For most cardinals κ<λ, $\kappa \leq \lambda$, it must happen that the infinite symmetric group Sκ$S_\kappa$ satisfies exactly the same first order theory as Sλ. $S_\lambda$. That is, the groups are elementarily equivalent. This is just because there are only continuum many theories in a countable language, but more cardinals than that.
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For most cardinals κ<λ, it must happen that the infinite symmetric group Sκ satisfies exactly the same first order theory as Sλ. That is, the groups are elementarily equivalent. This is just because there are only continuum many theories in a countable language, but more cardinals than that. |
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