7/27/2019 On Foundationalokay

1/3

On the Foundational Notions of Set Theory

9/17/2013 Philip Alexander White

Df. 1) A setS is a collection of objects.

Rmrk. 1) There can be a bag that contains objects, but it is something completely different to

specify what those objects are. Similarly we can have a set A and not specifyAs contents.

Df. 2) The cardinalityof a set A, written |A|, is the total number of distinct members in A.

Rmrk. 2) It is impossible to have |A| without knowing the number of objects in A. We cannot

have |A| and be unfamiliar withAs contents.

Rmrk. 3) A set S is epistemically distinct from |S|. |S| implies something about the members in S,

whereas S does not. This follows from Rmrk. 1 & 2.

Df. 3) The unionof two sets A and B, written AUB, is {all x, all y : x is in A, y is in B}

Df. 4)Cardinal addition, written +c, is summarized as follows: If |A|=a and |B|=b, then

|A|+c|B|= a+b.

Rmrk. 4) Notice that a+b in Df. 4 is a statement of traditional arithmetic concerning traditional

numbers, wheras |A|+c|B| is a statement about collections of objects and cardinalities.

Df. 5) For two sets A and B, U and +c give similar resultsif AUB=C and |A|+c|B|=|C|.

Rmrk 5) |A|+c|B| and AUB often give similar results. If A={a,b} and B={d,e}, then

|A|+c|B|= 2+2=4 and {a,b}U{d,e}={a,b,d,e}.

Lm. 1) Cardinal addition and union do not always give similar results. If A={a,b} and B={a,d}

then |A|+c|B|= 2+2=4 while {a,b}U{a,d}={a,b,d}.

7/27/2019 On Foundationalokay

2/3

Df. 6) A n-ary function F(A1,,An)=Y on n sets goes beyond what is explicitly given, if x is in Y but x is

not in any A.

Rmrk. 6) Obviously Df. 6 has a heavy naturalistic bent. It was designed to reflect the having of

actual distinct objects, and to mimic the physical actions one could do with those objects.

Rmrk. 7)To elaborate on Rmrk. 6, consider rocks {a,b} and rocks {c,d,e}. To union these

collections, move rocks {a,b} to be with rocks {c,d,e}. The result is the actual collection

{a,b,c,d,e}. I have not gone beyond what was explicitly given, and I have not used the

concept of number. (Rmrk. 1 & Df. 3)

Thrm. 1) Cardinal addition goes beyond what is explicitly given.Proof ) Recall the rocks from Rmrk. 7. To cardinal addition these two collections, the number of

rocks in {a,b} and {c,d,e} must first be known. So |{a,b}|=2 and |{c,d,e}|=3. This is no

longer the realm of rocks, this is the realm of numbers. However, when

presented with {a,b} and {c,d,e}, numbers were not implied. Therefore cardinal addition

steps beyond what is explicitly given. (Rmrk. 2, Lm. 1, & Df. 4)

Df. 7)The cardinal product, written Xc, can be summarized as follows. If |A|=a and |B|=b then

|A| Xc |B| = a * b.

Lm. 3) Cardinal products go beyond what is explicitly given. Use the sets from Rmrk. 7 and take

|{a,b}| Xc|{c,d,e}|= 2 * 3= 6 Thus, we go from the realm of rocks to the realm of

numbers. This is similar to Thrm. 1.

Df. 8) The cross product of two sets A and B, written A X B, is {(x,y) : x is in A, y is in B}.

Thrm. 3)The cross product goes beyond what is explicitly given.

Proof ) Once again, recall the rocks from Rmrk. 7, written as sets{a,b} and {c,d,e}. Take the cross

product like {a,b} X {c,d,e} = {(a,c),(a,d),(a,e),(b,c),(b,d),(b,e)}. However (a,d) is not a

member of either {a,b} or {c,d,e}, and by the Df. 6, {a,b} X {c,d,e} goes beyond what is

explicitly given.

7/27/2019 On Foundationalokay

3/3

Thrm. 4) For any two sets A and B, their union does not exceed what is explicitly given.

Proof ) The union of two sets A and B is {all x, all y : x is in A, y is in B}. Notice every object in

{all x, all y : x is in A, y is in B} is either in A or B, so AUB does not exceeded what is

explicitly given.

Scholium )

Is it not odd that cardinal addition, cardinal multiplication, and cross products all exceed what is

given? Is it not odd that union, out of all things, does not exceed what is given? As probably

noticed, I have had a strict naturalistic sense in mind, and each proof has a corresponding illustration

involving actual rocks. The rocks, I hold, have a naturalistic sense. Because cardinality, cardinal

addition, and cardinal multiplication, all extend the discourse to beyond our rocks, I am inclined tolabel these operations as unnatural. Union, on the other hand, seems to me the only natural

operation that can be performed on sets.