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For an infinite cardinal number, $\gimel (\kappa )={\kappa }^{\mathrm{cf}(\kappa )}$. The singular cardinals hypothesis predicts its value when $\kappa$ is a singular cardinal and $2^{\mathrm{cf}(\kappa )}<\kappa$. For a regular cardinal, its value is $2^{\kappa }$.

For every singular infinite cardinal number $\kappa$, the Gimel function has the smallest value permitted by König theorem for cardinal numbers and the exponent: ${\kappa }^{\mathrm{cf}(\kappa )}=\max \{{\kappa }^{+},2^{\mathrm{cf}(\kappa )}\}$. Where $2^{\mathrm{cf}(\kappa )}<\kappa$, this is the singular cardinals hypothesis. The hypothesis imposes no separate successor-power condition on regular cardinals.

Put $\mu =\mathrm{cf}(\lambda )$ and choose an increasing cofinal sequence of infinite cardinals $⟨{\lambda }_i:i<\mu ⟩$ below $\lambda$. Since $\lambda$ is a strong limit cardinal, $2^{{\lambda }_i}<\lambda$ for every $i$. Encode a subset $A\subseteq \lambda$ by its restrictions $⟨A\cap {\lambda }_i:i<\mu ⟩$. Cofinality makes this encoding injective, givingThe second inequality chooses an injection of each restriction set into $\lambda$. Conversely,since $\mu \le \lambda$ and an infinite cardinal satisfies $\lambda \cdot \mu =\lambda$. The two inequalities prove the singular strong limit power-set identityThis does not assume the singular cardinals hypothesis or determine either side as ${\lambda }^{+}$. The strong-limit assumption is what bounds each restricted power set.

The singular cardinals hypothesis is the assertion that for every infinite singular cardinal $\kappa$,In particular, for a singular strong-limit cardinal it gives $2^{\kappa }={\kappa }^{+}$. This is a restriction on singular-cardinal exponentiation; it does not assert the full Generalized continuum hypothesis at all cardinals.

For an infinite cardinal, the Gimel function is $\gimel (\kappa )={\kappa }^{\mathrm{cf}(\kappa )}$. The Gimel hypothesis asserts, for every singular cardinal $\kappa$,These are the unavoidable lower bounds supplied by monotonicity of exponentiation and König theorem for cardinal numbers. The hypothesis imposes the least allowed value at singular cardinals; it does not constrain the continuum function on regular cardinals to their successors. Thus it is weaker than Generalized continuum hypothesis. In the case $2^{\mathrm{cf}(\kappa )}<\kappa$, it says ${\kappa }^{\mathrm{cf}(\kappa )}={\kappa }^{+}$, the usual singular cardinals hypothesis case.

For an infinite cardinal number $\kappa$, the gimel function isThe singular cardinals hypothesis asserts that for every infinite singular cardinal $\kappa$,Here ${\kappa }^{+}$ is the successor cardinal. Equivalently, for every infinite singular cardinal,To see why the second formulation adds nothing in the other case, put $\theta =\mathrm{cf}(\kappa )$. If $\kappa \le 2^{\theta }$, then $2^{\theta }\le {\kappa }^{\theta }\le (2^{\theta }{)}^{\theta }=2^{\theta }$ by infinite cardinal arithmetic. Equality $\kappa =2^{\theta }$ is impossible here: the König theorem for cardinal numbers gives $\mathrm{cf}(2^{\theta })>\theta$, whereas $\mathrm{cf}(\kappa )=\theta$. Thus in this case $2^{\theta }\ge {\kappa }^{+}$, as required by the maximum formula. For a strong limit cardinal that is singular, the hypothesis also yields $2^{\kappa }={\kappa }^{+}$: restrictions of a subset of $\kappa$ to a cofinal sequence of smaller cardinals give $2^{\kappa }\le {\kappa }^{\theta }$, while the reverse inequality is immediate. The quantified implication above is the precise general statement.

For a singular strong limit cardinal $\lambda$, let $\mu =\mathrm{cf}(\lambda )$ and choose cofinal cardinals ${\lambda }_i<\lambda$. Subset restrictions inject $\mathcal{P}(\lambda )$ into ${\prod }_{i<\mu }\mathcal{P}({\lambda }_i)$, whose cardinal is at most ${\lambda }^{\mu }$ by the strong-limit property. Conversely ${\lambda }^{\mu }\le (2^{\lambda }{)}^{\mu }=2^{\lambda }$. This equality alone does not imply the singular cardinals hypothesis.