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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.

König theorem for cardinal numbers states that if ${\kappa }_i<{\lambda }_i$ for each $i\in I$, then ${\sum }_i{\kappa }_i<{\prod }_i{\lambda }_i$. Its cofinality formulation gives, for every infinite cardinal $\kappa$,Taking a cofinal sequence of smaller cardinals in $\kappa$ and comparing their sum with the product of their successors gives this inequality. Another standard consequence, for infinite $\lambda$ and $\kappa \ge 2$, isIndeed, if $\mu =\mathrm{cf}({\kappa }^{\lambda })\le \lambda$, the preceding inequality at ${\kappa }^{\lambda }$ would contradict $({\kappa }^{\lambda }{)}^{\mu }={\kappa }^{\lambda \mu }={\kappa }^{\lambda }$.

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.

This is König theorem for cardinal numbers. Since ${\kappa }_i<{\lambda }_i$, map $(i,\alpha )$ in the disjoint union to the element $s_{i,\alpha }$ of the product defined byHere $\alpha <{\kappa }_i$ implies $\alpha +1<{\lambda }_i$. The unique nonzero coordinate and its value recover $(i,\alpha )$, so this is an injection

For strictness, suppose $F$ mapped the disjoint union onto the product. For each $i$, the sethas cardinality at most ${\kappa }_i$, so it cannot exhaust ${\lambda }_i$. Choose $b_i\in {\lambda }_i\setminus A_i$. Then $b=(b_i{)}_{i\in I}$ differs from $F(i,\alpha )$ in coordinate $i$ for every $(i,\alpha )$, contradicting surjectivity. By Cantor-Schröder-Bernstein theorem, an injection in the reverse direction would combine with the displayed injection to give a bijection and hence a surjection. Therefore

Apply König theorem for cardinal numbers with $I=\omega$, ${\kappa }_i={\aleph }_i$, and ${\lambda }_i={\aleph }_{i+1}$. Standard infinite cardinal arithmetic giveswhereas every factor in the product is at most ${\aleph }_{\omega }$. Hence

Now suppose $2^{{\aleph }_{\omega }}={\aleph }_{\omega }$. The exponent laws for infinite cardinals would givecontradicting the strict inequality. Therefore