Gracious Living

Cardinal Arithmetic
December 8, 2010, 03:23
Filed under: Math, Set Theory | Tags: , , , ,

Let’s get tricky!  Unlike ordinal arithmetic, the arithmetic of cardinals isn’t really an extension of Peano arithmetic.  Rather, it’s a consequence of the set model of ordinals, with the sum of two ordinals being the order type of their disjoint union, and their product being the order type of their Cartesian product.

Part of the reason for this is that cardinals have a vastly different notion of “successor.”  See, the ordinals are well-ordered, so any subset has a least element, and thus we can identify each cardinal with the least ordinal of its cardinality.  So we’ll say $\aleph_0=\omega,\aleph_1=\omega_1,\dotsc$.  (Exercise: these are all limit ordinals.)  Then the cardinals, as a subset of the ordinals, are also well-ordered, and thus we can define the successor of a cardinal as the least cardinal greater than it, and the limit of a set of cardinals as their least upper bound.

Since the cardinals, as aleph numbers, are also indexed by the ordinals, we have $\aleph_{S\alpha}=S\aleph_\alpha,\aleph_{\lim\alpha}=\lim\aleph_\alpha$.  We can treat the map $\alpha\mapsto\aleph_\alpha$ as a function $\aleph:\mathbf{On}\rightarrow\mathbf{On}$; then this function is normal.  So in particular it has a fixed point (indeed, an unbounded class of them): this is an ordinal $\alpha$ such that $\omega_\alpha=\alpha$.  Even though the cardinals increase a lot “faster” than the ordinals, this number is simultaneously the $\alpha$th cardinal and the $\alpha$th ordinal!  The first such ordinal is $\omega_{\omega_{\omega_{\dotsc}}}$.

This fills me with the greatest glee.

We won’t be assuming the generalized continuum hypothesis for the first part of this, so we won’t mention the beth numbers (which are $\beth_0=\aleph_0,\beth_{S\alpha}=2^{\beth_\alpha},\beth_{\lim\alpha}=\lim\beth_\alpha$).  If you do assume it, the beth numbers are the aleph numbers; if you don’t, it’s difficult to say anything about them arithmetic-wise.

Like Devlin, I’ll use $\kappa,\lambda,\mu$ for cardinals, $\alpha,\beta,\gamma$ for ordinals.