The following is the introduction and conclusion to the report I wrote during my course in pure mathematics at Macquaire University 2020. I’ve omitted the bulk of the report due to mathematical formatting complications while attempting to translate LaTex into WordPress. The full report can be downloaded at the end of this post.
One fish, two fish, red fish, blue fish. There is something odd about this sequence as we read it in our heads. Beginning with one fish and then following with two fish feels natural, since if you see one fish and then you see another fish, you’ve seen two fish. However the ordering of red fish and then blue fish seems arbitrary, Dr Seuss may as well have said blue fish, red fish and no one would have batted an eye. What would be strange is to say you’ve seen two fish and then note you’ve seen one fish. This is given in the statement that you’ve seen two fish, since 2 = 1 + 1.
Natural numbers have properties that may feel obvious or trivial but we will see that this collection of properties can be uniquely characterised, up to isomorphism, using a few primitive notions. At first I will discuss the characterisation of the natural numbers using the Peano axioms and introduce the recursion theorem. Then a recursive definition of the arithmetic of the natural numbers will be given. Then I will use the recursion theorem and inductive methods to prove that any model (N, s′, 0′) of the Peano axioms will be structurally isomorphic to the natural numbers.
In Turin, Italy 1889, the mathematician Guiseppe Peano developed a way to describe the arithmetic of numbers using the primitive notions 0, number and ’successor of’. He used the following five postulates;
(P1) 0 is a number.
(P2) If a is a number then the successor of a is a number.
(P3) 0 is not the successor of any number.
(P4) If two numbers have the same successor, the two numbers are identical.
(P5) If N is a class containing 0 and the successor of every number belonging to N, then every number belongs to N.
It should be noted that Peano himself did not include 0 in the set of natural numbers and instead used 1 as the initial number. Hopefully by the end of this report it should be clear that these two conventions are equivalent to one another in that they both satisfy the Peano axioms. In fact we could start at any integer. This is alluded to in a quote by Peano in Bertrand Russell’s Principles of Mathematics.
All the systems which satisfy the primitive propositions have a one-one cor- respondence with the numbers. Number is what is obtained from all these systems by abstraction; in otherwords, number is the system which has all the properties enunciated in the primitive propositions, and only those.
Our task will be to find this so called one-one correspondence in the form of a bijection which preserves the arithmetic of these numbers and to do this we will introduce the recursion theorem… . . .
. . . … some mathematical induction … . . .
. . . … We’ve now seen that φ : N → N’ is a bijection which preserves addition, multiplication and order between the natural numbers (N,0,s) and any model of the Peano axioms (N’, 0′, s′). We used (N’, 0′, s′) for could have used for (N, 1, s) like Peano’s original axioms or even (N, 42, s). The recursion theorem always makes sure there exists a unique isomorphism between any two models of the Peano axioms.
Note: I have uploaded document which includes the report in full, containing a proof to the recursion theorem and a construction this one-one correspondence between the natural numbers and any model of the peano axioms.
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