NUMBER OPERATIONS BY RECURSIONS GRAPHED VIA BYPASS DIAGRAMS                                                                                                                                                
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        BYPASS TEDIOUS CARDINATION
          --------------------->
     SHIFT|                    ^LABEL
      FROM|                    |COUNTS
"TALLYING"|                    |FOR NATURAL
          |                    |NUMBER
          |                    |CARDINATION
          V-------------------->
          RECURSION ON COUNTING
 replace tedious TALLYING CARDINATION by
COUNTING - thus, |||| Þ 4 - GENERATED BY
RECURSION: INITIAL COUNT, 0 -- RECURSION
BY SUCCESSOR OPERATION, S( ) , s. t., FOR
ANY COUNT  n , S(n) º n + 1 , where  n + 1 is
SUCCESSOR of n - thus, S(0) = 1 -- S(1) = 2 --
S(2) = 3 - COUNTS ARE "NATURAL NUMBERS"
- EVERY NATURAL NUMBER HAS SUCCESSOR
so SUCCESSION IS A TOTAL OPERATION --
SUCCESSOR INDUCES INVERSE OPERATION,
PREDECESSOR, P( ), s. t. n is PREDECESSOR of
n + 1, i.e., P(n + 1) = n IF,AND ONLY IF, S(n) = n + 1
-- HOWEVER, 0 HAS NO PREDECESSOR, so P( )
IS A PARTIAL (NOT A TOTAL) OPERATION.
BYPASS TEDIOUS MULTIPLE COMBING OF COUNTS       
       ------------------------->
  SHIFT|                        ^ADDITION
   FROM|                        |OPERATION FOR
 COUNTS|                        |1ST PRIMARY
       |                        |OPERATION OF
       |                        |NATURAL NUMBER
       V----------------------- >SYSTEM
        RECURSION ON COUNTS FOR
         COMBINING
 A CHILD COUNT OFF 3 FINGERS, THEN 4
FINGERS, & COMBINES TO COUNT OFf THE
7 FINGERS - SUCH TEDIOUS COMBINING IS
REPLACED BY ADDITION CONSTRUCTED BY
RECURSION ON COUNTING OPERATION:
S(a) º a + 1, a + S(b) º S(a + b) - AS WITH
COUNTING, ADDITION ALWAYS YIELDS A
NATURAL NUMBER, SO ADDITION IS A TOTAL
OPERATION - PARTIAL LIMIT ON "COUNTING
BACKWARDS" PUTS LIMIT ON INVERSE OF
ADDITION: SUBTRACTION - BUT (NOTED
ELSEWHERE) MORE THAN INDUCTION ON
ADDING IS NEEDED TO DEFINE ITS INVERSE.
BYPASS TEDIOUS MULTIPLE COMBINING OF ADDENDS       
       ------------------------->
  SHIFT|                        ^MULTIPLICATION
   FROM|                        |OPERATION FOR
ADDENDS|                        |2ND PRIMARY
       |                        |OPERATION OF
       |                        |NATURAL NUMBER
       V----------------------- >SYSTEM
       RECURSION ON ADDENDS FOR
         COMBINING
 REPEATED DDITION, SUCH AS 5 + 5 + 5 + 5 IS
TEDIOUS -- BYPASSED BY DEFINING
MULTIPLICATION AS RECURSION ON
ADDITION: a * 1 º a, a * S(b) º a * b + a - LIKE
COUNTING AND ADDITION, MULTIPLICATION
IS TOTAL, ALWAYS YIELDING A NATURAL
NUMBER - BUT PARTIAL LIMITS ON INVERSE
OF COUNTING AND INVERSE OF ADDITION
PUT SUCH A LIMIT ON MULTIPLICATION.
BYPASS TEDIOUS MULTIPLE COMBINING OF PRODUCTS       
        ------------------------->
   SHIFT|                        ^EXPONENTIAL
    FROM|                        |OPERATION FOR
PRODUCTS|                        |3RD PRIMARY
        |                        |OPERATION OF
        |                        |NATURAL NUMBER
        V----------------------- >SYSTEM
       RECURSION ON PRODUCTS FOR
         COMBINING
 REPEATED MULTIPLICATION OF FACTORS,
SUCH AS 4*4*4*4 IS TEDIOUS - BYPASSED BY
DEFINING EXPONENTIATION, DENOTED
be = p, WHERE b IS BASE, e IS EXPONENT, p IS
POWER (eTH POWER OF b), DEFINED
RECURSIVELY BY MULTIPLICATION:
b0 º 1, bS(e) º (be)*b -- LIKE COUNTING,
ADDITION, MULTIPLICATION,
EXPONENTIATION IS A TOTAL OPERATION -
BUT LIMIT ON INVERSE OF, RSP., COUNTING,
ADDITION, MULTIPLICATION, PUTS LIMIT ON
INVERSE OF EXPONENTIATION.