Consciousness finitism

My 11-year-old has an interesting intuition, that it is impossible to have an infinite number of conscious beings. She is untroubled by Hilbert’s Hotel, and insists the intuition is specific to conscious beigs, but is unable to put her finger on what exactly bothers her about an infinity of conscious beings. It’s not considerations like “If there are infinitely many people, you probably have a near-duplicate.” Near-duplicates don’t bother her. It’s consciousness specifically. She is surprised that a consciousness-specific finitist intuition isn’t more common.

My best attempt at a defense of consciousness-finitism was that it seems reasonable to think of yourself as a uniformly randomly chosen member of the set of all conscious beings. But thinking of yourself as a uniformly randomly chosen member of a countably infinite set leads to the well-known paradoxes of countably infinite fair lotteries. So that may provide some sort of argument for consciousness-finitism. But my daughter insists that’s not where her intuition comes from.

Another argument for consciousness-finitism would be the challenges of aggregating utilities across an infinite number of people: If all the people are positioned at locations numbered 1,2,3,…, and you benefit the people at even-numbered locations, you benefit the same quantity of people as when you benefit the people whose locations are divisible by four, but clearly benefiting the people at the even-numbered locations is a lot better. I haven’t tried this family of arguments on my daughter, but I don’t think her intuitions come from thinking about well-being.

Still, I have a hard time believing in the impossibility of an infinite number of consciousnesses on the strength of such arguments. The main reason I have such a hard time is that it seems obvious that you could have a forward infinite regress of conscious beings, each giving birth to the next.

Infinite Dedekind finite sets

Most paradoxes of actual infinities, such as Hilbert’s Hotel, depend on the intuition that:

1. A collection is bigger than any proper subcollection.

A Dedekind infinite set is one that has the property that it is the same cardinality as some proper subset. In other words, a Dedekind infinite set is precisely one that violates (1).

In Zermelo-Fraenkel (ZF) set theory, it is easy to prove that any Dedekind infinite set is infinite. More interestingly, assuming the consistency of ZF, there are models of ZF with infinite sets that are Dedekind finite.

It is easy to check that if A is a Dedekind finite set, then A and every subset of A satisfies (1). Thus an infinite but Dedekind finite set escapes most if not all the standard paradoxes of infinity. Perhaps enemies of actual infinity, should thus only object to Dedekind infinities, not all infinities?

However, infinite Dedekind finite sets are paradoxical in their own special way: they have no countably infinite subsets—no subsets that can be put into one-to-one correspondence with the natural numbers. You might think this is absurd: shouldn’t you be able to take one element of an infinite Dedekind finite set, then another, then another, and since you’ll never run out of elements (if you did, the set wouldn’t be finite), you’d form a countably infinite sequence of elements? But, no: the problem is that repeating the “taking” requires the Axiom of Choice, and infinite Dedekind finite sets only live in set-theoretic universes without the Axiom of Choice.

In fact, I think infinite Dedekind finite sets are much more paradoxical than a run-of-the-mill Dedekind infinite sets.

Do we learn anything philosophical here? I am not sure, but perhaps. If infinite Dedekind finite sets are extremely paradoxical, then by the same token (1) seems an unreasonable condition in the infinite case. For Dedekind finitude is precisely defined by (1).

Infinity book progress

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

Infinite causal histories and causal loops

As I was thinking about causal finitism, the view that nothing can have an infinite causal past, I realized that there were structural similarities between the arguments for it on the basis of paradoxes like the Grim Reaper and Grandfather-like arguments against causal loops. And that led me to thinking whether there wasn't some way to generalize causal finitism so as to rule out both infinite causal pasts and causal loops.

There is. Here is one way. Say that a causal nexus is a network of nodes with partial-causation arrows between them, such that there is an arrow A→B if and only if A is a partial cause of B (or causally prior to? I think that's the same thing, but I'm not sure; or, if there is such a thing, directly causally prior to). Say that a monotonic sequence in a causal nexus is a finite sequence A₁,A₂,...,A_n of nodes such that each node is joined with an arrow to the next: A₁→A₂→...→A_n. The sequence culminates in A_n. Note that if there are causal loops, then a monotonic sequence can contain the same node multiple times.

The generalization of causal finitism now says:

• No metaphysically possible causal nexus contains a node that is the culmination of infinitely many monotonic sequences.

This rules out three kinds of causal nexuses:

1. Infinite regresses: longer and longer monotonic sequences of distinct nodes culminating in a given node.
2. Infinite cooperation: infinitely many arrows pointing to a single node (and hence infinitely many monotonic sequences of length two culminating in it).
3. Causal loops: longer and longer repeating monotonic sequences culminate in a given node (e.g., A→B, B→A→B, A→B→A→B, ...).

The possibility of handling infinite causal histories and causal loops--which I've long thought absurd--in the same framework makes me even more confident in causal finitism.

The twist

Consider the Truth-Teller Paradox:

1. This sentence is true.

You may not think this is a paradox at all. But ask yourself: is (1) true or false? No contradiction arises from either supposition, but it also should become plausible that there is no way of settling the question, and one may start to think that there just is no answer. We could stipulate a truth value, but stipulating a truth value will not give a meaning to the sentence. (I can stipulate that it is true that mimsy were the borogoves, but I have not thereby given the sentence meaning.) The sentence seems to, as it were, pull its content out of itself, and as such has no content. Now maybe you're not convinced by this—maybe you only feel a vague discomfort at (1). So now I apply the twist, by changing (1) slightly to something that is clearly paradoxical:

2. This sentence is false.

That (2) is paradoxical is clear (it's true if and only if it is false). The twist that took us from (1) to (2) made clearer that there is something fishy about the kind of self-dependence that (1) involves.

This kind of twist can be found in other cases. One might be vaguely worried about the set of all sets that contain themselves, and then make the twist and get the clearly paradoxical set of all sets that do not contain themselves. Or one might be worried about the causal loops that time travel would permit, say one's getting the plans for the time machine from one's future self, and then after building the time machine going back in time to hand those plans to one's then-past self. There is something fishy about such a causal loop. So you give it a twist, and you turn it into a clearly paradoxical story about shooting your grandfather before his children are conceived. I think one can argue that in this same way, Thomson's Lamp Paradox is a twist on Zeno's Achilles Paradox, and the Grim Reaper Paradox is a twist on Zeno's Dichotomy.

I wonder if there is anything interesting and general one can say about the logical structure of the twist. (There may be something in the literature.) In particular, I am curious whether one can infer the impossibility of the untwisted situation from the impossibility of the twisted situation.