# Peano Arithmetic

### Some articles on peano arithmetic, arithmetic:

Bew - Second Incompleteness Theorem - Implications For Consistency Proofs
... of proving, for example, the consistency of Peano arithmetic using any finitistic means that can be formalized in a theory the consistency of which is provable in Peano arithmetic ... the theory of primitive recursive arithmetic (PRA), which is widely accepted as an accurate formalization of finitistic mathematics, is provably consistent in PA ... Zermelo–Fraenkel set theory and T’ = primitive recursive arithmetic, the consistency of T’ is provable in T, and thus T’ can't prove the consistency of T by the above corollary of the second ...
List Of First-order Theories - Arithmetic
... The signature of a theory of arithmetic has The constant 0 The unary function, the successor function, here denoted by prefix S, or by prefix σ or postfix ′ elsewhere Two binary functions, denoted by infix + and ... Robinson arithmetic (also called Q) ... Robinson arithmetic can be thought of as Peano arithmetic without induction ...
Self-verifying Theories
... Self-verifying theories are consistent first-order systems of arithmetic much weaker than Peano arithmetic that are capable of proving their own consistency ... theorem, these systems cannot contain the theory of Peano arithmetic, and in fact, not even the weak fragment of Robinson arithmetic nonetheless, they can contain strong theorems for instance there are ... When the operations are expressed in this way, provability of a given sentence can be encoded as an arithmetic sentence describing termination of an analytic tableaux ...
The Paris–Harrington Theorem
... and Leo Harrington showed that the strengthened finite Ramsey theorem is unprovable in Peano arithmetic by showing that (in Peano arithmetic) it implies the consistency ... Since Peano arithmetic cannot prove its own consistency by Gödel's theorem, this shows that Peano arithmetic cannot prove the strengthened finite Ramsey theorem ... Its growth is so large that Peano arithmetic cannot prove it is defined everywhere, although Peano arithmetic easily proves that the Ackermann function is well defined ...
Ε₀
... numbers were introduced by Georg Cantor in the context of ordinal arithmetic they are the ordinal numbers ε that satisfy the equation , in which ω is the smallest transfinite ordinal ... Its use by Gentzen to prove the consistency of Peano arithmetic, along with Gödel's second incompleteness theorem, show that Peano arithmetic cannot prove the well-foundedness ...

### Famous quotes containing the word arithmetic:

O! O! another stroke! that makes the third.
He stabs me to the heart against my wish.
If that be so, thy state of health is poor;
But thine arithmetic is quite correct.
—A.E. (Alfred Edward)