conjunctive state HM of there being a hammer and a rock moving. Then HM is explained by the chunk of steel and my hand. But the chunk of steel and my hand constitute neither a material or not an efficient (nor any other) cause of HM. Thus, again, we have explanatory priority not corresponding to one of the four causes.

The above examples do, however, permit one to hold the following view:

1. All fundamental instances of explanatory priority are instances of the four causes.

Thus, the four causes would be like Aristotle’s four elements or three types of friendship: they combine to provide all the cases.

But now an interesting bit of heavy-duty metaphysics. Suppose that dense causal sequences are possible, i.e., causal sequences such that between any two items in the sequence there is an intermediate one. Then no instance of causation in the dense sequence will be fundamental. And hence (1) won’t tell us as much as it seems to. Indeed, given dense causal sequences, weakening the four cause thesis to (1) eviscerates the four cause thesis.

Thus we have an argument that if we want to take the four cause thesis seriously, we need to accept (1), and hence we need to reject dense causal sequences.

But if supertasks are possible, it seems like dense causal sequences should be possible. So, if we want to take the four cause thesis seriously, we need to reject supertasks.

It is, by the way, interesting to think about supertasks where the items in the task alternate between different types of causation.

Note that the above point applies to other sparse pluralisms about causation besides the four-cause one.

Shuffling an infinite deck of cards

Suppose I have an infinitely deep deck of cards, numbered with the positive integers. Can I shuffle it?

Given an infinite past, here is a procedure: n days ago, I perfectly fairly shuffle the top n cards in the deck.

When one reshuffles a portion of an already perfectly shuffled finite deck of cards, the full deck remains perfectly shuffled. So, the top n cards in the infinitely deep deck are perfectly shuffled for every finite n.

Can we argue that the thus-shuffled deck generates a countably infinite fair lottery, i.e., that if we pick cards off the top of the deck, all card numbers will be equally likely? At the moment I don’t know how to argue for that. But I can say that we get what I have called a countably infinite paradoxical lottery, i.e., one when any particular outcome has zero or infinitesimal probability.

For simplicity, let’s just consider picking the top card off the deck and consider a particular card number, say 100. For card 100 to be at the top of the deck, it had to be in the top n cards prior to the shuffling on day −n for each n. For instance, on day −1000, it had to to be in the top 1000 cards prior to the shuffling. The subsequent 1000 shufflings together perfectly shuffle the top 1000 cards. Thus, the probability that card 100 would end up at the top is 1/1000, given that it was in the top 1000 cards on day −1000. But it may not have been. So, all in all, the probability that card 100 would end up at the top is at most 1/1000. But the argument generalizes: for any n, the probability that card 100 would end up at the top is at most 1/n. Hence, the probability that card 100 would end up at the top is zero or infinitesimal.

If taking an infinite amount of time to shuffle is too boring, you can also do this with a supertask: one minute ago you shuffle the top card, 1.5 minutes ago you shuffle the top two cards, 1.75 minutes ago you shuffled the top three cards, and so on. Then you did the whole process in two minutes.

All the paradoxes of fair countably infinite lotteries reappear for any paradoxical countably infinite lottery. So, the above simple procedure is guaranteed to generate lots of fun paradoxes.

Here is a fun one. Carl shuffles the infinite deck. He now offers to pay Alice and Bob $20 each to play this game: they each take a card off the top of the deck, and the one with the smaller number has to pay $100 to the one with the bigger number. Alice and Bob happily agree to play the game. After all, they know the top two cards of the deck are perfectly shuffled, so they think it’s equally likely that each will win, and hence each calculates their expected payoff at 0.5×$100 − 0.5×$100 + $20 = $20. He puts them in separate rooms. As soon as each sees their own card (but not the other's), he now offers a new deal to them: if they each agree to pay him $80, he’ll broker a deal letting them swap their cards before determining who is the winner. Alice sees her card, and knows there are only finitely many cards with a smaller number, so she estimates her probability of being a winner at zero or infinitesimal. So she is nearly sure that if she doesn’t swap, she’ll be out $100, and hence it’s obviously worth swapping, even if it costs $80 to swap. Bob reasons the same way. So they each pay Carl $80 to swap. As a result, Carl makes $80+$80−$20−$20=$120 in each round of the game.

