THE CARDINAL NUMBERS REPRESENT "HOW MANY?". A CARDINAL NUMBER IS ASSIGNED TO A SET -- thus, 2 assigned to a set such as {a, b}. CARDINATION IS THE PROCESS OF MATCHING TWO SETS OR SELECTIONS TO SEE IF THEY ARE 1-1 CORRESPONDENT HENCE EQUAL (EQULLY CARDINAL). Thus, sets {a, b} and {j, k}, when CARDINATED, are found to be 1-1 CORRESPONDENT, hence, BOTH ARE REPRESENTSTIVES OF CARDINAL 2.

THE ORDINAL NUMBERS REPRESENT THE ORDERING OF ITEMS: FIRST, SECOND, THIRD, FOURTH, etc.

However, in an associated file, "The Paradox of Counting", I note that CHILDREN ARE TAUGHT TO COUNT (AN ORDINAL PROCESS) BY NAMING THE CARDINAL NUMBERS. Without, further council, this makes MUD of the DISTINCTION BETWEEN THE TWO DIFFERENT NUMBERS SYSTEMS!

The problem is confounded by the fact that THE DISTINCTION IS NO LONGER TAUGHT, as it was taught when I was a child. As a TEACHER OF (prospective and in-the-school) TEACHERS, I found that few knew of this DISTINCTION (and my colleagues objected to my teaching the distinction).

So what? The DIFFERENCE BETWEEN ORDINATION & CARDINATION is CRITICAL in passing from the FINITE to the INFINIIT & TRANSFINITE. The 5*4*3*2*1 = 120 different ways we can COUNT (ORDINATE) fingers of one hand yields the same CARDINALITY (CARDINATION). But DIFFERNCE CAN APPEAR BETWEEN ORDINATION & CARDINATION in the INFINITE & TRANFINITE, as in the DIFFERENTIAL & INTEGRAL CALCULUS, wherein DIFFERENT ORDERINGS OF INFINITE SEQUENCES RESULT IN DIFFERENT CARDINALS or no result! (Children should learn early that, without the CALCULUS we'd still be living in the MIDDLE AGES -- no autos, no steam engines, no electricity, no radio or TV or computers!) When I first encountered, in a Calculus class at Columbia U., that weird collisions of ORDINATION & CARDINATION can result in the INFINITE & TRANSFINITE, I was so traumatize I thought about quitting school! I want to plant essential ideas early to prevent such traumas later on -- PREVENTIVE EDUCATION.

And PRESCHOOL CHILDREN can INTERACTIVELY learn about CARDINALS and ORDINALS!

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BROWNPAGGING CARDINAL NUMBERS

1. MODEL STAGE
   • Distinguish a POSITION on a FLAT (say, table or floor) that is UNOCCUPIED, to REPRESENT CARDINAL ZERO. Shifting over a space, place a SINGLE BROWN BAG, TO REPRESENT CARDINAL ONE.
   • Shifting over, place a BROWN BAG AND ANOTHER BROWN BAG, TO REPRESENT CARDINAL TWO.
   • Shifting over, place a BROWN BAG, ANOTHER BAG, ANOTHER BAG, TO REPRESENT CARDINAL THREE.
   • Et cetera, until the "idea catches on".
2. ANOTHER MODEL STAGE: EMULATE ALL STEPS IN ABOVE STAGE.
3. ESTABLISH CORRESPONDENCE OF ONES OR TWOS OR THREES IN THE TWO MODELS.

. This CONVEYS THE NOTION OF CARDINALITY.

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BROWNPAGGING ORDINALS

1. On a flat, place a single brown bag, noting that it is EMPTY, to DENOTE FIRST, which contains ZEROTH (represented by emptiness):

   
   
   	|         |
           |         |
           |_________|

2. Shifting a little, place a BAG WITH A BAG INSIDE IT -- NOTING THAT THE "OUTSIDE" BAG CONTAINS A BAG, BUT THE "INSIDE" DOES NOT -- TO REPRESENT ORDINAL FIRST, NOTING THAT THE BAG OF THIS STAGE INCLUDES THE BAG OF THE PREVIOUS STAGE:

           |         |          | |
           |         |          | |
           |         |__________| |
           |______________________|
   Shifting slighlty, USE A BAG TO CONTAIN THE MODELS OF THE PREVIOUS STAGES, TO REPRESENT
   ORDINAL SECOND:
       |   |         |   |        |         | |   |
       |   |         |   |        |         | |   |
       |   |_________|   |        |_________| |   |
       |                 |____________________|   |                 
       |__________________________________________|
   Sparing myself some typing, imagine how this looks with a BAG CONTAINING THE "ZEROTH" MODEL
   BESIDE THE "FIRST" MODEL, to REPRESENT THE THIRD ORDINAL.
    Et cetera, so that the (n + 1)TH ORDINAL CONTAINS EACH OF THE n PRECEDING ORDINALS.

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The CONSTRUCTIONS ARE SO DIFFERENT AS TO PREVENT CONFUSING THE TWO DISTINCT CONCEPTS.

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But the child can see that TAKING APART AN ORDINAL MODEL CREATES A ROW OF CARDINAL MODELS!

This is why, as noted in the file, "The Paradox of Counting", ORDINATION MEASURES CARDINATION.

And, once this is clearly understood, we can ALLOW THE ABUSE OF LANGUAGE WHEREBY, IN COUNTING (AN ORDINAL PROCESS) WE CAN USE THE CARDINAL NAMES TO OBTAIN THE CARDINAL NAME OF A SET. Charlotte? The PARADOX IS NOT SLAIN UNTIL THE ABOVE DEMONSTRATION (OR ITS EQUIVALENT) IS COMPLETED.