Tuesday, November 3, 2015

An infinity between countable and uncountable

In my discrete math class we are talking about functions and cardinalities.

On of the outstanding results of formalizing set theory is the fact that there are different types of infinities,  that is, there is something bigger than infinity, or at least bigger than the usual infinity that we think about. 

One way of dealing with these infinities is by using cardinal numbers.  

These have arithmetic properties, that is, we can add, multiply, and even exponentiate. However, the way of understanding  these operations resort in set theoretical definitions. For example, adding two cardinals amounts to make the union of two disjoint sets, to multiply is to do the cartesian product, and exponentiation is related to finding all the functions from one set into the another. 

Since these operations are well behaved, that is, they satisfy the usual commutativity, associativity, and distributivity properties, we could think of mimicking the construction of rational numbers using cardinals.

Let $\mathcal{N}$ be the set of cardinals smaller than a given cardinal $\mathfrak{w}\geq 1$ and consider the equivalence relation $\sim$ on $\mathcal{N}\times\mathcal{N}$ given by:

$$(\mathfrak{a},\mathfrak{b})\sim(\mathfrak{c},\mathfrak{d}) \text{ iff } \mathfrak{a}\cdot\mathfrak{d}=\mathfrak{b}\cdot\mathfrak{c}\,.$$

We need to only consider the cardinals smaller than a given one since we cannot construct the set of all cardinals. Thus, with this, let $\mathcal{Q}=\mathcal{N}\times\mathcal{N}/\sim$ and define $+,\cdot$ as



These are well defined, and $[(1,1)]$ and $[(0,1)]$ serve as the multiplicative and additive identities respectively. Thus $\mathcal{Q}$ is a (commutative) ring. The order provided on the cardinals gets lifted to the quotient:

$$[(\mathfrak{a},\mathfrak{b})]\leq[(\mathfrak{c},\mathfrak{d})]\text{ iff } \mathfrak{a}\cdot\mathfrak{d}\leq\mathfrak{c}\cdot\mathfrak{b}\,.$$

Thus, we can also abstract the construction of the real numbers using Dedekind cuts using these cardinal rationals. 

Consider Dedekind type cuts of $\mathcal{Q}$, that is, a partition $\mathcal{A}\cup\mathcal{B}=\mathcal{Q}$ such that 

$$\forall\mathfrak{a}\in A\text{ and } \forall\mathfrak{b}\in\mathcal{B},\mathfrak{a}<\mathfrak{b}\,.$$

By considering all possible Dedekin type cuts, we can form a new set that behaves like the real numbers, in the sense that it becomes totally ordered and complete.

This suggests that we could do some sort of infinite calculus, with limits and derivatives, which would be nice.

PS: The relation $\sim$ defined above is not an equivalence relation (it fails to be transitive) and hence making the construction not to be a partition of $\mathcal{N}\times\mathcal{N}$, so a good way to make the construction work is by defining a better $\sim$ that makes this posible. If you can find it, maybe we can truly have infinities that behave like the real numbers. 

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