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] |
14423 | '0', 'number' and 'successor' cannot be defined by Peano's axioms [Russell] |
14422 | Any founded, non-repeating series all reachable in steps will satisfy 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] |
17853 | Number truths are said to be the consequence of PA - but it needs semantic consequence [Wright,C] |
13862 | There are five Peano axioms, which can be expressed informally [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] |