[Math] How would you define the set of even numbers in $\mathbb{Z}$ using a first order sentence


Given a language $\mathcal{L} = \{+,0\}$ and structure $\mathcal{M}$ with underlying universe $\mathbb{Z}$, I would like to write a formula or sentence $\phi$ so that the set

$$\Big\{ \bar{a} ~:~ \mathcal{M} \models \phi(\bar{a}) \Big\}$$

correspond to the set of even numbers. My current attempt is

$$\phi = \exists x \exists z ~ (z + z = x)$$

But it does not look correct to me. Could someone confirm improve my attempt?

Best Answer

Remember, you are trying to find a formula, not a sentence. Every time you put an even number (as a parameter), you now have a sentence which holds (with said parameter). It fails every time you put in an odd number.

So you are close! but the actual formula would be $$ \varphi(x) \equiv (\exists z)( z+ z =x)$$

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