Peano Axioms for the Number Line. Think of a number line that starts at 0. It only has the natural numbers as ticks, so 0, 1, 2, etc. It has no fractions marked and it has no signs positive or negative. Natural numbers are before signs. Signed numbers come later as a pair of a sign and a size.
- Zero is a unique tick on the number line.
- For each tick on the number line, there exists a unique tick immediately to the right of it.
- Zero is not a tick to the right of another tick.
- If the ticks to the right of two ticks are equal, then said two ticks are equal.
- If a set contains zero and each tick to the right of a tick, then it contains all the ticks on the number line.
The character count for the 5 Peano Axioms in number line form is 391 characters. This is within the capacity of first graders to commit to memory if desired.
The ticks on the number line are the natural numbers. In formal logic, we could set up a correspondence between ticks on the number line and the natural numbers in the usual version of the Peano Axioms. These types of correspondence were part of what Dedekind was thinking in terms of in 1888.
The number line is closed under going one tick to the right. The successor function is going one tick to the right.
If we want to distinguish ticks from numbers, we can use tick in the name, e.g. zero tick, one tick.
As a comparison, a version of the Peano Axioms is here:
- Is this college level?
- Are these statements, “obvious” to children?
- If they are obvious, is pointing them out too difficult?
- If they are not obvious, what are they missing when they use a number line?
- What are they learning with a number line if they don’t understand the above points?