Having trouble in understanding FOL structure

Question

Let $L$ be the language {$c,d,f,g,R$}. Given c,d are constant symbols. f,g are function symbols. R is relation symbol.

$N$={$\mathbb{N}$,0,1,+,·$\leq$}, in this structure, the formula ($\forall$ x)(c $\leq$ x $\land$ ($\neg$ c = x $\to$ d $\leq$ x)) expresses that every element is "non-negative" (i.e. at least 0) and that if an element is strictly positive, it greater or equal to 1.

I don't understand the second half, "if an element is strictly positive, it greater or equal to 1."

Answer 1

The interpretation of the language $L$ with the structure $N$ maps :

• the constant symbol $c$ on the number $0$,

• the constant symbol $d$ on the number $1$,

• the (binary) function symbols $f$ and $g$ on the operations of addition ($+$) and multiplication ($\times$) respectively,

• the (binary) relation symbol $R$ on the relation $\le$.

Thus, we have to compare the original formula :

$(∀ x)(R(c,x) ∧ (¬(c = x) → R(d,x)))$

and its interpretation in the said structure :

$(∀ x)(0 ≤ x ∧ (0 \ne x → 1 ≤ x))$.

With this interpretation, the formula means :

for all natural number $n$, $n$ is non-negative and if it is not zero then it is greater-or-equal to one,

that "sounds good" : every natural number is greater-or-equal to zero and if it is not zero (and thus it is positive) then it is greater-or-equal to one.