Initially attracted to *Less Wrong* by Eliezer Yudkowsky's intellectual boldness in his "infinite-sets atheism," I've waited patiently to discover its rationale. Sometimes it's said that our "intuitions" speak for infinity or against, but how could one, in a Kahneman-appropriate manner, arrive at intuitions about whether the cosmos is infinite? Intuitions about infinite sets might arise from an analysis of the *concept of *actually realized infinities. This is a distinctively philosophical form of analysis and one somewhat alien to *Less Wrong*, but it may be the only way to gain purchase on this neglected question. I'm by no means certain of my reasoning; I certainly don't think I've settled the issue. But for reasons I discuss in this skeletal argument, the conceptual—as opposed to the scientific or mathematical—analysis of "actually realized infinities" has been largely avoided, and I hope to help begin a necessary discussion.

**1. The actuality of infinity is a paramount metaphysical issue.**

*is*fundamentally conceptual.

*strangeness*of infinity’s manifestations, not to the

*incoherence*of its realization. The standard arguments against infinity, which predate Cantor, have been well-refuted, and I leave the mathematical critique of infinity to the mathematicians, who are mostly satisfied. (See Graham Oppy,

*Philosophical perspectives on infinity*(2006).)

**2. The principle of the identity of indistinguishables applies to physics and to sets, not to everything conceivable.**

*identity of indistinguishable*s, which holds that two separate things can’t have exactly the same properties. Things are constituted by their properties; if two things have exactly the same properties, nothing remains to make them different from one another. Physical objects do seem to conform to the identity of indistinguishables because physical objects are individuated by their positions in space and time, which are properties, but this is a physical rather than a metaphysical principle. Conceptually,

*brute distinguishability*, that is differing from all other things simply in being

*different*, is a property, although it provides us with no basis for identifying one thing and not another. There may be no way to use such a property in any physical theory, we may never learn of such a property and thus never have reason to believe it instantiated, but the property seems conceptually possible.

*determine*sets, so you can’t define a proper subset of brutely distinguishable things.

**3. Arguments against actually existing infinite sets.**

**A. Argument based on brute distinguishability.**

**B. Argument based on probability as limiting relative frequency.**

*brute distinguishability*. The following probability argument depends on different intuitions. Probabilities can be treated as idealizations at infinite limits. If you toss a coin, it will land heads roughly 50% of the time, and it gets closer to exactly 50% as the number of tosses “approaches infinity.” But if there can actually be an infinite number of tosses, contradiction arises. Consider the possibility that in an infinite universe or an infinite number of universes, infinitely many coin tosses actually occur. The frequency of heads and of tails is then infinite, so the relative frequency is undefined. Furthermore, the frequency of rolling a 1 on a die also equals the frequency of rolling 2 – 6: both are (countably) infinite. But if infinite quantities exist, then relative frequency should equal probability. Therefore, infinite quantities don’t exist.

**4. The nonexistence of actually realized infinite sets and the principle of the identity of indistinguishable sets together imply the Gold model of the cosmos.**

*ex nihilo*.