The uncountable spectral of countable theories


Let $T$ be a complete, first-order theory in a finite or countable language having infinite models. Let $I(T,\kappa)$ be the number of isomorphism types of models of $T$ of cardinality $\kappa$. We denote by $\mu$ (respectively $\hat\mu$) the number of cardinals (respectively infinite cardinals) less than or equal to $\kappa$.

THEOREM. $I(T,\kappa)$, as a function of $\kappa > \aleph_0$, is the minimum of $2^{\kappa}$ and one of the following functions:

1. $2^\kappa$;

2. the constant function $1$;

3. $\begin{cases}
|\hat\mu^n/{\sim_G}|-|(\hat\mu – 1)^n/{\sim_G}| & \hat\mu\lt \omega;\qquad \mbox{for some } 1\lt n\lt \omega\quad \mbox{and}\\
\quad\quad\qquad \; \hat\mu &\hat\mu\ge\omega;\qquad \mbox{some group } G\le \mathrm{Sym}(n)

4. the constant function $\beth_2$;

5. $\beth_{d+1}(\mu)$ for some infinite, countable ordinal $d$;

6. $\sum_{i=1}^d \Gamma(i)$ where $d$ is an integer greater than $0$ (the depth of $T$) and
\Gamma(i)\; \textit{ is either } \beth_{d-i-1}(\mu^{\hat\mu})\; \textit{ or }
\beth_{d-i}(\mu^{\sigma(i)} + \alpha(i)),$$
where $\sigma(i)$ is either $1$, $\aleph_0$ or $\beth_1$, and $\alpha(i)$ is $0$ or $\beth_2$; the first possibility for $\Gamma(i)$ can occur only when $d-i>0$.

The cases (2), (3) of functions taking a finite value were dealt with by Morley and Lachlan. Shelah showed (1) holds unless a certain structure theory (superstability and extensions) is valid. He also characterized (4) and (5) and showed that in all other cases, for large values of $\kappa$, the spectrum is given by $\beth_{d-1}(\mu^{<\sigma})$ for a certain $\sigma$, thespecial number of dimensions.”

The present paper shows, using descriptive set theoretic methods, that the continuum hypothesis holds for the special number of dimensions. Shelah’s superstability technology is then used to complete the classification of the all possible uncountable spectra.



Bradd Hart

Ehud Hrushovski

Michael C. Laskowski