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Continued from I
Sizes of infinity don’t stop there. Cantor discovered that any infinite set’s power set — the set of all subsets of its elements — has larger cardinality than it does. Every power set itself has a power set, so that cardinal numbers form an infinitely tall tower of infinities.
Standing at the foot of this forbidding edifice, Cantor focused on the first couple of floors. He managed to prove that the set formed from different ways of ordering natural numbers (from smallest to largest, for example, or with all odd numbers first) has cardinality ℵ1, one level up from the natural numbers. Moreover, each of these “order types” encodes a real number.
His continuum hypothesis asserts that this is exactly the size of the continuum — that there are precisely ℵ1 real numbers. In other words, the cardinality of the continuum immediately follow ℵ0, the cardinality of the natural numbers, with no sizes of infinity in between.
But to Cantor’s immense distress, he couldn’t prove it.
In 1900, the mathematician David Hilbert put the continuum hypothesis first on his famous list of 23 math problems to solve in the 20th century. Hilbert was enthralled by the nascent mathematics of infinity — “Cantor’s paradise,” as he called it — and the continuum hypothesis seemed like its lowest-hanging fruit.
To the contrary, shocking revelations last century turned Cantor’s question into a deep epistemological conundrum.
The trouble arose in 1931, when the Austrian-born logician Kurt Gödel discovered that any set of axioms that you might posit as a foundation for mathematics will inevitably be incomplete. There will always be questions that your list of ground rules can’t settle, true mathematical facts that they can’t prove.
As Gödel suspected right away, the continuum hypothesis is such a case: a problem that’s independent of the standard axioms of mathematics.
These axioms, 10 in all, are known as ZFC (for “Zermelo-Fraenkel axioms with the axiom of choice”), and they undergird almost all of modern math. The axioms describe basic properties of collections of objects, or sets. Since virtually everything mathematical can be built out of sets (the empty set {} denotes 0, for instance; {{}} denotes 1; {{},{{}}} denotes 2, and so on), the rules of sets suffice for constructing proofs throughout math.
In 1940, Gödel showed that you can’t use the ZFC axioms to disprove the continuum hypothesis. Then in 1963, the American mathematician Paul Cohen showed the opposite —you can’t use them to prove it, either. Cohen’s proof, together with Gödel’s, means the continuum hypothesis is independent of the ZFC axioms; they can have it either way.
In addition to the continuum hypothesis, most other questions about infinite sets turn out to be independent of ZFC as well. This independence is sometimes interpreted to mean that these questions have no answer, but most set theorists see that as a profound misconception.
They believe the continuum has a precise size; we just need new tools of logic to figure out what that is. These tools will come in the form of new axioms. “The axioms do not settle these problems,” said Magidor, so “we must extend them to a richer axiom system.” It’s ZFC as a means to mathematical truth that’s lacking — not truth itself.
Ever since Cohen, set theorists have sought to shore up the foundations of infinite math by adding at least one new axiom to ZFC. This axiom should illuminate the structure of infinite sets, engender natural and beautiful theorems, avoid fatal contradictions, and, of course, settle Cantor’s question.
Gödel, for his part, believed that the continuum hypothesis is false — that there are more reals than Cantor guessed. He suspected there are ℵ2 of them. He predicted, as he wrote in 1947, “that the role of the continuum problem in set theory will be this, that it will finally lead to the discovery of new axioms which will make it possible to disprove Cantor’s conjecture.”
Two rival axioms emerged that do just that. For decades, they were suspected of being logically incompatible. “There was always this tension,” Schindler said.
To understand them, we have to go back to Paul Cohen’s 1963 work, where he developed a technique called forcing. Starting with a model of the mathematical universe that included ℵ1 reals, Cohen used forcing to enlarge the continuum to include new reals beyond those of the model. Cohen and his contemporaries soon found that, depending on the specifics of the procedure, forcing lets you to add however many reals you like — ℵ2 or ℵ35, say. Aside from new reals, mathematicians generalized Cohen’s method to conjure up all manner of other possible objects, some logically incompatible with one another. This created a multiverse of possible mathematical universes. (continued)
Sizes of infinity don’t stop there. Cantor discovered that any infinite set’s power set — the set of all subsets of its elements — has larger cardinality than it does. Every power set itself has a power set, so that cardinal numbers form an infinitely tall tower of infinities.
