YOUR MONDAY MATH with Mathematician KP Hart: NATURAL NUMBERS Part 2

Here’s Mathematician KP Hart’s Math Question and Answer for Monday, November 16th!

What is a natural number Part 2

In What is a natural number – Part 1? we saw how Dedekind `created’ the natural numbers out of a Dedekind-infinite set.

Almost at the same time Giuseppe Peano made an attempt to ground the theory of the natural numbers completely in Logic. Next to a series of logical assumptions he formulated the following axioms that the set, N, of natural numbers should satisfy

1 belongs to N
if a belongs to N then so does a+1
is a and b belong to N then we have: a=b if and only if a+1=b+1
if a belongs to N then a+1≠1
if K is such that 1 belongs to K and for all x in K x+1 belongs to K too, then N is a subset ofK

This list is known as Peano’s Axioms for arithmetic. In Arithmetices principia, nova methodo exposita Peano showed how all of our elementary arithmetic come from these rules.

The difference with Dedekind’s approach is that the latter really wanted to create a set of natural numbers, whereas Peano only wrote down what properties the natural numbers should satisfy. The sets of Dedekind satisfy Peano’s requirements. What Peano and his contemporaries tried to do was use those axioms/requirements to determine the natural numbers up to isomorphism and thus to show that Dedekind’s sets were the only ones possible. That hope was dashed by later developments in mathematical logic: the natural numbers from non-standard analysis also satisfy the axioms but as a whole do not form a structure that is isomorphic to (any one of) Dedekind’s sets.

Next time we will see how we can make, within Set Theory, one canonical standard set of natural numbers.

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Read all of KP Harts math questions here!

About Dutch Mathematician KP Hart: In the beginning of this year the Dutch government opened a website, The Dutch Science Agenda, where everyone could post questions that they thought were of scientific interest. This was an attempt to involve the whole country in determining what the Dutch science agenda should be in the coming years.

I looked through the questions and searched for terms like `mathematics’, `infinity’ … to see what mathematical questions there were and I noticed various questions that already have answers (and have had for a long time). On a whim I decided to post answers to those questions, in Dutch. For your edification I will translate these posts into English.