Peano Axioms

This page will contain some mostly internal links to material on the Peano Axioms and some excerpts.

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

  1. Zero is a unique tick on the number line.
  2. For each tick on the number line, there exists a unique tick immediately to the right of it.
  3. Zero is not a tick to the right of another tick.
  4. If the ticks to the right of two ticks are equal, then said two ticks are equal.
  5. 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/

  1. Zero a tick.
  2. A tick then a tick.
  3. Zero no mom.
  4. Not zero, one mom.
  5. Zero green, green transfers, all green.

21 words 126 letters

http://newmathdoneright.com/2012/08/13/the-principle-of-math-induction-is-common-sense/

http://newmathdoneright.com/2012/08/13/the-principle-of-green-tree-induction/

http://newmathdoneright.com/2012/08/13/the-principle-of-green-induction/

The Principle of Green Induction is

1) If the first is green

2) If stage n is green then stage n’ is green

Then

All stages are green.

Another version of this is

The Principle of Green Induction is

1) If the first stage is green

2) If green transfers for any stage

Then

All stages are green.

http://newmathdoneright.com/2012/08/13/green-induction-recursive-proof-that-carry-less-than-2-in-addition-of-two-numbers/

http://newmathdoneright.com/2012/08/13/the-principle-of-badge-induction/

http://newmathdoneright.com/2012/08/13/the-marie-antoinette-principle-of-induction/

http://newmathdoneright.com/2012/08/13/the-principle-of-tweet-induction/

 

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