Peano was born today in 1858. Peano is best known for the Peano Axioms.

The definition of addition by induction was invented by the school teachers Hermmann Grassmann in 1861. Richard Dedekind wrote his 1888 book that created New Math for arithmetic. Dedekind’s framework had a flaw in the existence of an infinite set proof.

Peano’s axiomatic framework built on Dedekind and Grassmann. It was, however, actually a step backward from Dedekind’s 1888 book in terms of teaching the concepts of numbers and their meaning in terms of order and counting.

Peano made addition an axiomatic notion instead of one defined by recursion. One of Dedekind’s most important contribution was the Recursion Theorem that justifies the use of definition by recursion aka definition by induction.

==Some versions of the Peano Axioms

http://newmathdoneright.com/2012/05/14/peano-axioms-number-line/

- 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.

http://newmathdoneright.com/2012/08/16/lucky-duck-induction/

Zero is a lucky duck.

Each lucky duck is a mom lucky duck.

Zero is not a baby lucky duck.

Each baby lucky duck has exactly one mom lucky duck.

If a farm has the zero lucky duck and the baby lucky duck of

each mom lucky duck on the farm, then the farm has all the lucky ducks.

http://newmathdoneright.com/2012/08/13/peano-axioms-in-21-words-126-characters/

- Zero a tick.
- A tick then a tick.
- Zero no mom.
- Not zero, one mom.
- Zero green, green transfers, all green.

21 words 126 letters

==

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The book has 178 Examples, 463 Exercises, 98 definitions, 45 Lemmas with proofs, references to over 96 webpages, 58 chapters, and is 391 pages when formatted as pdf with Latex. The book is however not a pdf and is 100 percent html with no pdf, jpeg or pngs.

Typical treatments of the Peano Axioms cover the same material as in the book in about 20 pages or less. They usaully have few examples and the few exercises are as difficult as the theory.

In contrast, this text has many examples too trivial for the current texts on the Peano Axioms to cover. Building at a very slow pace with many numerical examples, the reader is taken through the Peano Axioms themselves, simple consequences, order of natural numbers, and simple identities used to prove the properties of addition. This build up includes many very simple proofs by mathematical induction.

There are no quadratic or higher algebraic formulas in the book. Complicated algebra formulas are the main stumbling block to learning mathematical induction. None are in the book, yet there are many worked out proofs of simple relationship using mathematical induction and simple problems for students to do.

The e-books in this series are 100 percent html. No pdf and no jpegs or pngs. They are completely in html and will resize on any device used to read them. They are fully searchable. You can take notes on any field or equation.