Causal Finitism, of course, says that you can’t have an infinite causal history, so you can’t have done the infinite number of shufflings.

A ray of Newtonian particles

Imagine a Newtonian universe consisting of an infinite number of equal masses equidistantly arranged at rest along a ray pointing to the right. Each mass other than first will experience a smaller gravitational force to the left and a greater (but still finite, as it turns out) gravitational force to the right. As a result, the whole ray of masses will shift to the right, but getting compressed as the masses further out will experience less of a disparity between the left-ward and right-ward forces. There is something intuitively bizarre about a whole collection of particles starting to move in one direction under the influence of their mutual gravitational forces. It sure looks like a violation of conservation of momentum. Not that such oddities should surprise us in infinitary Newtonian scenarios.

Σ10 alethic Platonism

Here is an interesting metaphysical thesis about mathematics: Σ₁⁰ alethic Platonism. According to Σ₁⁰ alethic Platonism, every sentence about arithmetic with only one unbounded existential quantifier (i.e., an existential quantifier that ranges over all natural numbers, rather than all the natural numbers up to some bound), i.e., every Σ₁⁰ sentence, has an objective truth value. (And we automatically get Π₁⁰ alethic Platonism, as Π₁⁰ sentences are equivalent to negations of Σ₁⁰ sentences.)

Note that Σ₁⁰ alethic Platonism is sufficient to underwrite a weak logicism that says that mathematics is about what statements (narrowly) logically follow from what recursive axiomatizations. For Σ₁⁰ alethic Platonism is equivalent to the thesis that there is always a fact of the matter about what logically follows from what recursive axiomatization.

Of course, every alethic Platonist is a Σ₁⁰ alethic Platonist. But I think there is something particularly compelling about Σ₁⁰ alethic Platonism. Any Σ₁⁰ sentence, after all, can be rephrased into a sentence saying that a certain abstract Turing machine will halt. And it does seems like it should be possible to embody an abstract Turing machine as a physical Turing machine in some metaphysically possible world with an infinite future and infinite physical resources, and then there should be a fact of the matter whether that machine would in fact halt.

There is a hitch in this line of thought. We need to worry about worlds with “non-standard” embodiments of the Turing machine, embodiments where the “physical Turing machine” is performing an infinite task (a supertask, in fact an infinitely iterated supertask). To rule those worlds out in a non-arbitrary way requires an account of the finite and the infinite, and that account is apt to presuppose Platonism about the natural numbers (since the standard mathematical definition of the finite is that a finite set is one whose cardinality is a natural number). We causal finitists, however, do not need to worry, as we think that it is impossible for Turing machines to perform infinite tasks. This means that causal finitists—as well as anyone else who has a good account of the difference between the finite and the infinite—have good reason to accept Σ₁⁰ alethic Platonism.

I haven't done any surveys, but I suspect that most mathematicians would be correctly identified as at least being Σ₁⁰ alethic Platonists.

Infinity, Causation, and Paradox -- cheaper [expired]

I notice that Amazon has two hardcover copies of the book for $25 shipped, which is about half of the regular price. [Expired]

Recognizing the finite

We have a simple procedure for recognizing finite sequences. We start at the beginning and go through the sequence one item at a time (e.g., by scanning with our eyes). If we reach the end, we are confident the sequence was finite. This procedure can be relied on if and only if there are no supertasks—i.e., if and only if it is impossible to have an infinite sequence of tasks started and completed.

How do we know that there are no supertasks? Either empirically or a priori. To know it empirically, we would have to know that the various tasks we’ve completed were finite. But how would we know of any tasks we’ve completed that it’s finite if not by the above procedure?

So we have to know it a priori.

And the only story I know of how we could do that is by a priori cognizing some anti-infinity principle like Causal Finitism.

I am not sure how strong the above argument is. It is a little too close to standard sceptical worries for comfort.

Do we have to know that seven is finite to know that three is finite?

