Arithmetization of analysis

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The arithmetization of analysis was a research program in the foundations of mathematics carried out in the second half of the 19th century.


Kronecker originally introduced the term arithmetization of analysis, by which he meant its constructivization in the context of the natural numbers (see quotation at bottom of page). The meaning of the term later shifted to signify the set-theoretic construction of the real line. Its main proponent was Weierstrass, who argued the geometric foundations of calculus were not solid enough for rigorous work.

Research program[edit]

The highlights of this research program are:


An important spinoff of the arithmetization of analysis is set theory. Naive set theory was created by Cantor and others after arithmetization was completed as a way to study the singularities of functions appearing in calculus.

The arithmetization of analysis had several important consequences:


  • "God created the natural numbers, all else is the work of man." — Kronecker