Find the number less than 2, in FOL

Question

How to express that "the number less than 2 is odd" in first order logic, using only these symbols: 2, *, <, =, $\lnot$, $\land$, $\to$, $\exists$ and $\forall$ ?

Answer 1

By saying "the number less than 2" they are implicitly saying that there is exactly one number less than 2 (it's called a definite description). So, I would simply add this to the symbolization:

$$\exists x (x < 2 \land \forall y \ (y < 2 \rightarrow y = x) \land \neg \exists z\ 2 \cdot z=x )$$

Answer 2

"All numbers less than 2 are odd" (a false statement if the universe is $\Bbb Z$) would be $$\forall x\colon (x<2\to\neg\exists k\colon x=2\cdot k) $$ "All numbers immediately preceding 2 in the order are odd" would be $$\forall x\colon ((x<2\land \neg \exists z\colon (x<z\land z<2))\to\neg \exists k\colon x=2\cdot k )$$