Concept
Peano axioms
Overview
axiomatic system for the natural numbers
Overview adapted from Wikipedia (CC BY-SA 4.0; see the article history for authors).
Possible passages
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One great advance, from the point of view of correctness, has been made by introducing points as they are required, and not starting, as was formerly done, by assuming the whole of space. This method is due partly to Peano, partly to another Italian named Fano. To those unaccustomed to it, it has an air of somewhat wilful pedantry. In this way, we begin with the following axioms: (1) There is a class of entities…
Bertrand Russell, Mysticism and Logic and Other Essays
The author is tied to Peano axioms in the knowledge graph.
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 Peano axioms is; the author is tied to Peano axioms in the knowledge graph.
_Arithmetic_ I reach what M. Couturat calls the _ordinal theory_ which is the foundation of arithmetic properly so called. M. Couturat begins by stating Peano's five assumptions, which are independent, as has been proved by Peano and Padoa. 1. Zero is an integer. 2. Zero is not the successor of any integer. 3. The successor of an integer is an integer. To this it would be proper to add, Every integer has a…
Henri Poincaré, The Foundations of Science
Close in meaning to Peano axioms; the term itself is not named.
Philosophers who discussed Peano axioms
- Kurt Gödel
- Crispin Wright
- Willard Van Orman Quine
- George Boolos
- David Lewis (philosopher)
- Alfred Tarski
- Charles Sanders Peirce
Related ideas
- First-order logic
- Random House Webster's Unabridged Dictionary
- Gödel's completeness theorem
- Giuseppe Peano
- Integer
- Compactness theorem
- Zermelo–Fraenkel set theory
- Finitism
- John von Neumann
- Kurt Gödel
- Harvard University
- Function (mathematics)
- Axiomatic system
- Mathematical induction
- Group (mathematics)
- Mathematical logic
- Model theory
- Reflexive relation
- Natural number
- Number theory
- Relation (mathematics)
- Richard Dedekind
- Axiom of infinity
- Recursion
Place in the tradition
Part of Analytic philosophy.