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1874-1891 CEReputable source · 2 sourcesWell documented

Cantor Proves Infinities Come in Different Sizes

A diagonal argument breaks mathematics open, and its own discoverer can barely believe what he has found

On the timeline · around 1874-1891 CE · Modern MathematicsModern MathematicsCantor Proves Infinities Come in Different Sizes18251850187519001925

Quick facts

Cantor's dates
1845-1918
Key correspondence
Letter to Dedekind, 1877
Chief opponent
Leopold Kronecker
Unresolved problem
The continuum hypothesis

What happened

Georg Cantor, born in 1845 in St Petersburg, developed set theory in the 1870s and 1880s and proved that infinite sets can differ in size, showing that the real numbers cannot be placed in one-to-one correspondence with the whole numbers even though both sets are infinite. Cantor was startled by his own results: writing to fellow mathematician Richard Dedekind in 1877 about a related discovery, a one-to-one correspondence between spaces of different dimensions, he wrote I see it, but I don't believe it, and elsewhere acknowledged he was placing himself in a certain opposition to views widely held concerning the mathematical infinite. His former teacher Leopold Kronecker rejected the entire approach, and the resulting bitter antagonism between the two men became public at a Mathematical Association meeting in 1891. Cantor spent much of his later career trying and repeatedly failing to resolve the continuum hypothesis, the question of whether any infinite set exists with a size strictly between that of the integers and that of the real numbers, at times believing he had solved it only to discover an error the next day.

Why it matters

Cantor's set theory became the foundation on which nearly all of modern mathematics is built, but it also opened the door to the paradoxes and foundational questions that would occupy mathematicians and logicians, including Godel, for the following half-century, and the continuum hypothesis Cantor could never resolve was later shown to be formally undecidable within standard set theory.

How we know

Cantor's papers on set theory and his correspondence with Dedekind survive and have been extensively studied, and the conflict with Kronecker, including its public dimension at professional meetings, is documented in contemporary accounts from mathematicians who witnessed it.

Sources

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