Concept
Set-theoretic definition of natural numbers
Overview
constructions of the whole numbers from sets
Overview adapted from Wikipedia (CC BY-SA 4.0; see the article history for authors).
Possible passages
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Taking any one of the natural numbers, say 29, it is easy to see that it must have all inductive properties. For since such properties belong to 0 and are hereditary, they belong to 1; therefore, since they are hereditary, they belong to 2, and so on; by twenty-nine repetitions of such arguments we show that they belong to 29. We may _define_ the "inductive" numbers as _all those that possess all inductive…
Bertrand Russell, Our Knowledge of the External World as a Field for Scientific Method in Philosophy
States what Set-theoretic definition of natural numbers is; the author is tied to Set-theoretic definition of natural numbers in the knowledge graph.
A whole number is thus a collection of 1s, but the properties of the whole number are something quite different from that of the elements through which it is constituted, just as an atom may be composed of electrons and yet, in valency, possess a property that is not the direct analogue of any property possessed by electrons not so organized.
John Dewey, Creative Intelligence
Close in meaning to Set-theoretic definition of natural numbers; the term itself is not named.
Thus there arises a classification of modes of formation, or in the language commonly used by mathematicians, a classification of Functions. Any number, considered as formed from any other number, is called a function of it; and there are as many kinds of functions as there are modes of formation. The simple functions are by no means numerous, most functions being formed by the combination of several of the…
John Stuart Mill, A System of Logic, Ratiocinative and Inductive
States what Set-theoretic definition of natural numbers is.
Philosophers who discussed Set-theoretic definition of natural numbers
Related ideas
- Zermelo–Fraenkel set theory
- Axiom of infinity
- Peano axioms
- Edward N. Zalta
- Set (mathematics)
- Set theory
- Hume's principle
- Willard Van Orman Quine
- George Boolos
- Type theory
- Structure (mathematical logic)
- Bertrand Russell
- Russell's paradox
- Gottlob Frege
- Natural deduction
- Category theory
- Cardinal number
- Equivalence relation
- New Foundations
- Empty set
- Ordinal number
Place in the tradition
Part of Type theory.
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