Three is a finite number. How do we know this?

Here’s a proof that three is finite:

1. 0 is finite. (Axiom)

2. For all n, if n is finite, then n + 1 is finite. (Axiom)

3. 3=0+1+1+1. (Axiom)

4. So, 0+1 is finite. (By a and b)

5. So, 0+1+1 is finite. (By b and d)

6. So, 0+1+1+1 is finite. (By b and e)

7. So, 3 is finite. (By c and f)

Let’s assume we can answer the difficult question of how we know axioms (a) and (b), and allow that (c) is just true by definition.

I want to raise a different issue. To know that three is finite by means of the above argument, it seems we have to know that the argument is a proof.

One might think this is easy: a proof is a sequence of statements such that each non-axiomatic statement logically follows from the preceding ones, and it’s clear that (d)-(g) each follow from the previous by well-established rules of logic.

One could ask about how we know these rules of logic to be correct—but I won’t do that here. Instead, I want to note that it is false that every sequence of statements such that each non-axiomatic statement logically follows from the preceding ones is a proof. This is the case only for finite sequences of statements. The following infinite sequence of statements is not a proof, even though every statement follows from preceding ones: “…, so I am Napoleon, so I am Napoleon, so I am Napoleon.”

Very well, so to know that (a)-(g) is a proof, I need to know that (a)-(g) are only finitely many statements. OK, let’s count: (a)-(g) are seven statements. So it seems we have to know that seven is finite (or something just as hard to know) in order to use the proof to know that three is finite.

This, of course, would be paradoxical. For to use a proof analogous to (a)-(g) to show that seven is finite, we would need a proof of eleven steps, and so we would need to know that eleven is finite to know that the proof is a proof.

Maybe we can just see that seven is finite? But then we gain nothing by (a)-(g), since the knowledge-by-proof will depend on just seeing that seven is finite, and it would be simpler and more reliable just to directly see that three is finite.

It might be better to say that we can just see that the proof exhibited above, namely (a)-(g), is finite.

It seems that knowledge-by-proof in general depends on recognition of the finite. Or else on causal finitism.

Nitpicking about the causal exclusion argument

Exclusion arguments against dualism, and sometimes against nonreductive physicalism, go something like this.

1. Every physical effect has a sufficient microphysical cause.

2. Some microphysical effects have non-overdetermined mental causes.

3. If an event E has two distinct causes A and B, with A sufficient, it is overdetermined.

4. So, some mental causes are identical to microphysical causes.

But (3) is just false as it stands. It neglects such cases of non-overdetermining distinct causes A and B as:

5. A is a sufficient cause of E and B is a proper part of A, or vice versa. (Example: E=window breaking; A=rock hitting window; B=front three quarters of rock hitting window.)

6. A is a sufficient cause of B and B is a sufficient cause of E, or vice versa, with these instances of sufficient causation being transitive. (Example: E=window breaking; A=Jones throwing rock at window; B=rock impacting window.)

7. B is an insufficient cause of A and A is a sufficient cause of B, with these instances of causation being transitive. (Example: E=window breaking; B=Jones throwing rock in general direction of window; A=rock impacting window.)

8. A and B are distinct fine-grained events which correspond to one coarse-grained event.

To take care of (6) and (7), we could replace “cause” with “immediate cause” in the argument. This would require the rejection of causation by a dense sequence of causes (e.g., the state of a Newtonian system at 3 pm is caused by its state at 2:30 pm, its state at 2:45 pm, at 2:52.5 pm, and so on, with no “immediate” cause). I defend such a rejection in my infinity book. But the price of taking on board the arguments in my infinity book is that one then has very good reason to accept the Kalaam argument, and hence to deny (1) (since the first physical state will then have a divine, and hence non-microphysical, cause).

We could take care of (5) and (8) by replacing “distinct” with “non-overlapping” in (3). But then the conclusion of the argument becomes much weaker, namely that some mental causes overlap microphysical causes. And that’s something that both the nonreductive physicalist and hylomorphic dualist can accept for different reasons: the nonreductive physicalist may hold that mental causes totally overlap with microphysical causes; the hylomorphist will say that the form is a part of both the mental cause and of the microphysical cause. Maybe we still have an argument against substance dualism, though.

