# The modernity of Dedekind’s anticipations contained in *What are numbers and what are they good for?*

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## Abstract

We show that Dedekind, in his proof of the principle of definition by mathematical recursion, used implicitly both the concept of an inductive cone from an inductive system of sets and that of the inductive limit of an inductive system of sets. Moreover, we show that in Dedekind’s work on the foundations of mathematics one can also find specific occurrences of various profound mathematical ideas in the fields of universal algebra, category theory, the theory of primitive recursive mappings, and set theory, which undoubtedly point towards the mathematics of twentieth and twenty-first centuries.

## Mathematics Subject Classification

01A55 11-03 03B30 03D20 03E30 08A60 18A30## Notes

### Acknowledgements

We would like to thank all who helped us by reading parts of the manuscript and suggesting mathematical improvements: Enric Cosme, Emily Riehl, Roberto Rubio, Sergiu Rudeanu, and Rafael Sivera. Our thanks are further due to our dear friend José García Roca—example of intelligence, goodness, and integrity—for correcting the English and suggesting improvements. And also for his tireless encouragement and invaluable continued moral support. Moreover, we are greatly indebted to the reviewer for his helpful comments and important stylistic suggestions.

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