Past exam of the mathematics course of the University of Cambridge / 2013 / iii / Paper 19 / 2 / i

The axiom of constructibility asserts every set belongs to the constructible universe, or $V=L$. Define $L_0=\varnothing$, $L_{\alpha +1}=\mathrm{Def}(L_{\alpha })$, and $L_{\eta }={\bigcup }_{\alpha <\eta }{L}_{\alpha }$ at limit stages. Here $\mathrm{Def}(X)$ consists of subsets of $X$ first-order definable over $(X,\in )$ with finitely many parameters from $X$. Then $L={\bigcup }_{\alpha \in \mathrm{Ord}}{L}_{\alpha }$. The assertion is $\forall x\exists \alpha (x\in L_{\alpha })$, rather than a statement that every set is parameter-free definable.

The Bukovský-Hechler theorem states: if $\kappa$ is singular and there are a cardinal $\rho <\kappa$ and a cardinal $\tau$ such that $2^{\mu }=\tau$ for every cardinal $\mu$ with $\rho \le \mu <\kappa$, thenThus an eventual plateau of the power-set function below a singular cardinal continues at that cardinal. One can see the mechanism by decomposing $\kappa$ into $\mathrm{cf}(\kappa )$ bounded pieces: $2^{\kappa }=(2^{<\kappa }{)}^{\mathrm{cf}(\kappa )}$. On the plateau choose $\mu \ge \mathrm{cf}(\kappa )$, so ${\tau }^{\mathrm{cf}(\kappa )}=(2^{\mu }{)}^{\mathrm{cf}(\kappa )}=2^{\mu }=\tau$.

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.