Infinity, Causation, and Paradox

Amazon USA finally has it in stock.

Preorders open on Infinity, Causation, and Paradox

Preorder here. Amazon lists a release date of August 30.

You can also look at the table of contents.

Causal countability

Say that a set S is causally countable if and only if it is metaphysically possible for someone to causally think through all the items in S. To causally think through the xs is to engage in a step-by-step sequential process of thinking about individual xs such that:

1. Every individual one of the xs is thought about in precisely one step of the process.

2. Each step in the process has at most one successor step.

3. With at most one exception, each step in the process is the successor of exactly one step.

4. The successor of a step causally depends on it.

Causal finitism then ensures that any causally countable set is countable in the mathematical sense. And, conversely, given some assumptions about reality being rational, any countable set is causally countable.

However, causal countability escapes the Skolem paradox, because of causal finitism and how it is anchored in the non-mathematical notion of causation.

A slightly different causal finitist approach to finitude

The existence of non-standard models of arithmetic makes defining finitude problematic. A finite set is normally defined as one that can be numbered by a natural number, but what is a natural number? The Peano axioms sadly underdetermine the answer: there are non-standard models.

Now, causal finitism is the metaphysical doctrine that nothing can have an infinite causal history. Causal finitism allows for a very neat and pretty intuitive metaphysical account of what a natural number is:

• A natural number is a number one can causally count to starting with zero.

Causal counting is counting where each step is causally dependent on the preceding one. Thus, you say “one” because you remember saying “zero”, and so on. The causal part of causal counting excludes a case where monkeys are typing at random and by chance type up 0, 1, 2, 3, 4. If causal finitism is false, the above account is apt to fail: it may be possible to count to infinite numbers, given infinite causal sequences.

While we can then plug this into the standard definition of a finite set, we can also define finitude directly:

• A finite set or plurality is one whose elements can be causally counted.

One of the reasons we want an account of the finite is so we get an account of proof. Imagine that every day of a past eternity I said: “And thus I am the Queen of England.” Each day my statement followed from what I said before, by reiteration. And trivially all premises were true, since there were no premises. Yet the conclusion is false. How can that be? Well, because what I gave wasn’t a proof, as proofs need to be finite. (I expect we often don’t bother to mention this point explicitly in logic classes.)

The above account of finitude gives an account of the finitude of proof. But interestingly, given causal finitism, we can give an account of proof that doesn’t make use of finitude:

• To causally prove a conclusion from some assumptions is to utter a sequence of steps, where each step’s being uttered is causally dependent on its being in accordance with the rules of the logical system.

• A proof is a sequence of steps that could be uttered in causally proving.

My infinite “proof” that I am the Queen of England cannot be causally given if causal finitism is true, because then each day’s utterance will be causally dependent on the previous day’s utterance, in violation of causal finitism. However, interestingly, the above account of proof does not guarantee that a proof is finite. A proof could contain an infinite number of steps. For instance, uttering an axiom or stating a premise does not need to causally depend on previous steps, but only on one’s knowledge of what the axioms and premises are, and so causal finitism does not preclude having written down an infinite number of axioms or premises. However, what causal finitism does guarantee is that the conclusion will only depend on a finite number of the steps—and that’s all we need to make the proof be a good one.

What is particularly nice about this approach is that the restriction of proofs to being finite can sound ad hoc. But it is very natural to think of the process of proving as a causal process, and of proofs as abstractions from the process of proving. And given causal finitism, that’s all we need.

From the finite to the countable

Causal finitism lets you give a metaphysical definition of the finite. Here’s something I just noticed. This yields a metaphysical definition of the countable (phrased in terms of pluralities rather than sets):

1. The xs are countable provided that it is possible to have a total ordering on the xs such if a is any of the xs, then there are only finitely many xs smaller (in that ordering) than x.

