Given a $\sigma $-finite measure space $(X,\mu )$, a Young function $\Phi $, and a one-parameter family of Young functions $\{\Psi _q\}$, we find necessary and sufficient conditions for the associated Orlicz norms of any function $f\in L^\Phi (X,\mu )$ to satisfy

$$\begin{align*}\lim_{q\rightarrow \infty}\|f\|_{L^{\Psi_q}(X,\mu)}=C\|f\|_{L^\infty(X,\mu)}. \end{align*}$$

The constant C is independent of f and depends only on the family $\{\Psi _q\}$. Several examples of one-parameter families of Young functions satisfying our conditions are given, along with counterexamples when our conditions fail.

Kripke recently suggested viewing the intuitionistic continuum as an expansion in time of a definite classical continuum. We prove the classical consistency of a three-sorted intuitionistic formal system IC, simultaneously extending Kleene’s intuitionistic analysis I and a negative copy C° of the classically correct part of I, with an “end of time” axiom ET asserting that no choice sequence can be guaranteed not to be pointwise equal to a definite (classical or lawlike) sequence. “Not every sequence is pointwise equal to a definite sequence” is independent of IC. The proofs are by Crealizability interpretations based on classical ω-models ${\cal M}$ = $\left( {\omega ,{\cal C}} \right)$ of C°.