+ \omega = \omega \not= \omega+1$. Thus, the infinite realm of Cantor’s ordinals differs significantly from the finite: in the finite realm, commutativity rules, whereas in the infinite, it does not not. Thus the transfer of properties between these two realms fails.

Similarly, there are many infinitary frameworks where the law of continuity fails to hold. For example, consider the Conway–Alling surreal framework; see e.g., [@Al85]. Here one can’t extend even such an elementary function as $\sin(x)$ from ${{\mathbb R}}$ to the surreals. Even more strikingly, $\sqrt{2}$ turns out to be (sur)rational; see [@Co01 chapter 4]. The surnaturals don’t satisfy the Peano Arithmetic. Therefore transfer from the finite to the infinite domain fails also for the domain of the surreals.

On the other hand, the combined insight of [@He48], [@Lo55], and [@Ro61] was that there does exist an infinitary framework where the law of continuity can be interpreted in a meaningful fashion. This is the ${{\mathbb R}}\subset{}^\ast{{\mathbb R}}$ framework. While it is not much of a novelty that many infinitary systems don’t obey a law of continuity/transfer, the novelty is that there is one that does, as shown by Hewitt, [Ł]{}o[ś]{}, and Robinson, in the context of first-order logic.

Throughout 