Here’s an intuitive argument that this definition fits with the usual mathematical one if we have an independently adequate notion of nautral numbers. Let N be the natural numbers. Then if the xs are countable, for any a among the xs, define f(a) to be the number of xs smaller than a. Since all finite pluralities are numbered by the natural numbers, f(a) is a natural number. Moreover, f is one-to-one. For suppose that a ≠ b are both xs. By total ordering, either a is less than b or b is less than a. If a is less than b, there will be fewer things less than a than there are less than b, since (a) anything less than a is less than b but not conversely, and (b) if you take something away from a finite collection, you get a smaller collection. Thus, if a is less than b, then f(a)<f(b). Conversely, if b is less than a, then f(b)<f(a). In either case, f(a)≠f(b), and so f is one-to-one. Since there is a one-to-one map from the xs to the natural numbers, there are only countably many xs.

This means that if causal finitism can solve the problem of how to define the finite, we get a solution to the problem of defining the countable as a bonus.

One of the big picture things I’ve lately been thinking about is that, more generally, the concept of the finite is foundationally important and prior to mathematics. Descartes realized this, and he thought that we needed the concept of God to get the concept of the infinite in order to get the concept of the finite in turn. I am not sure we need the concept of God for this purpose.

Infinity book progress

I've just sent off the final contracted-for manuscript of Infinity, Causation and Paradox.

Infinite proofs

Consider this fun “proof” that 0=1:

      …

• So, 3=4

• So, 2=3

• So, 1=2

• So, 0=1.

What’s wrong with the proof? Each step follows from the preceding one, after all, and the only axiom used is an uncontroversial axiom of arithmetic that if x + 1 = y + 1 then x = y (by definition, 2 = 1 + 1, 3 = 1 + 1 + 1, 4 = 1 + 1 + 1 + 1 and so on).

Well, one problem is that intuitively a proof should have a beginning and an end. This one has an end, but no beginning. But that’s easily fixed. Prefix the above infinite proof with this infinite number of repetitions of “0=0”, to get:

• 0=0

• So, 0=0

• So, 0=0

• So, 0=0

      …

      …

• So, 3=4

• So, 2=3

• So, 1=2

• So, 0=1.

Now, there is a beginning and an end. Every step in the proof follows from a step before it (in fact, from the step immediately before it). But the conclusion is false. So what’s wrong?

The answer is that there is a condition on proofs that we may not actually bother to mention explicitly when we teach logic: a proof needs to have a finite number of steps. (We implicitly indicate this by numbering lines with natural numbers. In the above proof, we can’t do that: the “second half” of the proof would have infinite line numbers.)

So, our systems of proof depend on the notion of finitude. This is disquieting. The concept of finitude is connected to arithmetic (the standard definition of a finite set is one that can be numbered by a natural number). So is arithmetic conceptually prior to proof? That would be a kind of Platonism.

Interestingly, though, causal finitism—the doctrine that nothing can have an infinite causal history—gives us a metaphysical verificationist account of proof that does not presuppose Platonism:

• A proof is a sequence of steps such that it is metaphysically possible for an agent to verify that each one followed by the rules from the preceding steps and/or the axioms by observation of each step.

For, given causal finitism, only a finite number of steps can be in the causal history of an act of verification of a proposition. (God can know all the steps in an infinite chain, but God isn’t an observer: an observer’s observational state is caused by the observations.)

A causal finitist definition of the finite

Causal finitism says that nothing can have infinitely many causes. Interestingly, we can turn causal finitism around into a definition of the finite.

Say that a plurality of objects, the xs, is finite if and only if it possible for there to be a plurality of beings, the ys, such that (a) it is possible for the ys to have a common effect, and (b) it is possible for there to be a relation R such that whenever x₀ one of the xs, then there is exactly one of a y₀ among the xs such that Rx₀y₀.

Here's a way to make it plausible that the definition is extensionally correct if causal finitism is true. First, if the definition holds, then clearly there are no more of the xs than of the ys, and causal finitism together with (a) ensures that there are finitely many of the ys, so anything that the definition rules to be finite is indeed finite. Conversely, suppose the xs are a finite plurality. Then it should be possible for there to be a finite plurality of persons each of which thinks about a different one of the xs in such a way that each of the xs is thought about by one of the ys. Taking being thought about as the relation R makes the definition be satisfied.

