## 580 -Cardinal arithmetic (4)

February 11, 2009

2. Silver’s theorem.

From the results of the previous lectures, we know that any power $\kappa^\lambda$ can be computed from the cofinality and gimel functions (see the Remark at the end of lecture II.2). What we can say about the numbers $\gimel(\lambda)$ varies greatly depending on whether $\lambda$ is regular or not. If $\lambda$ is regular, then $\gimel(\lambda)=2^\lambda.$ As mentioned on lecture II.2, forcing provides us with a great deal of freedom to manipulate the exponential function $\kappa\mapsto 2^\kappa,$ at  least for $\kappa$ regular. In fact, the following holds:

Theorem 1. (Easton). If ${\sf GCH}$ holds, then for any definable function $F$ from the class of infinite cardinals to itself such that:

1. $F(\kappa)\le F(\lambda)$ whenever $\kappa\le\lambda,$ and
2. $\kappa<{\rm cf}(F(\kappa))$ for all $\kappa,$

there is a class forcing ${\mathbb P}$ that preserves cofinalities and such that in the extension by ${\mathbb P}$ it holds that $2^\kappa=F^V(\kappa)$ for all regular cardinals $\kappa;$ here, $F^V$ is the function $F$ as computed prior to the forcing extension. $\Box$

For example, it is consistent that $2^\kappa=\kappa^{++}$ for all regular cardinals $\kappa$ (as mentioned last lecture, the same result is consistent for all cardinals, as shown by Foreman and Woodin, although their argument is significantly more elaborate that Easton’s). There is almost no limit to the combinations that the theorem allows: We could have $2^\kappa=\kappa^{+16}$ whenever $\kappa=\aleph_\tau$ is regular and $\tau$ is an even ordinal, and $2^\kappa=\kappa^{+17}$ whenever $\kappa=aleph_\tau$ for some odd ordinal $\tau.$ Or, if there is a proper class of weakly inaccessible cardinals (regular cardinals $\kappa$ such that $\kappa=\aleph_\kappa$) then we could have $2^\kappa=$ the third weakly inaccessible strictly larger than $\kappa,$ for all regular cardinals $\kappa,$ etc.

Morally, Easton’s theorem says that there is nothing else to say about the gimel function on regular cardinals, and all that is left to be explored is the behavior of $\gimel(\lambda)$ for singular $\lambda.$ In this section we begin this exploration. However, it is perhaps sobering to point out that there are several weaknesses in Easton’s result.

## 305 -4. Fields.

February 11, 2009

Definition 1. Let ${\mathbb F}$ be a set. We say that the quintuple $({\mathbb F},+,\times,0,1)$ is a field iff the following conditions hold:

1. $+:{\mathbb F}\times {\mathbb F}\to {\mathbb F}.$
2. $\times:{\mathbb F}\times {\mathbb F}\to {\mathbb F.}$ (We say that ${\mathbb F}$ is closed under addition and multiplication.)
3. $0,1\in{\mathbb F}.$
4. $0\ne1.$
5. Properties 1–9 of the Theorem from last lecture hold with elements of ${\mathbb F}$ in the place of complex numbers, ${}0$ in the place of $\hat0,$ and ${}1$ in the place of $\hat1.$