For context:
Consider the predicate logic language with the natural numbers {0, 1, 2, 3, . . .} as constants; the predicates < (x < y means x is smaller than y); = (x = y means x is equal to y)....
Apologies for this very basic question, but I'm slightly confused. For this formula:
$∀x(x≠0 → 0<x)$
≠ doesn't exist in our language which only includes {¬,→,∧,∨,∀,∃}.
So should it be expressed like: $∀x(x=¬0 → 0<x)$ or as $∀x(x¬=0 → 0<x)$
Would appreciate some guidance.
Usually the $\neq$ predicate is considered to be an abbreviation: "$t\neq u$" is simply a space-saving (and, arguably, more readable) way to write $\neg(t=u)$.
Neither "$t=\neg u$" nor "$t \mathrel{{\neg}{=}} u$" are well-formed, though: The $\neg$ symbol must always apply to a complete formula.
In most formal developments, $\neg(t=u)$ is again an abbreviation of something like $\neg(p_0^2(t,u))$, where $p_0^2$ is the predicate letter one has chosen to use for equality -- but that's a different story.