Toward the end of Kunen's Models of Set Theory section in his most recent Set Theory text, after talking about relativization, he begins to mention the idea of relative consistency proofs. I've been looking at the following exercise in this section.

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> Give a purely finitistic proof that the theory $DTO$ of dense total orders without first or last elements is consistent. Here, $\mathcal{L} = \{ < \}$.
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There is a hint for the exercise: Give a purely finitistic definition of the notion $\mathbb{Q} \models \varphi[s]$, and then prove that $\mathbb{Q} \models \varphi$ whenever $\varphi$ is a sentence that is provable from $DTO$. 

Not working with consistency proofs before, nor seeing the theory $DTO$ anywhere in the previous sections of the text, I'm not entirely sure what to do. Also, what does he mean by "purely finitistic?" How would I go about giving a purely finitistic definition of $\mathbb{Q} \models \varphi[s]$? I would greatly appreciate it if anyone would be able to help me with this one.