Head sets of natural numbers starting from 0 are sets like

0

0,1

0,1,2

etc.

Tail sets are all the numbers starting from and after that number.

0 to infinity, including 0

1 to infinity including 1 but not 0

2 to infinity including 2 but not 0, 1

etc.

Dedekind in his 1888 book teaches tail sets first because the logic is easier. However, for students head sets are more concrete. Modern axiomatic set theory books teach head sets first, calling them initial segments. However, those books typically don’t do as good a job isolating head sets as a concept and teaching their properties.

If math profs understood their own literature from Dedekind onwards, they would consider whether it was better to teach head sets first or tail sets first. However, they are unaware that Dedekind teaches tail sets first and modern treatments like von Neumann construction teach head sets first.

This tells us that even math profs don’t understand this very well and don’t understand the historical literature. This then carries over to math profs not knowing how to teach math majors or even math grad students this material. If math profs don’t even know that Dedekind taught tail sets first and axiomatic set theory books teach head sets first, it shows they themselves don’t know Peano axioms very well.

This carries over to the whole teaching of pre-service math teachers in college. This is done by math profs who don’t know math foundations very well or how to teach math foundations even to math majors/math grad students. So they don’t know how to teach the substance to pre-service math teachers. They also don’t know how to teach the pre-service teachers how to teach it to the students.

The same applies to the textbook makers or to worksheet makers or to Khan Academy or other video makers. At least until Khan starts copying from my webpage. Which seems the only way that anyone will ever use it.

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## About New Math Done Right

Author of Pre-Algebra New Math Done Right Peano Axioms. A below college level self study book on the Peano Axioms and proofs of the associative and commutative laws of addition.
President of Mathematical Finance Company. Provides economic scenario generators to financial institutions.

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