# Infinity

## Philosophy 20607 01 (32582)

### University of Notre Dame                                                                              Spring 2023

Instructor: Joel David Hamkins, O’Hara Professor of Philosophy and Mathematics
3:30-4:45 Tuesdays + Thursdays, DeBartolo Hall 208

Course Description. This course will be a mathematical and philosophical exploration of infinity, covering a wide selection of topics illustrating this rich, fascinating concept—the mathematics and philosophy of the infinite.

Along the way, we shall find paradox and fun—and all my favorite elementary logic conundrums and puzzles. It will be part of my intention to reveal what I can of the quirky side of mathematics and logic in its connection with infinity, but with a keen eye open for when issues happen to engage with philosophically deeper foundational matters.

The lectures will be based on the chapters of my forthcoming book, The Book of Infinity, currently in preparation, and currently being serialized and made available on the Substack website as I explain below.

Topics. Among the topics we shall aim to discuss will be:

• The Book of Numbers
• Largest number contest
• The googol plex chitty bang stack hierarchy
• Galileo’s Salviati on infinity
• Hilbert’s Grand Hotel
• The uncountable
• How to count (to infinity and beyond!)
• Slaying the Hydra
• Transfinite recursion
• The continuum hypothesis
• The axiom of choice
• Orders of infinity
• The lattice of subsets of ℕ
• Potential versus actual infinity
• Confounding puzzles of infinity
• Infinite liars
• Infinite utilitarianism
• Infinite computation
• Infinite games
• Indescribable numbers
• Extremely remote events of enormous consequence
• The sand reckoner
• The outer limits of reason
• Puzzles of epistemic logic and the problem of common knowledge

Mathematical background. The course will at times involve topics and concepts of a fundamentally mathematical nature, but no particular mathematical background or training will be assumed. Nevertheless, it is expected that students be open to mathematical thinking and ideas, and furthermore it is a core aim of the course to help develop the student’s mastery over various mathematical concepts connected with infinity.

Readings. The lectures will be based on readings from the topic list above that will be made available on my Substack web page, Infinitely More. Readings for the topic list above will be gradually released there during the semester. Each reading will consist of a chapter essay my book-in-progress, The Book of Infinity, which is being serialized on the Substack site specifically for this course. In some weeks, there will be supplemental readings from other sources.

Student access. I will issue subscription invitations to the Substack site for all registered ND students using their ND email, with free access to the site during the semester, so that students can freely access the readings.  Students are free to manage their subscriptions however they see fit. Please inform me of any access issues. There are some excellent free Substack apps available for Apple iOS and Android for reading Substack content on a phone or other device.

Discussion forum. Students are welcome to participate in the discussion forums provided with the readings to discuss the topics, the questions, to post answer ideas, or engage in the discussion there. I shall try to participate myself by posting comments or hints.

Homework essays. Students are expected to engage fully with every topic covered in the class. Every chapter concludes with several Questions for Further Thought, with which the students should engage. It will be expected that students complete approximately half of the Questions for Further thought. Each question that is answered should be answered essay-style with a mini-essay of about half a page or more.

Extended essays. A student may choose at any time to answer one of the Questions for Further Thought more fully with a more extended essay of two or three pages, and in this case, other questions on that particular topic need not be engaged. Every student should plan to exercise this option at least twice during the semester.

Final exam.  There will be a final exam consisting of questions similar to those in the Questions for Further Thought, covering every topic that was covered in the course. The final grade will be based on the final exam and on the submitted homework solutions.

Open Invitation. Students outside of Notre Dame are welcome to follow along with the Infinity course, readings, and online discussion. Simply subscribe at Infinitely More, keep up with the readings and participate in the discussions we shall be having in the forums there.

# The axiom of determinacy for small sets

I should like to argue that the axiom of determinacy is true for all games having a small payoff set. In particular, the size of the smallest non-determined set, in the sense of the axiom of determinacy, is the continuum; every set of size less than the continuum is determined, even when the continuum is enormous.

We consider two-player games of perfect information. Two players, taking turns, play moves from a fixed space $X$ of possible moves, and thereby together build a particular play or instance of the game $\vec a=\langle a_0,a_1,\ldots\rangle\in X^\omega$. The winner of this instance of the game is determined according to whether the play $\vec a$ is a member of some fixed payoff set $U\subset X^\omega$ specifying the winning condition for this game. Namely, the first player wins in the case $\vec a\in U$.

