Can We Choose Our Own Mathematical Universe, or Is There a Right One?

A Never-Ending Game with No Clear Rulebook

Suppose you and a friend play a simple game. You each take turns naming a whole number: you say 7, they say 3, you say 0, they say 9, and so on, forever. After infinitely many turns you’ve together built an endless sequence — a real number. Before the game, you agree on a target set A. If the final sequence lands in A, you win; otherwise, your friend wins. Does one of you have a winning strategy — a plan that guarantees victory no matter what the other does?

Some sets make it easy. If A is the set of all possible sequences, you win trivially. If A is empty, your friend wins. If A is countable, your friend can win by a clever diagonal trick. But in 1962, mathematicians Jan Mycielski and Hugo Steinhaus noticed something surprising: if you assume the Axiom of Choice, you can construct a set A so bizarre that neither player has a winning strategy. The Axiom of Determinacy (AD) says that every set of reals is determined — one player always has a winning strategy. AD contradicts the Axiom of Choice, so most mathematicians don’t accept it for all sets.

There is, however, a more modest idea: definable determinacy. It claims that sets described in a simple, step-by-step way are determined. The simplest definable sets are Borel sets — built from open sets by countable unions and intersections. In 1974, Donald Martin proved that all Borel games are determined, using only the usual rules of mathematics, ZFC. But as we move to more complex projective sets, determinacy goes beyond ZFC. So we face a choice: do we accept stronger determinacy axioms, like PD (all projective sets are determined) or AD^L(ℝ) (all sets in the smallest universe containing all reals are determined)? This is not just a game; it opens a window into the deepest question of mathematical truth.

A Staircase of Theories, and the Problem of Choosing

Mathematicians organize theories by their strength. One theory is interpretable in another if everything the first says can be translated into the language of the second. This creates a vast hierarchy. At the bottom sits Robinson Arithmetic (Q). By adding principles — like “exponentiation is total” or “large cardinals exist” — we climb upward.

When a statement φ is independent of a theory T, T can’t prove φ or its negation. There are two kinds. A sentence like “ZFC is consistent” (Con(ZFC)) is vertically independent: adding it gives a strictly stronger theory (T < T + φ). A sentence like the Continuum Hypothesis (CH) is horizontally independent: adding it or its negation gives theories that are mutually interpretable with T (T ≡ T + CH and T ≡ T + ¬CH). So horizontally independent statements don’t change the strength; they just create parallel universes at the same level.

This leads to the question of pluralism. A pluralist says that for horizontally independent statements, there is no correct answer — we can freely choose which universe to work in, like picking a game rule. A non-pluralist insists that only one path leads to truth, and we need new axioms to find it. Some limitative views draw a line: strict finitists accept only Q, finitists go up to primitive recursive arithmetic, predicativists stop at ATR₀. Beyond that line, they become pluralists. Our story focuses on the level of ZFC and the question of whether we can settle statements about sets of real numbers — the realm of second-order arithmetic. The central battleground is between pluralists who say we can’t choose between ZFC + “all projective sets have nice properties” and ZFC + “they don’t,” and non-pluralists who argue that one theory is objectively right.

Two Roads: Determinacy and Giant Cardinals

Two powerful families of new axioms emerged. The first is definable determinacy. PD says every projective set of reals is determined. Under PD, all projective sets have the perfect set property (they are either countable or contain a perfect subset), the Baire property (they are “almost open”), and are Lebesgue measurable. Even the behavior of set-uniformization (a choice-function property) falls into a neat, periodic pattern. This beautiful structure theory gave mathematicians a strong reason to like PD.

