Models of ZFC as Ordered Fields

Question

Question: Let $M$ be a set such that $\langle M,\in\rangle\vDash ZFC$. Consider the partial order $\langle M,\subseteq\rangle$. Using order extension principle there is a linear order $\subseteq^{*}$ such that $\subseteq^*$ extends $\subseteq$. Are there definable operators $+:M\times M:\longrightarrow M$, $\cdot:M\times M\longrightarrow M$, $-:M\longrightarrow M$, $^{-1}:M\longrightarrow M$ and elements $\overline{0}, \overline{1}\in M$ such that the structure $\langle M,\subseteq^*, +,\cdot, -,^{-1}, \overline{0}, \overline{1}\rangle$ is an ordered field? In the other words, is there a natural field structure on a model of ZFC which is compatible with its $\in$ - structure ($\subseteq$ - order)?

Answer 1

I don't think so. The critical thing is that the order has a least element, $\emptyset$. Let us assume that $\emptyset \lt \overline 0$ We know $\overline 1 \gt \overline 0$ in an ordered field, so what is $-(-(\emptyset)+\overline 1)$? If $\emptyset \ge \overline 0$ what is $\emptyset + \overline 1$?