In Whitehead&Russell's PM's ✳210, how can the product of $\lambda$ be not a member of $\lambda$?

Question

Take ✳210.23 for example:

Assuming $\kappa$ is a classes of classes such that, of any two, one is contained in the other, i.e. $\alpha, \beta \in \kappa .\supset_{\alpha, \beta} : \alpha \subset \beta . \vee . \beta \subset \alpha $

✳210.23 $\vdash:Hp✳210.2. \lambda \subset \kappa. p‘\lambda\in \kappa-\lambda.\supset. p‘\lambda=prec_Q‘\lambda=tl_Q‘\lambda$

I wonder, in the hypothesis, if the product of $\lambda$ can ever NOT be a member of $\lambda$. In other words, can this ever happen? $\lambda \subset \kappa.p‘\lambda\in \kappa-\lambda$.

If you can come up with a concrete example, please help me out.

Thanks!

Answer 1

See page 592, vol.II :

if $\lambda$ is any sub-class of $\kappa$, the logical product $p‘\lambda$ is the minimum of $\lambda$ if it is a member of $\lambda$, and the lower limit of $\lambda$ if it is a member of $\kappa$ but not of $\lambda$.

SECTION B deals with sections (sic !) and the Dedekindian treatment of the continuum.

I think that the discussion is related to the Completeness property of the real line.

Consider as $\kappa$ the set $\mathbb R$ of real numbers, with the usual ordering, and consider as $\lambda$ the set $\{ r \in \mathbb Q^+ | 2 \le r^2 \}$.

I.e. $\lambda$ is the set of positive rational numbers whose square is greater-or-equal than two; clearly, the lower limit of $\lambda$ is $\sqrt 2$, which is not rational.

Thus, it is a member of $\kappa$ but not of $\lambda$, and so it is not the minimum of $\lambda$.

If instead we consider $\lambda' = \{ r \in \mathbb R^+ | 2 \le r^2 \}$ [i.e. the set of positive real numbers whose square is greater-or-equal than two], we have that $\sqrt 2$ is the minimum of $\lambda'$, because - being a real - it is a member of $\lambda'$.