torpedoes (suggested new terminology)

Definition 1. The 0-contrapositive of a statement of the form ‘every A is a B’ is the statement ‘the number of A’s that are not a B is 0’.

Definition 2. A torpedo is a 0-count contrapositive form of the customary statement of a theorem of the form ‘every A is a B’.


1. The number of integers greater than 1 having no prime divisor is 0.

2. The number of triangles that can’t be inscribed in a circle is 0.

3. The number of squares having a diagonal commensurable with its side is 0.

4. The number of real numbers x for which the square of x is less than 0 is 0.

5. The number of finite division rings that are not a field is 0.

6. The number of groups that are not a permutation group is 0.

7. The number of non-constant bounded entire functions is 0.

8. The number of non-constant polynomials not having a root is 0.

9. The number of differentiable functions that are not continuous is 0.

10. The number of maps requiring more than 4 colors is 0.

11. The number of perfect point sets that are countable is 0.

12. The number of positive integers not representable as the sum of 4 squares is 0.

13. The number of proper arithmetic progressions that contain no prime is 0.

14. The number of primes having no primitive root is 0.

The value of torpedoes is that they allow the early introduction of concepts to the young, by omitting all secondary verbiage. The young can absorb these concepts on the fly in the same way that they absorb their native language. Memorizing the answer to ‘the number of maps requiring more than 4 colors’ is not much more difficult than memorizing the answer to ‘7 times 8’. This early familiarity later serves as a springboard to discussion of the topics.






[suppressing terminology]