# Is order relation necessary for true arithmetic? [closed]

For example, I want to get $$Th(\mathbb Z_p),Th(\mathbb Q_p),Th(\mathbb C_p),Th(\mathbb C),Th(\mathbb H)$$...But they are not ordered fields.

Thinking the other way around, it would be implausible if their true arithmetic did not exist.

• @AlexKruckman Thank you. So order relation is completely irrelevant? Mar 28, 2022 at 3:24
• @AlexKruckman Thank you very much, it seems that I misunderstood some words when I read the literature, English is not my native language... Mar 28, 2022 at 3:38

## 1 Answer

For any language $$L$$ and any $$L$$-structure $$M$$, the complete theory of $$M$$, denoted $$\mathrm{Th}(M)$$, is the set of all $$L$$-sentences true in $$M$$.

In the special case when $$L=\{<,+,\times,0,1\}$$ (the language of ordered rings / language of arithmetic) and $$M=\mathbb{N}$$, we call $$\mathrm{Th}(M)$$ true arithmetic.

You're welcome to consider $$\mathrm{Th}(\mathbb{Z}_p)$$, etc. in the language of rings (without the order) but it would be confusing to call this theory "true arithmetic". Not because the order is absent, but because "true arithmetic" specifically refers to the theory of $$\mathbb{N}$$.