Completeness and incompleteness · Definition

Arithmetic

logicfoundationsarithmetic

Definition

Arithmetic studies numbers, their operations and their properties.

\[\mathbb{N},\quad +,\quad \cdot,\quad \le\]

Arithmetic studies numbers, operations and elementary properties such as addition, multiplication, divisibility and order.

Besides computing, arithmetic makes it possible to formulate precise statements about numbers. That is why it appears as part of the minimal language that some formal systems need to express their own operations and proofs.

It is worth distinguishing the operation from its properties: adding integers, for example, is an operation; that addition is associative and commutative are properties that can later be proved.

Think about it

Why does arithmetic appear in the context of Gödel?

Because a sufficiently expressive formal system must be able to represent arithmetic operations and statements in order to build the coding of the result.

Exercises 10

  1. Arithmetic mainly studies...
  2. Which equality expresses the commutativity of addition?
  3. The additive neutral element of the integers is...
  4. If a,b ∈ ℤ, then a+b belongs to...
  5. Which operation is not closed in ℤ?
  6. Select operations that are part of elementary arithmetic.
  7. Which statement about multiplication of integers is correct?
  8. If a≠0 and a divides b, then there is an integer k such that...
  9. Select true properties of the usual addition in ℤ.
  10. Why is the usual division not a binary operation on ℤ?

References

  • Herbert B. Enderton, A Mathematical Introduction to Logic, 2nd edition, Academic Press, 2001.