Re: What is the 1st order formal system known as PA?

Rupert wrote:
> Usually there is no predicate "is a number".

Then there's a fair amount of terminological slippage among writers.
When I look up 'Peano arithmetic' or 'Peano axioms', even among
contemporary sources, I usually get a list of five axioms that are
pretty close to five of the nine axioms in Peano's paper (the other
four are axioms about identity), as I cast this in first order logic
with identity (I omit leading univesal quantifiers):

0 0-place function symbol
S 1-place function symbol
N 1-place predicate symbol

Axioms:

N0

Nn -> NSn

~(Nn & 0=Sn)

(Nn & Nk & Sn=Sk) -> n=k)

(phi[0] & An(Nn -> (phi[n] -> phi[Sn]))) -> An(Nn -> phi[n])

[I'm confused about Peano's own formulation as it is given on page 94
of van Heijenoort's 'From Frege To Godel'. There, axiom 7 does not seem
to state that S is 1-1. Is this the result of typos? Should the second
'=' be the symbol for material implication instead, and the antecedent
and consequent reversed?]

But Shoenfield on page 204 of 'Mathematical Logic' specifies Peano
arithmetic as axiomatized in first order logic with identity and
omitting leading universal quantifiers (cf. page 22 also):

0 0-place function symbol
S 1-place function symbol
+ 2-place function symbol
* 2-place function symbol
< 2-place predicate symbol

Sn not= 0

Sn=Sk -> n=k

n+0=n

n+Sk=S(n+k)

n*0=0

n*Sk=(n*k)+n

n not< 0

n<Sk -> (n<k v n=k)

(phi[0] & An(phi(n) -> phi[Sn])) -> An phi[n]

So am I correct to take it that in current discussions, by 'Peano
arithmetic', people mean the theory axiomatized by the above axioms?
(It could be a different set of axioms as long as it axiomatizes the
same theory.) Also, that each of those axioms is independent of the set
of the others? But, as to independence of primitive symbols, couldn't
we take '<' as defined by:

Df. n < k <-> Ej(j not= 0 & n+j=k)

> > Also, in set theory we prove that any two Peano systems are isomorphic.
> > And, if I'm not mistaken, we prove that any two completely ordered
> > fields are isomorphic. Yet, Lowenheim-Skolem tells us that for the
> > axioms of Peano arithmetic there are non-isomorphic models, and for the
> > axioms for a complete ordered field, there are non-isomorphic models.
>
> The axioms for a complete ordered field are second-order.
> Loewenheim-Skolem applies to first-order theories.

Yes, I overlooked that. To speak of bounds on subsets of the domain, we
have to speak of sets, so that we're in set theory or we can cast the
matter in second-order. But, to express subsets of the domain couldn't
we use an axiom schema with phi, for formulas, instead of using sets or
quantifying over second-order predicate symbols; thus we could express
the axioms of a complete ordered field as a first order theory with
axioms and one axiom schema?

And, by noting for myself that indeed 'Peano arithmetic' denotes
different things depending on the writer or context, I understood the
rest of your comments.

Thanks for you remarks. They have been helpful.

MoeBlee

.

Relevant Pages

  • Re: What is the 1st order formal system known as PA?
    ... > Then there's a fair amount of terminological slippage among writers. ... > When I look up 'Peano arithmetic' or 'Peano axioms', ... I usually get a list of five axioms that are ... > 0 0-place function symbol ...
    (sci.logic)
  • Re: Can ZFC prove Addition is Associative?
    ... The Axiom of Infinity is Induction. ... The N of Peano world does not exist in the ZFC world. ... We can't get this far since you can't add the N given in the Peano ... axioms to the the sets given in the ZFC axioms. ...
    (sci.logic)
  • Re: Can ZFC prove Addition is Associative?
    ... The Axiom of Infinity is Induction. ... ZFC says a certain set exists. ... I say that Peano's axioms cannot give it a name. ... how can Peano give a name to a certain set in ZFC if ZFC ...
    (sci.logic)
  • Re: Cantor Confusion
    ... >> That a infinite string of decimal ... >> of decimal digits does not define a natural number. ... When Peano defines the natural numbers, does he talk about what they ... The Peano axioms for the natural numbers are a bunch of axioms, ...
    (sci.math)
  • Re: Can STRONG induction for N be derived from Peanos Axioms?
    ... But Peano did include 0 in his axioms. ... >>staements and proofs considerably if you consider it part of the naturals. ... break anything (it'll prevent lots of thms), ...
    (sci.logic)