A strategy in such a game is a function $\sigma:X^{<\omega}\to X$ that instructs a particular player how to move next, given the sequence of partial play, and such a strategy is a winning strategy for that player, if all plays made against it are winning for that player. (The first player applies the strategy $\sigma$ only on even-length input, and the second player only to the odd-length inputs.) The game is determined, if one of the players has a winning strategy.

It is not difficult to see that if $U$ is countable, then the game is determined. To see this, note first that if the space of moves $X$ has at most one element, then the game is trivial and hence determined; and so we may assume that $X$ has at least two elements. If the payoff set $U$ is countable, then we may enumerate it as $U=\{s_0,s_1,\ldots\}$. Let the opposing player now adopt the strategy of ensuring on the $n^{th}$ move that the resulting play is different from $s_n$. In this way, the opposing player will ensure that the play is not in $U$, and therefore win. So every game with a countable payoff set is determined.

Meanwhile, using the axiom of choice, we may construct a non-determined set even for the case $X=\{0,1\}$, as follows. Since a strategy is function from finite binary sequences to $\{0,1\}$, there are only continuum many strategies. By the axiom of choice, we may well-order the strategies in order type continuum. Let us define a payoff set $U$ by a transfinite recursive procedure: at each stage, we will have made fewer than continuum many promises about membership and non-membership in $U$; we consider the next strategy on the list; since there are continuum many plays that accord with that strategy for each particular player, we may make two additional promises about $U$ by placing one of these plays into $U$ and one out of $U$ in such a way that this strategy is defeated as a winning strategy for either player. The result of the recursion is a non-determined set of size continuum.

So what is the size of the smallest non-determined set? For a lower bound, we argued above that every countable payoff set is determined, and so the smallest non-determined set must be uncountable, of size at least $\aleph_1$. For an upper bound, we constructed a non-determined set of size continuum. Thus, if the continuum hypothesis holds, then the smallest non-determined set has size exactly continuum, which is $\aleph_1$ in this case. But what if the continuum hypothesis fails? I claim, nevertheless, that the smallest non-determined set still has size continuum.

Theorem. Every game whose winning condition is a set of size less than the continuum is determined.

Proof. Suppose that $U\subset X^\omega$ is the payoff set of the game under consideration, so that $U$ has size less than continuum. If $X$ has at most one element, then the game is trivial and hence determined. So we may assume that $X$ has at least two elements. Let us partition the elements of $X^\omega$ according to whether they have exactly the same plays for the second player. So there are at least continuum many classes in this partition. If $U$ has size less than continuum, therefore, it must be disjoint from at least one (and in fact from most) of the classes of this partition (since otherwise we would have an injection from the continuum into $U$). So there is a fixed sequence of moves for the second player, such that any instance of the game in which the second player makes those moves, the result is not in $U$ and hence is a win for the second player. This is a winning strategy for the second player, and so the game is determined. QED

This proof generalizes the conclusion of the diagonalization argument against a countable payoff set, by showing that for any winning condition set of size less than continuum, there is a fixed play for the opponent (not depending on the play of the first player) that defeats it.

The proof of the theorem uses the axiom of choice in the step where we deduce that $U$ must be disjoint from a piece of the partition, since there are continuum many such pieces and $U$ had size less than the continuum. Without the axiom of choice, this conclusion does not follow. Nevertheless, what the proof does show without AC is that every set that does not surject onto $\mathbb{R}$ is determined, since if $U$ contained an element from every piece of the partition it would surject onto $\mathbb{R}$. Without AC, the assumption that $U$ does not surject onto $\mathbb{R}$ is stronger than the assumption merely that it has size less the continuum, although these properties are equivalent in ZFC.  Meanwhile, these issues are relevant in light of the model suggested by Asaf Karagila in the comments below, which shows that it is consistent with ZF without the axiom of choice that there are small non-determined sets. Namely, the result of Monro shows that it is consistent with ZF that $\mathbb{R}=A\sqcup B$, where both $A$ and $B$ have cardinality less than the continuum. In particular, in this model the continuum injects into neither $A$ nor $B$, and consequently neither player can have a strategy to force the play into their side of this partition. Thus, both $A$ and $B$ are non-determined, even though they have size less than the continuum.