structure for 'Mathematics'    |     alphabetical list of themes    |     expand these ideas

6. Mathematics / B. Foundations for Mathematics / 3. Axioms for Number / d. Peano arithmetic

[set of arithmetic axioms proposed by Dedekind and Peano]

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