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

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.

Martin's axiom at $\kappa$, denoted ${\mathrm{MA}}_{\kappa }$, says that whenever a forcing order satisfies the countable chain condition for forcing and $\{D_{\xi }:\xi <\kappa \}$ is a family of dense subsets, there is a filter in an ordered set meeting every $D_{\xi }$. Equivalently, allow any family of at most $\kappa$ dense subsets. The filter in an ordered set is directed towards stronger common extensions and closed towards weaker conditions. Full Martin's axiom requires ${\mathrm{MA}}_{\kappa }$ for every $\kappa <2^{{\aleph }_0}$. The bound below the continuum is part of the usual full axiom, explaining its role in part (iv)(b).

A well-pruned set-theoretic tree of height $\kappa$ has the property that every node $t$ extends to every higher level below $\kappa$: if $\mathrm{ht}(t)<\beta <\kappa$, there is $u$ of height $\beta$ with $t{<}_{T}u$. Equivalently, the heights of extensions of every node are unbounded in $\kappa$, since taking predecessors then gives an extension at any prescribed intermediate level. This is stronger than merely having no terminal nodes. For a kappa-tree we use the usual regular uncountable height cardinal and levels of size less than that cardinal.

The Beth numbers form a strictly increasing continuous sequence: strictness comes from Cantor theorem, and at a limit $\delta$ one has ${\beth }_{\delta }={\sup }_{\xi <\delta }{\beth }_{\xi }$. A cofinal sequence in $\delta$ of length $\mathrm{cf}(\delta )$ therefore induces a cofinal sequence in ${\beth }_{\delta }$, giving $\mathrm{cf}({\beth }_{\delta })\le \mathrm{cf}(\delta )$.

Conversely, let $\{a_i:i<\mathrm{cf}({\beth }_{\delta })\}$ be cofinal in ${\beth }_{\delta }$. For each $i$ choose ${\xi }_i<\delta$ with $a_i<{\beth }_{{\xi }_i}$. The indices must be cofinal in $\delta$; otherwise all $a_i$ would be bounded by a single smaller Beth number. Thus $\mathrm{cf}(\delta )\le \mathrm{cf}({\beth }_{\delta })$, andThis is the cofinality of a continuous cardinal hierarchy at a nonzero limit index; the limit hypothesis is essential to the supremum argument.

We give an explicit Gimel recursion for cardinal exponentiation. First recover the continuum function by induction on infinite cardinals. At a regular $\kappa$,because $2^{\kappa }\le {\kappa }^{\kappa }\le (2^{\kappa }{)}^{\kappa }=2^{\kappa }$. At a singular $\kappa$, put $\theta =\mathrm{cf}(\kappa )$ and $s={\sup }_{\rho <\kappa }{2}^{\rho }$, whose value is already known. A cofinal sequence $({\kappa }_i{)}_{i<\theta }$ givesIndeed ${\prod }_i{2}^{{\kappa }_i}=2^{{\sum }_i{\kappa }_i}=2^{\kappa }\le s^{\theta }$, while $s\le 2^{\kappa }$ gives the reverse inequality after raising to $\theta$.

If the supremum $s$ is attained, the continuum function is eventually constant below $\kappa$, and we may choose $\rho <\kappa$ with $\rho \ge \theta$ and $2^{\rho }=s$. Then $s^{\theta }=(2^{\rho }{)}^{\theta }=s$. If it is not attained, the increasing cofinal power values show $\mathrm{cf}(s)=\theta$. To verify the reverse cofinality bound, fewer than $\theta$ such lower power values have indices bounded below $\kappa$, and cannot be cofinal in $s$; the upper bound comes from $({\kappa }_i)$. ConsequentlyEvery singular-stage value is therefore determined by the previously computed powers and the given Gimel function.

