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### 6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / d. Peano arithmetic

#### [set of arithmetic axioms proposed by Dedekind and Peano]

25 ideas
 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 [Korsgaard 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]