The Curie-Weiss model is an exchangeable probability measure $\mu$ on $\{0,1\}^n.$
It has two parameters -- the external magnetic field $h$ and the interaction $J$.
A natural problem is to determine when this measure extends to an exchangeable measure
on $\{0,1\}^{\infty}$. We will discuss two approaches to the following result:
$\mu$ can be (infinitely) extended if and only if $J\geq 0$. One of these
approaches relies on the classical Hausdorff moment problem. When $Jn$ can $\mu$ be extended to an exchangeable measure on $\{0,1\}^l$. Our approach
to this question involves an apparently new type of moment problem, which we will
solve. We then take $J=-c/l$, and determine the values of $c$ for which $l$-extendibility
is possible for all large $l$. This is joint work with Jeff Steif and Balint Toth.