I'm having some trouble translating these sentences into first-order logic - any idea where to begin? Thanks!
"Consider a first-order language, with relation symbols < and =, and the constant symbol 0. Consider the interpretation: the natural numbers, with 'less than' and 'equality', and the distinguished element 0. Formalise the following statements:
- There is a smallest number
- There is no largest number
- Every number has an immediate successor
- Every number is greater than some other number
- There is some number that every other number is greater than"
Here some comments and hints:
$\neg\exists x(x<0)$ means: "There is no number smaller than 0". If you want the exact translation of "There is a smallest number", it should be: $\exists x\forall y (x\leq y)$. That is: "There is a $x$ such that, for every $y$, $x$ is smaller or equal to $y$".
Your second answer is correct: $\neg\exists x\forall y (y\leq x)$.
For the third, I will only give you a hint: since you don't have a successor function, consider this: For every x, there is a y greater than x, such that there is nothing else between them. Figure out yourself how to express that.
Fourth: this is very easy, you don't need a hint. Note that this is false: 0 is not greater than any other number, since all our objects are natural numbers. But a false statement is still a statement.
Fifth: this is similar to the first.