The second family is large cardinal axioms. These assert the existence of enormous infinite numbers — measurable cardinals, Woodin cardinals, supercompact cardinals — that tower above the ordinary infinite. They seemed fantastical, but they have deep consequences for definable sets. In 1965, Robert Solovay showed that a measurable cardinal makes all Σ̰¹₂ sets have the regularity properties. In 1970, Martin proved that a measurable cardinal actually implies Π̰¹₁-determinacy. The pattern continued: more Woodin cardinals give more determinacy. A landmark result by Martin, John Steel, and W. Hugh Woodin in 1985 showed that if there are ω‑many Woodin cardinals with a measurable above them, then AD^L(ℝ) holds — full determinacy in the universe L(ℝ).

Even more striking, these two roads are equivalent: deep theorems show that AD^L(ℝ) is equivalent to the existence of certain inner models of large cardinals. So accepting determinacy at this level and accepting large cardinals is essentially the same choice. The two rival camps merged into one.

The Case for a ‘Good’ Mathematical Universe

How would you decide which axioms to adopt? You can’t prove AD^L(ℝ) from ZFC. The non-pluralist points to fruitful consequences. Under AD^L(ℝ), every set of reals in L(ℝ) is nicely behaved: they all have the Baire property, are Lebesgue measurable, and have the perfect set property. Uniformization extends beautifully. L(ℝ) becomes a paradise for analysts.

But a pluralist might say, “Maybe another theory gives the same nice results.” Remarkably, Woodin proved a recovery theorem: if you assume that all sets in L(ℝ) are Lebesgue measurable, have the Baire property, and satisfy Σ²₁-uniformization, then AD^L(ℝ) must be true. The good properties actually force the determinacy axiom.

Another piece of evidence is generic absoluteness. Large cardinals that imply AD^L(ℝ) “freeze” the theory of L(ℝ): no matter how you expand the mathematical universe by forcing, the truth of statements about L(ℝ) stays the same. This is like having a rock-solid foundation. Moreover, many different strong theories — some incompatible with each other, like the Proper Forcing Axiom and the existence of an ω₁-dense ideal — separately imply AD^L(ℝ). So AD^L(ℝ) sits at the overlapping consensus of a huge range of natural theories. As in science, when many independent lines of evidence converge on one conclusion, the case becomes compelling.

So, Is There a Right Answer, or Can We Pick?

Now we return to the pluralist challenge. Pluralists like Solomon Feferman (1928–2016), Akihiro Kanamori, and Saharon Shelah accept all the beautiful mathematics. But they argue that such results give only practical reasons to adopt determinacy, not evidence of an objective truth. For them, mathematics beyond ZFC is like choosing a rulebook for a game — you pick the one that makes the most interesting matches, but no rulebook is “the true one.” At best, there are many equally legitimate mathematical universes.

Non-pluralists like Donald Martin, John Steel, and Hugh Woodin disagree. They say the extrinsic case — the convergence of structure, recovery theorems, and consensus — provides theoretical reasons to believe AD^L(ℝ) holds in the one true universe of sets. They draw a parallel to physics: we accept the atomic theory not because it is self‑evident, but because it explains a wide range of data and makes the bigger picture simpler. Just as we infer that atoms exist because that theory is fruitful, we can infer that AD^L(ℝ) is true because it makes the theory of definable sets so powerfully coherent.

Where does this leave you? Even if you never touch advanced set theory, the question matters. It asks whether there are mathematical facts that exist independently of us, waiting to be discovered, or whether mathematics is a human invention with many possible forms. The next time you solve an equation, ask yourself: is the answer true no matter what, or did we just agree on the rules that make it so? The debate is far from over, but it shows that even in the most abstract corners of thought, we find the same puzzle: what is truth, and how do we find it?

Think about it

1. If someone invented a new arithmetic where 2+2=5, and it was perfectly consistent, would you say that arithmetic is wrong, or that you’re just playing a different game?
2. Suppose a new rule makes a puzzle neat and solves many old problems, but you can’t prove it from the rules you already know. Is it ever okay to adopt it as a new rule? When would you draw the line?
3. Could there be a question about numbers that has no true answer, only opinions? What would that mean for a test in your math class?