∞ + 1 = ∞

The word on the left of that equation names two different things, and the equation is true of one and false of the other. Cardinal numbers answer how many. Ordinal numbers answer in what order. For finite quantities the two coincide so completely that everyday language never separates them, which is why the small print is doing real work here.

As cardinals, the equation holds and the proof is a pairing. The cardinality of the natural numbers is ℵ₀. Add one element and you can still match the new set one-to-one against the naturals: send the new element to 0, and each old n to n+1. Nothing is left over on either side, so the two sets have the same cardinality. That is the whole content of ℵ₀ + 1 = ℵ₀, and it is the same argument that fills room 1 of Hilbert's hotel at 004.

As ordinals it fails, and it fails in an asymmetric way that is easy to miss. Ordinal addition is concatenation of well-ordered sequences, and concatenation is not commutative. Put one item in front of the natural numbers and you get 1 + ω, which is order-isomorphic to ω itself — you have simply relabeled. Put one item after all of them and you get ω + 1, a sequence with a last element, which ω does not have. No relabeling can produce a last element from an order that has none, so ω + 1 is strictly greater than ω.

So the honest statement is that 1 + ω = ω while ω + 1 > ω, and that ℵ₀ + 1 = ℵ₀ throughout. The shirt is not hedging when it says SOMETIMES. It is naming the exact condition under which a widely repeated piece of pop mathematics is correct.

There is a third system worth mentioning so the claim is not read too broadly. The extended real line adds symbols +∞ and −∞ for the purposes of limits, and there ∞ + 1 = ∞ holds by definition — but ∞ is not a number in that system, and expressions like ∞ − ∞ are left undefined precisely because no consistent value can be assigned. Three systems, three different answers, one symbol.

Cantor built the transfinite ordinals and cardinals in the 1870s and 1880s and was attacked for it; Kronecker reportedly called Cantor a corrupter of youth. The distinction that the small print protects is now the first thing taught in any set theory course, and the last thing most people learn about infinity outside one.