Can someone explain to me the definition of definability in first-order logic in simple terms and with an example? I would appreciate this. I just want to really understand this.
Thank you.
Here is the definition I have.
Definability in an Interpretation
Let $ I = (D,(\cdot)^{I}) $ be a first-order interpretation and $ \phi $ be a first-order formula. A set $ B $ of $ k $-tuples over $ D $, i.e. $ B \subseteq D^{k} $, is defined by the formula $ \phi $ if $ B = \{ (\theta(x_{1}),\theta(x_{2}),\ldots,\theta(x_{k})) \,|\, (I,\theta) \models \phi \} $.
A set $ B $ is definable in first-order logic if it is defined by some first-order formula $ \phi $.