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A Basic Model for the Peano Axioms For a more concrete definition of the natural numbers, as well as a limited proof of the consistency of the Peano axioms, we can construct a model of the nonnegative integers.

Axioms of the Integers

The foundation of the model is the object we will use as zero , along with a "successor" function, which we will call succ. Given any number, the successor function provides us with the next number; we use it to inductively define all the rest of the natural numbers.

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Satisfaction of the Peano Axioms We'll now prove the Peano axioms as theorems within the model. Consider any element of the model, a. Call its successor b. To prove P. Furthermore, note that each member is defined as a set and the number of elements in each member is equal to the number represented by that member.

Number Theory/Axioms

So, 0 contains 0 elements, 1 contains 1 element, 2 contains 2 elements, and so forth. Call the number of elements n. Consider a subset of the natural numbers, S , which contains 0 and which is closed under the succ operation, and assume that it does not contain all the natural numbers. Choose a number which is not in S ; call that number x. Since x is a natural number, it's possible to get from 0 to x with a finite chain of succ operations.

2. The ring structure

Certainly, 0 is, by assumption. But some member on the list must not be in S , since, at least, x is not. Upon encountering the first number which is not a member of S , back up.

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Well 12 divided by 2 is 6, which is a whole number, so in this case we get a whole number result. If we divide two whole numbers we cannot guarantee that the result will still be a whole number. So the set of whole numbers is not closed under the operation of division.

Positive whole numbers are closed under addition - you always get a positive whole number in the result.

lecture 4 - W HOLE NUMBERS 1 Axioms for positive integers...

To decide whether a set of numbers is closed under some operation or other, look for cases where the result is no longer in the set you started with. In the case of real numbers, which include positive, negative, fractional, and irrational like sqrt 2 numbers, the operations of addition, multiplication, division and subtraction are all closed apart from division by zero which is not defined.

But taking square roots is not closed because if, for example, we try sqrt -5 , we no longer get a real number as a result. In fact, we have moved into the realm of complex numbers. The idea of 'closure' does not apply only to operations on sets of numbers. We can have operations on vectors and matrices, for example, which might or might not be closed.

RA1.3. Peano Axioms and Induction