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Single Idea 14423

[filed under theme 6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / d. Peano arithmetic ]

Full Idea

That '0', 'number' and 'successor' cannot be defined by means of Peano's five axioms, but must be independently understood.

Gist of Idea

'0', 'number' and 'successor' cannot be defined by Peano's axioms

Source

Bertrand Russell (Introduction to Mathematical Philosophy [1919], I)

Book Ref

Russell,Bertrand: 'Introduction to Mathematical Philosophy' [George Allen and Unwin 1975], p.9

Related Idea

Idea 18133 The usual definitions of identity and of natural numbers are impredicative [Bostock]

The 25 ideas with the same theme [set of arithmetic axioms proposed by Dedekind and Peano]:

18096 Zero is a member, and all successors; numbers are the intersection of sets satisfying this [Dedekind, by Bostock]
13949 All models of Peano axioms are isomorphic, so the models all seem equally good for natural numbers [Cartwright,R on Peano]
18113 PA concerns any entities which satisfy the axioms [Peano, by Bostock]
17634 Peano axioms not only support arithmetic, but are also fairly obvious [Peano, by Russell]
5897 0 is a non-successor number, all successors are numbers, successors can't duplicate, if P(n) and P(n+1) then P(all-n) [Peano, by Flew]
14422 Any founded, non-repeating series all reachable in steps will satisfy Peano's axioms [Russell]
14423 '0', 'number' and 'successor' cannot be defined by Peano's axioms [Russell]
7530 Russell tried to replace Peano's Postulates with the simple idea of 'class' [Russell, by Monk]
18246 Dedekind failed to distinguish the numbers from other progressions [Shapiro on Russell]
18097 The Peano Axioms describe a unique structure [Bostock]
10068 Natural numbers have zero, unique successors, unending, no circling back, and no strays [Smith,P]
16902 Peano arithmetic requires grasping 0 as a primitive number [Burge]
17441 Wright thinks Hume's Principle is more fundamental to cardinals than the Peano Axioms are [Wright,C, by Heck]
13862 There are five Peano axioms, which can be expressed informally [Wright,C]
17853 Number truths are said to be the consequence of PA - but it needs semantic consequence [Wright,C]
17854 What facts underpin the truths of the Peano axioms? [Wright,C]
10058 No two numbers having the same successor relies on the Axiom of Infinity [Musgrave]
17792 1st-order PA is only interesting because of results which use 2nd-order PA [Mayberry]
13657 First-order arithmetic can't even represent basic number theory [Shapiro]
10202 Natural numbers just need an initial object, successors, and an induction principle [Shapiro]
17459 Frege's Theorem explains why the numbers satisfy the Peano axioms [Heck]
17882 It is remarkable that all natural number arithmetic derives from just the Peano Axioms [Potter]
17887 PA is consistent as far as we can accept, and we expand axioms to overcome limitations [Koellner]
16343 The global reflection principle seems to express the soundness of Peano Arithmetic [Halbach]
16321 The compactness theorem can prove nonstandard models of PA [Halbach]