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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
Zero is a member, and all successors; numbers are the intersection of sets satisfying this [Dedekind, by Bostock]
     Full Idea: Dedekind's idea is that the set of natural numbers has zero as a member, and also has as a member the successor of each of its members, and it is the smallest set satisfying this condition. It is the intersection of all sets satisfying the condition.
     From: report of Richard Dedekind (Nature and Meaning of Numbers [1888]) by David Bostock - Philosophy of Mathematics 4.4
All models of Peano axioms are isomorphic, so the models all seem equally good for natural numbers [Cartwright,R on Peano]
     Full Idea: Peano's axioms are categorical (any two models are isomorphic). Some conclude that the concept of natural number is adequately represented by them, but we cannot identify natural numbers with one rather than another of the isomorphic models.
     From: comment on Giuseppe Peano (Principles of Arithmetic, by a new method [1889], 11) by Richard Cartwright - Propositions 11
     A reaction: This is a striking anticipation of Benacerraf's famous point about different set theory accounts of numbers, where all models seem to work equally well. Cartwright is saying that others have pointed this out.
PA concerns any entities which satisfy the axioms [Peano, by Bostock]
     Full Idea: Peano Arithmetic is about any system of entities that satisfies the Peano axioms.
     From: report of Giuseppe Peano (Principles of Arithmetic, by a new method [1889], 6.3) by David Bostock - Philosophy of Mathematics 6.3
     A reaction: This doesn't sound like numbers in the fullest sense, since those should facilitate counting objects. '3' should mean that number of rose petals, and not just a position in a well-ordered series.
Peano axioms not only support arithmetic, but are also fairly obvious [Peano, by Russell]
     Full Idea: Peano's premises are recommended not only by the fact that arithmetic follows from them, but also by their inherent obviousness.
     From: report of Giuseppe Peano (Principles of Arithmetic, by a new method [1889], p.276) by Bertrand Russell - Regressive Method for Premises in Mathematics p.276
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]
     Full Idea: 1) 0 is a number; 2) The successor of any number is a number; 3) No two numbers have the same successor; 4) 0 is not the successor of any number; 5) If P is true of 0, and if P is true of any number n and of its successor, P is true of every number.
     From: report of Giuseppe Peano (works [1890]) by Antony Flew - Pan Dictionary of Philosophy 'Peano'
     A reaction: Devised by Dedekind and proposed by Peano, these postulates were intended to avoid references to intuition in specifying the natural numbers. I wonder if they could define 'successor' without reference to 'number'.
Any founded, non-repeating series all reachable in steps will satisfy Peano's axioms [Russell]
     Full Idea: Given any series which is endless, contains no repetitions, has a beginning, and has no terms that cannot be reached from the beginning in a finite number of steps, we have a set of terms verifying Peano's axioms.
     From: Bertrand Russell (Introduction to Mathematical Philosophy [1919], I)
'0', 'number' and 'successor' cannot be defined by Peano's axioms [Russell]
     Full Idea: That '0', 'number' and 'successor' cannot be defined by means of Peano's five axioms, but must be independently understood.
     From: Bertrand Russell (Introduction to Mathematical Philosophy [1919], I)
Russell tried to replace Peano's Postulates with the simple idea of 'class' [Russell, by Monk]
     Full Idea: What Russell tried to show [at this time] was that Peano's Postulates (based on 'zero', 'number' and 'successor') could in turn be dispensed with, and the whole edifice built upon nothing more than the notion of 'class'.
     From: report of Bertrand Russell (The Principles of Mathematics [1903]) by Ray Monk - Bertrand Russell: Spirit of Solitude Ch.4
     A reaction: (See Idea 5897 for Peano) Presumably you can't afford to lose the notion of 'successor' in the account. If you build any theory on the idea of classes, you are still required to explain why a particular is a member of that class, and not another.
Dedekind failed to distinguish the numbers from other progressions [Shapiro on Russell]
     Full Idea: Dedekind's demonstrations nowhere - not even where he comes to cardinals - involve any property distinguishing numbers from other progressions.
     From: comment on Bertrand Russell (The Principles of Mathematics [1903], p.249) by Stewart Shapiro - Philosophy of Mathematics 5.4
     A reaction: Shapiro notes that his sounds like Frege's Julius Caesar problem, of ensuring that your definition really does capture a number. Russell is objecting to mathematical structuralism.
The Peano Axioms describe a unique structure [Bostock]
     Full Idea: The Peano Axioms are categorical, meaning that they describe a unique structure.
     From: David Bostock (Philosophy of Mathematics [2009], 4.4 n20)
     A reaction: So if you think there is nothing more to the natural numbers than their structure, then the Peano Axioms give the essence of arithmetic. If you think that 'objects' must exist to generate a structure, there must be more to the numbers.
Natural numbers have zero, unique successors, unending, no circling back, and no strays [Smith,P]
     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.
     From: Peter Smith (Intro to Gödel's Theorems [2007], 01.1)
     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.
Peano arithmetic requires grasping 0 as a primitive number [Burge]
     Full Idea: In the Peano axiomatisation, arithmetic seems primitively to involve the thought that 0 is a number.
     From: Tyler Burge (Frege on Apriority (with ps) [2000], 5)
     A reaction: Burge is pointing this out as a problem for Frege, for whom only the logic is primitive.
Wright thinks Hume's Principle is more fundamental to cardinals than the Peano Axioms are [Wright,C, by Heck]
     Full Idea: Wright is claiming that HP is a special sort of truth in some way: it is supposed to be the fundamental truth about cardinality; ...in particular, HP is supposed to be more fundamental, in some sense than the Dedekind-Peano axioms.
