Realizing the totally unordered structure of ordinals

We present tools for analysing ordinals in realizability models of classical set theory built using Krivine's technique for realizability. This method uses a conservative extension of $ZF$ known as $ZF_{\varepsilon}$, where two membership relations co-exist, the usual one denoted $\in$ and a stricter one denoted $\varepsilon$ that does not satisfy the axiom of extensionality; accordingly we have two equality relations, the extensional one $\simeq$ and the strict identity $=$ referring to sets that satisfy the same formulas. We define recursive names using an operator that we call reish, and we show that the class of recursive names for ordinals coincides extensionally with the class of ordinals of realizability models. We show that reish$(\omega)$ is extensionally equal to omega in any realizability model, thus recursive names provide a useful tool for computing $\omega$ in realizability models. We show that on the contrary $\varepsilon$-totally ordered sets do not form a proper class and therefore cannot be used to fully represent the ordinals in realizability models. Finally we present some tools for preserving cardinals in realizability models, including an analogue for realizability algebras of the forcing property known as the $\kappa$-chain condition.