Laver and set theory

In this commemorative article, the work of Richard Laver is surveyed in its full range and extent.Accepted manuscrip

Chromatic number of the product of graphs, graph homomorphisms, Antichains and cofinal subsets of posets without AC

We have observations concerning the set theoretic strength of the following combinatorial statements without the axiom of choice. 1. If in a partially ordered set, all chains are finite and all antichains are countable, then the set is countable. 2. If in a partially ordered set, all chains are finite and all antichains have size $\aleph_{\alpha}$, then the set has size $\aleph_{\alpha}$ for any regular $\aleph_{\alpha}$. 3. CS (Every partially ordered set without a maximal element has two disjoint cofinal subsets). 4. CWF (Every partially ordered set has a cofinal well-founded subset). 5. DT (Dilworth's decomposition theorem for infinite p.o.sets of finite width). 6. If the chromatic number of a graph $G_{1}$ is finite (say $k<\omega$), and the chromatic number of another graph $G_{2}$ is infinite, then the chromatic number of $G_{1}\times G_{2}$ is $k$. 7. For an infinite graph $G=(V_{G}, E_{G})$ and a finite graph $H=(V_{H}, E_{H})$, if every finite subgraph of $G$ has a homomorphism into $H$, then so has $G$. Further we study a few statements restricted to linearly-ordered structures without the axiom of choice.Comment: Revised versio

Joint Laver diamonds and grounded forcing axioms

I explore two separate topics: the concept of jointness for set-theoretic guessing principles, and the notion of grounded forcing axioms. A family of guessing sequences is said to be joint if all of its members can guess any given family of targets independently and simultaneously. I primarily investigate jointness in the case of various kinds of Laver diamonds. In the case of measurable cardinals I show that, while the assertions that there are joint families of Laver diamonds of a given length get strictly stronger with increasing length, they are all equiconsistent. This is contrasted with the case of partially strong cardinals, where we can derive additional consistency strength, and ordinary diamond sequences, where large joint families exist whenever even one diamond sequence does. Grounded forcing axioms modify the usual forcing axioms by restricting the posets considered to a suitable ground model. I focus on the grounded Martin's axiom which states that Martin's axioms holds for posets coming from some ccc ground model. I examine the new axiom's effects on the cardinal characteristics of the continuum and show that it is quite a bit more robust under mild forcing than Martin's axiom itself.Comment: This is my PhD dissertatio

Force to change large cardinal strength

This dissertation includes many theorems which show how to change large cardinal properties with forcing. I consider in detail the degrees of inaccessible cardinals (an analogue of the classical degrees of Mahlo cardinals) and provide new large cardinal definitions for degrees of inaccessible cardinals extending the hyper-inaccessible hierarchy. I showed that for every cardinal $\kappa$, and ordinal $\alpha$, if $\kappa$ is $\alpha$-inaccerssible, then there is a $\mathbb{P}$ forcing that $\kappa$ which preserves that $\alpha$-inaccessible but destorys that $\kappa$ is $(\alpha+1)$-inaccessible. I also consider Mahlo cardinals and degrees of Mahlo cardinals. I showed that for every cardinal $\kappa$, and ordinal $\alpha$, there is a notion of forcing $\mathbb{P}$ such that $\kappa$ is still $\alpha$-Mahlo in the extension, but $\kappa$ is no longer $(\alpha +1)$-Mahlo. I also show that a cardinal $\kappa$ which is Mahlo in the ground model can have every possible inaccessible degree in the forcing extension, but no longer be Mahlo there. The thesis includes a collection of results which give forcing notions which change large cardinal strength from weakly compact to weakly measurable, including some earlier work by others that fit this theme. I consider in detail measurable cardinals and Mitchell rank. I show how to change a class of measurable cardinals by forcing to an extension where all measurable cardinals above some fixed ordinal $\alpha$ have Mitchell rank below $\alpha.$ Finally, I consider supercompact cardinals, and a few theorems about strongly compact cardinals. Here, I show how to change the Mitchell rank for supercompactness for a class of cardinals

Joint Laver Diamonds and Grounded Forcing Axioms

In chapter 1 a notion of independence for diamonds and Laver diamonds is investigated. A sequence of Laver diamonds for κ is joint if for any sequence of targets there is a single elementary embedding j with critical point κ such that each Laver diamond guesses its respective target via j. In the case of measurable cardinals (with similar results holding for (partially) supercompact cardinals) I show that a single Laver diamond for κ yields a joint sequence of length κ, and I give strict separation results for all larger lengths of joint sequences. Even though the principles get strictly stronger in terms of direct implication, I show that they are all equiconsistent. This is contrasted with the case of θ-strong cardinals where, for certain θ, the existence of even the shortest joint Laver sequences carries nontrivial consistency strength. I also formulate a notion of jointness for ordinary ◊κ-sequences on any regular cardinal κ. The main result concerning these shows that there is no separation according to length and a single ◊κ-sequence yields joint families of all possible lengths. In chapter 2 the notion of a grounded forcing axiom is introduced and explored in the case of Martin\u27s axiom. This grounded Martin\u27s axiom, a weakening of the usual axiom, states that the universe is a ccc forcing extension of some inner model and the restriction of Martin\u27s axiom to the posets coming from that ground model holds. I place the new axiom in the hierarchy of fragments of Martin\u27s axiom and examine its effects on the cardinal characteristics of the continuum. I also show that the grounded version is quite a bit more robust under mild forcing than Martin\u27s axiom itself

Numerical study of strain rate effects on stress strain response of soils

Imperial Users onl