     From: report of Crispin Wright (Frege's Concept of Numbers as Objects [1983]) by Richard G. Heck - Cardinality, Counting and Equinumerosity 1
     A reaction: Heck notes that although PA can be proved from HP, HP can be proven from PA plus definitions, so direction of proof won't show fundamentality. He adds that Wright thinks HP is 'more illuminating'.
There are five Peano axioms, which can be expressed informally [Wright,C]
     Full Idea: Informally, Peano's axioms are: 0 is a number, numbers have a successor, different numbers have different successors, 0 isn't a successor, properties of 0 which carry over to successors are properties of all numbers.
     From: Crispin Wright (Frege's Concept of Numbers as Objects [1983], Intro)
     A reaction: Each statement of the famous axioms is slightly different from the others, and I have reworded Wright to fit him in. Since the last one (the 'induction axiom') is about properties, it invites formalization in second-order logic.
Number truths are said to be the consequence of PA - but it needs semantic consequence [Wright,C]
     Full Idea: The intuitive proposal is the essential number theoretic truths are precisely the logical consequences of the Peano axioms, ...but the notion of consequence is a semantic one...and it is not obvious that we possess a semantic notion of the requisite kind.
     From: Crispin Wright (Frege's Concept of Numbers as Objects [1983], Intro)
     A reaction: (Not sure I understand this, but it is his starting point for rejecting PA as the essence of arithmetic).
What facts underpin the truths of the Peano axioms? [Wright,C]
     Full Idea: We incline to think of the Peano axioms as truths of some sort; so there has to be a philosophical question how we ought to conceive of the nature of the facts which make those statements true.
     From: Crispin Wright (Frege's Concept of Numbers as Objects [1983], Intro)
     A reaction: [He also asks about how we know the truths]
No two numbers having the same successor relies on the Axiom of Infinity [Musgrave]
     Full Idea: The axiom of Peano which states that no two numbers have the same successor requires the Axiom of Infinity for its proof.
     From: Alan Musgrave (Logicism Revisited [1977], §4 n)
     A reaction: [He refers to Russell 1919:131-2] The Axiom of Infinity is controversial and non-logical.
1st-order PA is only interesting because of results which use 2nd-order PA [Mayberry]
     Full Idea: The sole theoretical interest of first-order Peano arithmetic derives from the fact that it is a first-order reduct of a categorical second-order theory. Its axioms can be proved incomplete only because the second-order theory is categorical.
     From: John Mayberry (What Required for Foundation for Maths? [1994], p.412-1)
First-order arithmetic can't even represent basic number theory [Shapiro]
     Full Idea: Few theorists consider first-order arithmetic to be an adequate representation of even basic number theory.
     From: Stewart Shapiro (Foundations without Foundationalism [1991], 5 n28)
     A reaction: This will be because of Idea 13656. Even 'basic' number theory will include all sorts of vast infinities, and that seems to be where the trouble is.
Natural numbers just need an initial object, successors, and an induction principle [Shapiro]
     Full Idea: The natural-number structure is a pattern common to any system of objects that has a distinguished initial object and a successor relation that satisfies the induction principle
     From: Stewart Shapiro (Philosophy of Mathematics [1997], Intro)
     A reaction: If you started your number system with 5, and successors were only odd numbers, something would have gone wrong, so a bit more seems to be needed. How do we decided whether the initial object is 0, 1 or 2?
Frege's Theorem explains why the numbers satisfy the Peano axioms [Heck]
     Full Idea: The interest of Frege's Theorem is that it offers us an explanation of the fact that the numbers satisfy the Dedekind-Peano axioms.
     From: Richard G. Heck (Cardinality, Counting and Equinumerosity [2000], 6)
     A reaction: He says 'explaining' does not make it more fundamental, since all proofs explain why their conclusions hold.
It is remarkable that all natural number arithmetic derives from just the Peano Axioms [Potter]
     Full Idea: It is a remarkable fact that all the arithmetical properties of the natural numbers can be derived from such a small number of assumptions (as the Peano Axioms).
     From: Michael Potter (Set Theory and Its Philosophy [2004], 05.2)
     A reaction: If one were to defend essentialism about arithmetic, this would be grist to their mill. I'm just saying.
PA is consistent as far as we can accept, and we expand axioms to overcome limitations [Koellner]
     Full Idea: To the extent that we are justified in accepting Peano Arithmetic we are justified in accepting its consistency, and so we know how to expand the axiom system so as to overcome the limitation [of Gödel's Second Theorem].
     From: Peter Koellner (On the Question of Absolute Undecidability [2006], 1.1)
     A reaction: Each expansion brings a limitation, but then you can expand again.
The global reflection principle seems to express the soundness of Peano Arithmetic [Halbach]
     Full Idea: The global reflection principle ∀x(Sent(x) ∧ Bew[PA](x) → Tx) …seems to be the full statement of the soundness claim for Peano arithmetic, as it expresses that all theorems of Peano arithmetic are true.
     From: Volker Halbach (Axiomatic Theories of Truth [2011], 22.1)
     A reaction: That is, an extra principle must be introduced to express the soundness. PA is, of course, not complete.
The compactness theorem can prove nonstandard models of PA [Halbach]
     Full Idea: Nonstandard models of Peano arithmetic are models of PA that are not isomorphic to the standard model. Their existence can be established with the compactness theorem or the adequacy theorem of first-order logic.
     From: Volker Halbach (Axiomatic Theories of Truth [2011], 8.3)