Of course, on this account of finitude, causal finitism is trivial, for if a plurality of objects has an effect, then they satisfy the above definition if we take R to be identity. But what then becomes non-trivial is that our usual platitudes about the finite are correct.

Mathematical Platonist Universalism, consistency, and causal finitism

Mathematical Platonists say that sets and numbers exist. But there is a standard epistemological problem: How do we have epistemic access to the sets to the extent of knowing some of the axioms they satisfy? There is a solution to this epistemological problem, mathematical Platonist universalism (MPU): for any consistent collection of mathematical axioms, there are Platonic objects that satisfy these axioms. MPU looks to be a great solution to the epistemological problems surrounding mathematical Platonism. How did evolved creatures like us get lucky enough to have axioms of set theory or arithmetic that are actually true of the sets? It didn’t take much luck: As soon as we had consistent axioms, it was guaranteed that there would be a plurality of objects that satisfied them, and if the axioms fit with our “set intuitions”, we could call the members of any such plurality “sets” while if they fit with our “number intuitions”, we could call them “natural numbers”. And the difficult questions about whether things like the Axiom of Choice are true are also easily resolved: the Axiom of Choice is true of some pluralities of Platonic objects and is false of others, and unless we settle the matter by stipulation, no one of these pluralities is the sets. (The story here is somewhat similar to Joel Hamkins’ set theoretic multiverse, but I don’t know if Hamkins has the kind of far-reaching epistemological application in mind that I am thinking about.)

This story has a serious problem. It is surely only the consistent axioms that are satisfied by a plurality of objects. Axioms are consistent, by definition, provided that there is no proof of a contradiction from them. But proofs are themselves mathematical objects. In fact, we’ve learned from Goedel that proofs can be thought of as just numbers. (Just write your proof in ASCII, and encode it as a binary number.) Hence, a plurality of axioms is consistent if and only if there does not exist a number with a certain property, namely the property of encoding a proof of a contradiction from these axioms. But on MPU there is no unique plurality of mathematical objects deserving to be called “the numbers”. So now MPU faces a very serious problem. It said that any consistent plurality of axioms is true of some plurality of Platonic objects, and there are no privileged pluralities of “numbers” or “sets”. But consistency is itself defined by means of “the numbers”. And the old epistemological problems for Platonism resurface at this level. How do we have access to “the numbers” and the axioms they satisfy so as to have reason to think that the facts about consistency of axioms are as we think they are?

One could try making the same move again. There is no privileged notion of consistency. There are many notions of consistency, and for any axioms that are consistent with respect to any notion of consistency there exists a plurality of Platonic satisfiers. But now this literally threatens incoherence. But unless we specify some boundaries on the notion of consistency, this is going to literally let square circles into Platonic universalism. And if we specify the boundaries, then epistemological problems that MPU was trying to solve will come back.

At my dissertation defense, Robert Brandom offered a very clever suggestion for how to use my causal powers account of modality to account for provability: q can be proved from p provided that it is causally possible for someone to write down a proof of q from p. This can be used to account for consistency: axioms are consistent provided that it is not causally possible to write down a proof of a contradiction from them. There is a bit of a problem here, in that proofs must be finite strings of symbols, so one needs an account of the finite, and a plurality is finite if and only if its count is a natural number, and so this account seems to get us back to needing privileged numbers.

But if one adds causal finitism (the doctrine that only finite pluralities can together cause something) to the mix, we get a cool account of proof and consistency. Add the stipulation that the parts of a “written proof” need to have causal powers such that they are capable of together causing something (e.g., causing someone to understand the proof). Causal finitism then guarantees that any plurality of things that can work together to cause an effect is finite.

So, causal finitism together with the causal powers account of modality gives us a metaphysical account of consistency: axioms are consistent provided that it is not causally possible for someone to produce a written proof of a contradiction from them.

Progress report on books

My Necessary Existence book with Josh Rasmussen is right now in copyediting by Oxford.

