The difficulty of the Peano Axioms is not greater either in abstraction or in formulas from algebra I. Natural numbers are defined by the Peano Axioms below.
To give away the answer, they are 0,1,2,.. Note the natural numbers are not positive numbers. Natural numbers are without sign. The confusion of unsigned numbers and unsigned sizes with positive numbers is a major mistake in math. This makes it overly complicated to say many things that are more simply said in terms of sizes than positive numbers.
- There is a unique natural number zero.
- Every natural number has a unique successor.
- Zero is not the successor of any natural number.
- If the successors of two numbers are equal, then said numbers are equal.
- If zero is a member of a set, and the set contains all its successors, then the set contains all the natural numbers.
A set that contains all its successors means that for any element of the set, the successor of said element is a member of the set. This is called being a closed set under the successor function.
If a function contains all the natural numbers and it is stated to contain only natural numbers, then it is the set of all natural numbers. This follows from basic principles of logic.
The axioms in effect give us a function, successor, and elements, natural numbers, and these together are what is needed to get started and to carry us through. That is, together with some set theory axioms and some equality axioms.
Of these five axioms, the first 4 are “obvious”. They can’t be considered too difficult for students of algebra one. That leaves axiom 5. Stated as it is here, it also is not more difficult than algebra one, since algebra one uses sets like natural numbers, signed whole numbers, rational numbers and real numbers.
An example of these stated in mathematical algebraic notation is at this link for Carl Lee of the University of Kentucky notes.
A.7 The student will investigate and analyze function (linear and quadratic) families and their characteristics both algebraically and graphically, including
a) determining whether a relation is a function;
b) domain and range;
c) zeros of a function;
d) x– and y-intercepts;
e) finding the values of a function for elements in its domain; and
f) making connections between and among multiple representations of functions including concrete, verbal, numeric, graphic, and algebraic.
A.8 The student, given a situation in a real-world context, will analyze a relation to determine whether a direct or inverse variation exists, and represent a direct variation algebraically and graphically and an inverse variation algebraically.
If we denote the successor function by S(), then S(0) = 1, and S(n) = n+1.
Actually, we define addition in terms of S() but for now we just give this simple equivalence.
The domain of S is the set of natural numbers, i.e. 0,1,2,… Call this N.
The range of S is the set of natural numbers not including 0, so 1,2,3, … Call this N’.
A typical element of S is the ordered pair (n,n+1).
The graph of S is the set of ordered pairs (n,n+1). This is a formal definition of graph as opposed to a visual representation.
We can plot some of the points (n,n+1).
The first point is (0,0+1) or (0,1). This is the 1 tick on the Y-axis. This is the Y-intercept.
The successor function has no X-intercept. This is what axiom one tells us.
Another point is (1,2). Some points are (2,3), (3,4), (4,5), and (5,6).
These points lie on a line that goes one over and one up starting from the 1 tick on the Y-axis. So this is a 45 degree line. But it only includes points whose X-coordinates are natural numbers.
So students in algebra one are supposed to be able to handle a function like successor.
This “proves” that the Peano Axioms can be taught in Algebra 1.
If we have the Peano Axioms as a word problem and asked the student to find the domain, range, and graph the function, they should be able to do it as a homework problem. This might be challenging, but it would not seem out of line to students as unreasonable. This is an interesting experiment that actual algebra one and other courses might want to try and see what happens. Don’t announce it as the Peano Axioms. Just give the 5 axioms and call them 5 conditions. Then ask them to find the domain, range and graph the function. See if they say this is college level material or if they just do it.
We can rewrite this to make it seem like a normal algebra one relation or function problem.
Students, you are given the following information on a relation. Use this information to answer the questions that follow.
- Zero is in the domain of the relation.
- For any given number in the domain, the relation contains the ordered pair of said number as first coordinate and said number plus one as the second coordinate.
- There is no number before zero in the domain.
- For any two ordered pairs in the relation, if the first elements are the same, then so are the second elements.
- Zero is in the domain and if any number is in the domain, then that number plus one is in the domain. All numbers in the domain are reachable by adding one consecutively to zero.
Questions on the Relation
- What is the domain of the relation?
- What is the range of the relation?
- Give a few ordered pairs that belong to the relation.
- Is the relation a function?
- Does the relation have an X-intercept? If so, what is it (or What are they)?
- Does the relation have a Y-intercept. If so, what is it (or What are they)?
- What does the graph of the relation look like?
Try giving this to algebra one students and see what their reaction is. Give them the following survey. I will try to set this up in Survey Monkey or something similar.
- Is this problem hard?
- Does this problem belong in this class?
- Is this problem college level?
- Could you have done this problem before algebra one?
- What grade level do you think you could have done this problem?
- Would this problem be too hard on a test?