Now fix an infinite exponent $\lambda$ and recurse on the infinite base $\kappa$. If $2\le \kappa \le \lambda$, then ${\kappa }^{\lambda }=2^{\lambda }$, already known. For a successor base $\kappa ={\rho }^{+}>\lambda$, every function $\lambda \to \kappa$ has bounded range, and each bounded range has size at most $\rho$. HenceThis is the Hausdorff formula for cardinal exponentiation. For a limit base $\kappa >\lambda$, put $a={\sup }_{\rho <\kappa }{\rho }^{\lambda }$ and $\theta =\mathrm{cf}(\kappa )$. If $\theta >\lambda$, all ranges are bounded and counting over those bounds gives ${\kappa }^{\lambda }=a$ (here $a\ge \kappa$).

If $\theta \le \lambda$, then ${\kappa }^{\lambda }=a^{\theta }$. To see the nontrivial upper bound, use a cofinal sequence of bounds ${\kappa }_i$. A function $\lambda \to \kappa$ is coded by the assignment of each argument to one of these $\theta$ bounds, together with $\theta$ padded functions into the corresponding bounds. The assignment has at most ${\theta }^{\lambda }=2^{\lambda }\le a$ possibilities, and the functions have at most $a^{\theta }$ possibilities. Conversely $a\le {\kappa }^{\lambda }$ and $({\kappa }^{\lambda }{)}^{\theta }={\kappa }^{\lambda }$, giving the lower bound. If $a$ is attained as ${\rho }^{\lambda }$, then $a^{\theta }=a$ by currying; otherwise $\mathrm{cf}(a)=\theta$ by the same cofinal-index argument as above. ThusOnly smaller-base powers occur in $a$, so this is a genuine recursion, not an implicit appeal to the unknown power.

Finally, finite positive exponents give ${\kappa }^n=\kappa$ for infinite $\kappa$; finite bases at infinite exponents satisfy $m^{\lambda }=2^{\lambda }$ for $m\ge 2$. The cases with base $0$ or $1$, exponent $0$, or both arguments finite are elementary, with $0^0=1$ under the empty-function convention. The Gimel function therefore determines both requested class functions completely.

For a regular uncountable kappa-tree, keep exactly the nodes whose extensions have unbounded heights:This set is predecessor-closed. At any level $\alpha$, if no node survived, the extension heights above each of its fewer than $\kappa$ nodes would be bounded. Regularity would give a single bound for their union, contradicting the height of the original set-theoretic tree. Thus every level of $T^{\ast }$ is nonempty and still has size less than $\kappa$.

If $t\in T^{\ast }$ and $\beta >\mathrm{ht}(t)$, consider its extensions at level $\beta$. If none survived, fewer than $\kappa$ bounded extension sets would again bound every extension of $t$, a contradiction. Therefore a surviving level-$\beta$ extension exists. HenceThis unbounded-extension kernel of a regular tree uses regularity essentially. If one permits singular-height set-theoretic trees in the term “$\kappa$-tree”, the unrestricted assertion is false: take fewer than $\kappa$ disjoint branches with lengths cofinal in a singular $\kappa$. The levels are small and the height is $\kappa$, but no node has unbounded extensions. A common root can be added without creating a well-pruned subtree. The usual regular-height convention is therefore the one used here.

Prune an ${\aleph }_1$-Suslin tree as in part (a), then use its nodes as forcing conditions, with extensions stronger. Two conditions are compatible exactly when comparable, so the absence of uncountable tree antichains is the forcing countable chain condition for forcing. For each $\alpha <{\omega }_1$, the set $D_{\alpha }$ of nodes of height at least $\alpha$ is dense, by well-pruned set-theoretic tree.

If $2^{{\aleph }_0}>{\aleph }_1$, full Martin's axiom includes ${\mathrm{MA}}_{{\aleph }_1}$. It would provide a filter in an ordered set meeting all these dense subsets of a forcing order. Directedness makes that filter in an ordered set a chain in a partial order, and meeting every $D_{\alpha }$ makes its heights unbounded, producing an uncountable branch. This contradicts the Suslin-tree property. A Suslin set-theoretic tree together with failure of Continuum hypothesis therefore implies failure of Martin's axiom. This is the Suslin-tree obstruction to Martin's axiom.