I am making final revisions to the manuscript of Infinity, Causation and Paradox, with a deadline in mid October. As of right now, I've finished revising five out of ten chapters.

I am toying with one day writing a book on the ethics of love.

Supertasks and empirical verification of non-measurability

I have this obsession with probability and non-measurable events—events to which a probability cannot be attached. A Bayesian might think that this obsession is silly, because non-measurable events are just too wild and crazy to come up in practice in any reasonably imaginable situation.

Of course, a lot depends on what “reasonably imaginable” means. But here is something I can imagine, though only by denying one of my favorite philosophical doctrines, causal finitism. I have a Thomson’s Lamp, i.e., a lamp with a toggle switch that can survive infinitely many togglings. I have access to it every day at the following times: 10:30, 10:45, 10:52.5, and so on. Each day, at 10:00 the lamp is off, and nobody else has access to the machine. At each time when I have access to the lamp, I can either toggle or not toggle its switch.

I now experiment with the lamp by trying out various supertasks (perhaps by programming a supertask machine), during which various combinations of toggling and not toggling happen. For instance, I observe that if I don’t ever toggle the switch, the lamps stays off. If I toggle it a finite number of times, it’s on when that number is odd and off when that number is even. I also notice the following regularities about cases where an infinite number of togglings happens:

1. The same sequence (e.g., toggle at 10:30, don’t toggle at 10:45, toggle at 10:52.5, etc.) always produces the same result.

2. Reversing a finite number of decisions in a sequence produces the same outcome when an even number of decisions is reversed, and the opposite outcome when an odd number of decisions is reversed.

(Of course, 1 is a special case of 2.) How fun! I conclude that 1 and 2 are always going to be true.

Now I set up a supertask machine. It will toss a fair coin just prior to each of my lamp access times, and it will toggle the switch if the coin is heads and not toggle it if it is tails.

Question: What is the probability that the lamp will be on at 11?

“Answer:” Given 1 and 2, the event that the lamp will be on at 11 is not measurable with respect to the standard (completed) product measure on a countable infinity of coin tosses. (See note 1 here.)

So, given supertasks (and hence the falsity of causal finitism), we could find ourselves in a position where we would have to deal with a non-measurable set.

Yet another infinite lottery machine

In a number of posts over the past several years, I’ve explored various ways to make a countably infinite fair lottery machine (assuming causal finitism is false), typically using supertasks in some way.

Here’s another, slightly simplified from a construction in Norton. Suppose we toss a countably infinite number of fair coins to make an array with infinitely many infinite rows that could look like this:

HTHTHHHHHHHTTT...
THTHTHTHTHHHHH...
HHHHHTHTHTHTHT...
...

Make sure that nobody looks at the coins after they are tossed. Here’s something that could happen: each row of the array contains one and only one tails. This is unlikely (probability zero; Norton originally said it's nonmeasurable, but that was a mistake, and we're coauthoring a correction to his paper) but possible. Have a robot scan the array—a supertask will be needed—to verify whether this unlikely event has happened. If not, we have failed to make the machine. But if yes, our array will look relevantly like:

HHTHHHHHHHHHHH...
HHHHHTHHHHHHHH...
HHTHHHHHHHHHHH...
...

Continue making sure nobody looks at the coins. Put a robot at the beginning of the first row. Now, you have an countably infinite fair lottery machine that you can use over and over. To use it, just tell the robot to scan the row it’s at, announce the position of the lone tails, and move to the beginning of the next row. Applied to the above array, you will get the sequence of results 3,6,3,….

Of course, it’s very unlikely that we will succeed in making the machine (the probability is zero). But we might. And once we do, we can run as many paradoxes of infinity as we like. And we might even find ourselves lucky enough to be in a universe where some natural random process has already generated such a lucky array, in which case we don’t even have to flip the coins.

Once we have the machine, we can have lots of fun with it. For instance, it seems antecedently really unlikely that the first hundred times you run the machine, the numbers you get will be in increasing order. But no matter how many numbers you've pulled from the machine, you are all but certain that the next number will be bigger than any of them.