Standing at the foot of this forbidding edifice, Cantor focused on the first couple of floors. He managed to prove that the set formed from different ways of ordering natural numbers (from smallest to largest, for example, or with all odd numbers first) has cardinality ℵ1, one level up from the natural numbers. Moreover, each of these “order types” encodes a real number.
His continuum hypothesis asserts that this is exactly the size of the continuum — that there are precisely ℵ1 real numbers. In other words, the cardinality of the continuum immediately follow ℵ0, the cardinality of the natural numbers, with no sizes of infinity in between.
But to Cantor’s immense distress, he couldn’t prove it.
In 1900, the mathematician David Hilbert put the continuum hypothesis first on his famous list of 23 math problems to solve in the 20th century. Hilbert was enthralled by the nascent mathematics of infinity — “Cantor’s paradise,” as he called it — and the continuum hypothesis seemed like its lowest-hanging fruit.
To the contrary, shocking revelations last century turned Cantor’s question into a deep epistemological conundrum.
The trouble arose in 1931, when the Austrian-born logician Kurt Gödel discovered that any set of axioms that you might posit as a foundation for mathematics will inevitably be incomplete. There will always be questions that your list of ground rules can’t settle, true mathematical facts that they can’t prove.
As Gödel suspected right away, the continuum hypothesis is such a case: a problem that’s independent of the standard axioms of mathematics.
These axioms, 10 in all, are known as ZFC (for “Zermelo-Fraenkel axioms with the axiom of choice”), and they undergird almost all of modern math. The axioms describe basic properties of collections of objects, or sets. Since virtually everything mathematical can be built out of sets (the empty set {} denotes 0, for instance; {{}} denotes 1; {{},{{}}} denotes 2, and so on), the rules of sets suffice for constructing proofs throughout math.
In 1940, Gödel showed that you can’t use the ZFC axioms to disprove the continuum hypothesis. Then in 1963, the American mathematician Paul Cohen showed the opposite —you can’t use them to prove it, either. Cohen’s proof, together with Gödel’s, means the continuum hypothesis is independent of the ZFC axioms; they can have it either way.
In addition to the continuum hypothesis, most other questions about infinite sets turn out to be independent of ZFC as well. This independence is sometimes interpreted to mean that these questions have no answer, but most set theorists see that as a profound misconception.
They believe the continuum has a precise size; we just need new tools of logic to figure out what that is. These tools will come in the form of new axioms. “The axioms do not settle these problems,” said Magidor, so “we must extend them to a richer axiom system.” It’s ZFC as a means to mathematical truth that’s lacking — not truth itself.
Ever since Cohen, set theorists have sought to shore up the foundations of infinite math by adding at least one new axiom to ZFC. This axiom should illuminate the structure of infinite sets, engender natural and beautiful theorems, avoid fatal contradictions, and, of course, settle Cantor’s question.
Gödel, for his part, believed that the continuum hypothesis is false — that there are more reals than Cantor guessed. He suspected there are ℵ2 of them. He predicted, as he wrote in 1947, “that the role of the continuum problem in set theory will be this, that it will finally lead to the discovery of new axioms which will make it possible to disprove Cantor’s conjecture.”
Two rival axioms emerged that do just that. For decades, they were suspected of being logically incompatible. “There was always this tension,” Schindler said.
To understand them, we have to go back to Paul Cohen’s 1963 work, where he developed a technique called forcing. Starting with a model of the mathematical universe that included ℵ1 reals, Cohen used forcing to enlarge the continuum to include new reals beyond those of the model. Cohen and his contemporaries soon found that, depending on the specifics of the procedure, forcing lets you to add however many reals you like — ℵ2 or ℵ35, say. Aside from new reals, mathematicians generalized Cohen’s method to conjure up all manner of other possible objects, some logically incompatible with one another. This created a multiverse of possible mathematical universes. (continued)