User:Farkle Griffen/Cardinal Arithmetic

Basic arithmetic can be done on cardinal numbers in a very natural way, by extending the theorems for finite combinatorial principles above. The intuitive principle that is ${\displaystyle A}$ and ${\displaystyle B}$ are disjoint then addition of these sets is simply taking their union, written as ${\displaystyle |A\cup B|=|A|+|B|}$. Thus if ${\displaystyle A}$ and ${\displaystyle B}$ are infinite, cardinal addition is defined as ${\displaystyle |A|+|B|:=|A\sqcup B|}$ where ${\displaystyle \sqcup }$ denotes disjoint union. Similarly, by the multiplication of two sets is intuitively the ways to pair their elements (as in the multiplication principle), therefore cardinal as ${\displaystyle |A|\cdot |B|:=|A\times B|.}$ These definitions can be shown to satisfy the basic properties of standard arithmetic:

• Associativity: ${\displaystyle A\sqcup (B\sqcup C)\sim (A\sqcup B)\sqcup C}$, and ${\displaystyle A\times (B\times C)\sim (A\times B)\times C}$
• Commutativity: ${\displaystyle A\sqcup B\sim B\sqcup A}$, and ${\displaystyle A\times B\sim B\times A}$
• Distributivity: ${\displaystyle A\times (B\sqcup C)\sim (A\times B)\sqcup (A\times C)}$

Cardinal exponentiation ${\displaystyle |A|^{|B|}}$ is defined via set exponentiation, the set of all functionctions ${\displaystyle f:B\mapsto A}$, that is, ${\displaystyle |A|^{|B|}:=|A^{B}|.}$ For finite sets this can be shown to coincide with standard natural number exponentiation, but includes as a corollary that zero to the power of zero is one ${\displaystyle (0^{0}=1)}$ since there is exactly one function from the empty set to itself: the empty function. A combinatroial argument can be used to show ${\displaystyle 2^{|A|}=|{\mathcal {P}}(A)|}$. In general, cardinal exponentiation is not as well behaved as cardinal addition and multiplication. For example, even though it can be proven that the expression ${\displaystyle 2^{\aleph _{0}}}$does indeed correspond to some aleph, it is unprovable from standard set theories which aleph it corresponds to. The following properties of exponentiation can be shown by currying:

• ${\displaystyle (A^{B})^{C}\sim A^{B\times C}}$

• ${\displaystyle A^{B\,\sqcup \,C}\sim A^{B}\times A^{C}}$
• ${\displaystyle (A\times B)^{C}\sim A^{C}\times B^{C}}$