Problem saying:
Let $T=\{\forall x(x\neq S^{n}x)|1\leq n\}\cup\{\sigma\}$ in $\mathcal{L}=\{S\}$ , where $S$ is a unary function and $S^{n}$ abbreviates $\underbrace{S\dots S}_{n}$ , and $\sigma$ is a sentence saying that $S$ is bijective.Show that $T$ is complete.
So, I tried to do so by using Los-Vaught Test. So, I want to show that
1) $T $ is consistent(satisfiable) and
2) If is $\kappa$-categorical for some infinite cardinal larger than $Card(\mathcal L)$.
So, I show 1) with model $M=(\mathbb N,S)$ where S is a successor function. Then $M\models T$.
Thus, I have to show that it is $\kappa$-categorical for some cardinal $\kappa$.
But How can I?
edit) In my model, S is not bijective. So, I also fail to show that T is consistent.
To repair your consistency proof, replace $\mathbb N$ with $\mathbb Z$. To prove categoricity in all uncountable powers, show that any model of $T$ is a disjoint union of some isomorphic copies of the $\mathbb Z$ example. If the model has uncountable cardinality $\kappa$, then there have to be exactly $\kappa$ copies, so the model is unique up to isomorphism. ($T$ is not $\aleph_0$-categorical, because the number of copies of $\mathbb Z$ in a countably infinite model of $T$ can be any positive integer or $\aleph_0$.) By the way, to finish the proof of completeness, you also have to check that $T$ has no finite models, but that's pretty easy.