Showing that $\lambda_n \vdash \sigma \Leftrightarrow \sigma$ holds in all models with at least $n$ elements.

Question

As the title says, I’m trying to show that $\lambda_n \vdash \sigma \Leftrightarrow \sigma$ holds in all models with at least $n$ elements. It’s from Logic and Structure, van Dalen (2013 edition).

We’re allowed to use both soundness and completeness.

This is the definition of $\lambda_n$ (taking $\bigwedge$ to mean $\bigwedge_{i\neq j, 1 \leq i \leq n, 1 \leq i \leq n}$):

\begin{equation} \lambda_n = \exists x_1 … \exists x_n \bigwedge x_i \neq x_j \end{equation}

Lemma 3.4.5 in van Dalen states (among other things) that:

(i) $\mathfrak{A} \vDash \varphi \wedge \psi \Leftrightarrow \mathfrak{A} \vDash \varphi$ and $\mathfrak{A} \vDash \psi$.

(vii) $\mathfrak{A} \vDash \exists x\varphi \Leftrightarrow \mathfrak{A} \vDash \varphi[\overline{a}/x]$, for some $a \in |\mathfrak{A}|$.

So far, I have tried the following.

($\Rightarrow$): Suppose that $\lambda_n \vdash \sigma$. By soundness, $\lambda_n \vDash \sigma$. Let $\mathfrak{A}$ be an arbitrary model such that $|\mathfrak{A}| \geq n$. $\mathfrak{A} \vDash \lambda_n$ iff for some $a_1, …, a_n \in |\mathfrak{A}|$, we have $\mathfrak{A} \vDash \bigwedge x_i \neq x_j[\overline{a}_1 / x_1][\overline{a}_2 / x_2]…[\overline{a}_n / x_n]$. By case (i) and (vii) of Lemma 3.4.5 (van Dalen), it follows that $\mathfrak{A} \vDash \exists x_1 … \exists x_n \bigwedge \overline{x}_i \neq \overline{x}_j$. But $\lambda_n = \exists x_1 … \exists x_n \bigwedge \overline{x}_i \neq \overline{x}_j$. So $\mathfrak{A} \vDash \lambda_n$. But then $\mathfrak{A} \vDash \sigma$. Since $\mathfrak{A}$ was arbitrary, $\sigma$ holds in all models with at least $n$ elements.

I’m sure I did something wrong here, but I just can’t spot it.

($\Leftarrow$): Suppose that $\mathfrak{A} \vDash \sigma$ for all $\mathfrak{A}$ such that $|\mathfrak{A}| \geq n$. ???

Here I am simply stuck. I don’t understand van Dalen’s explanations very well. All help is greatly appreciated, so long as the complete solution isn’t revealed. Thanks in advance!

Answer 1

The Completeness theorem states that for any theory $T$ (so any set of statements), $T\vdash \sigma$ iff $\sigma$ holds in all models of $T$.

Apply that with $T = \{ \lambda_n \}$, showing that a model of $T$ is just a model with at least $n$ elements.