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Single Idea 10068

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

Full Idea

The sequence of natural numbers starts from zero, and each number has just one immediate successor; the sequence continues without end, never circling back on itself, and there are no 'stray' numbers, lurking outside the sequence.

Gist of Idea

Natural numbers have zero, unique successors, unending, no circling back, and no strays

Source

Peter Smith (Intro to Gödel's Theorems [2007], 01.1)

Book Ref

Smith,Peter: 'An Introduction to Gödel's Theorems' [CUP 2007], p.1


A Reaction

These are the characteristics of the natural numbers which have to be pinned down by any axiom system, such as Peano's, or any more modern axiomatic structures. We are in the territory of Gödel's theorems.


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

Zero is a member, and all successors; numbers are the intersection of sets satisfying this [Dedekind, by Bostock]
All models of Peano axioms are isomorphic, so the models all seem equally good for natural numbers [Cartwright,R on Peano]
PA concerns any entities which satisfy the axioms [Peano, by Bostock]
Peano axioms not only support arithmetic, but are also fairly obvious [Peano, by Russell]
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]
Any founded, non-repeating series all reachable in steps will satisfy Peano's axioms [Russell]
'0', 'number' and 'successor' cannot be defined by Peano's axioms [Russell]
Russell tried to replace Peano's Postulates with the simple idea of 'class' [Russell, by Monk]
Dedekind failed to distinguish the numbers from other progressions [Shapiro on Russell]
The Peano Axioms describe a unique structure [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 [Wright,C, by Heck]
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]