Past exam of the mathematics course of the University of Cambridge / 2014 / iii / Paper 19 / 3 / i / a

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.