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PA (Definition)

Peano Arithmetic (PA) is the restriction of Peano's axioms to a first order theory of arithmetic. The only change is that the induction axiom is replaced by induction restricted to arithmetic formulas:

$\displaystyle \phi(0)\wedge \forall x(\phi(x)\rightarrow\phi(x'))\rightarrow \forall x\phi(x))$   where $\displaystyle \phi$ is arithmetical

Note that this replaces the single, second-order, axiom of induction with a countably infinite schema of axioms.

Appropriate axioms defining $ +$, $ \cdot$, and $ <$ are included. A full list of the axioms of PA looks like this (although the exact list of axioms varies somewhat from source to source):

  • $ \forall x (x'\neq 0)$ (0 is the first number)
  • $ \forall x,y (x'=y'\rightarrow x=y)$ (the successor function is one-to-one)
  • $ \forall x (x+0=x)$ (0 is the additive identity)
  • $ \forall x,y(x+y'=(x+y)')$ (addition is the repeated application of the successor function)
  • $ \forall x(x\cdot 0=0)$
  • $ \forall x,y(x\cdot(y')=x\cdot y+x)$ (multiplication is repeated addition)
  • $ \forall x(\neg (x<0))$ (0 is the smallest number)
  • $ \forall x,y(x<y'\leftrightarrow x<y\vee x=y)$
  • $ \phi(0)\wedge \forall x(\phi(x)\rightarrow\phi(x'))\rightarrow \forall x\phi(x))$   where $ \phi$ is arithmetical



"PA" is owned by Henry.
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See Also: Peano arithmetic

Other names:  Peano arithmetic, first order Peano arithmetic

Pronunciation (guide):
 Peano: /pay-ah'noh/
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Cross-references: identity, additive, one-to-one, function, successor, source, axioms, countably infinite, axiom of induction, second-order, arithmetic formulas, restricted, induction, induction axiom, first order theory, Peano's axioms, restriction
There are 12 references to this entry.

This is version 7 of PA, born on 2002-08-17, modified 2002-08-25.
Object id is 3301, canonical name is PeanoArithmeticFirstOrder.
Accessed 8025 times total.

Classification:
AMS MSC03F30 (Mathematical logic and foundations :: Proof theory and constructive mathematics :: First-order arithmetic and fragments)

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