What are the necessary components to finding structures for a suitable language?

Question

The Question (and its Answer)

We are given the following:

$$\sigma: (\forall x)(\exists y)[x < y \rightarrow x + 1 \neq y]$$

Now, in order to find the structure of a language that makes $\sigma$ true, we say the following:

We will work in $\mathcal{L}_{NT}$, and let: $$ \mathfrak{} = (ℕ , 0 , , + , , < )$$

be the standard structure.

We will let $s$ be a variable assignment that will assign $v_i$ to number $i$. We can see that this makes $\sigma$ true.

The Problem:

It seems a bit too simplistic to be true that the aforesaid answer is sufficient structure of a language specifications to make $\sigma$ true. It feels like there must me other moving parts involved as well, and I'm not sure what those would be. Any ideas?

Answer 1

If we want to have a "fully formal" proof that $\mathfrak N \vDash \sigma$, we have to apply the clauses of the satisfaction definition [see Leary, Def.1.7.4, page 29]:

$\mathfrak N \vDash \sigma \text { iff for every } n \in \mathbb N : \mathfrak N \vDash (∃v_2)[v_1 < v_2 → v_1+1 ≠ v_2][s(v_1 \mid n)]$,

where I've adopted the convention that $x$ is $v_1$ and $y$ is $v_2$.

This in turn implies:

$$\mathfrak N \vDash (∃v_2)[v_1 < v_2 → v_1+1 ≠ v_2][s(v_1 \mid n)] \text { iff for some } m \in \mathbb N : \mathfrak N \vDash (v_1 < v_2 → v_1+1 ≠ v_2)[s'(v_2 \mid m)]$$

where $s'(v_2 \mid m)$ is a "variant" of $s(v_1 \mid n)$.

What does it mean? Having chosen $n \in \mathbb N$ whatever, can we always find a suitable $m$ such that:

$(n < m → n+1 ≠ m)$

holds in $\mathbb N$?

The answer is: Yes. For $s(v_1)=n$ it is enough to choose $s'(v_2)=n+2$.

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The variable assignment function [see Leary, Def.1.7.1, page 28] is a function that maps variables of the language to elements of the domain. Spercigfically, for the above question we have:

$s: \text {Var} \to \mathbb N$.

The symbols $s(v_1 \mid n)$ [see Def.1.7.2] is simply: $s(v_1)=n$.