One of the standard ways of postulating large cardinal axioms is to consider elementary embeddings,j, from the universe,V, into some transitive submodel,M. See Reinhardt–Solovay [7] for more details. Ifjis not the identity, andκis the first ordinal moved byj, thenκis a measurable cardinal. Conversely, Scott [8] showed that wheneverκis measurable, there is suchjandM. If we had assumed, in addition, that, thenκwould be theκth measurable cardinal; in general, the wider we assumeMto be, the largerκmust be.

T. K. Menas [4, pp. 225-234] introduced a combinatorial property χ (μ) of a measure μ on a supercompact cardinal κ and proved that measures with this property also have the partition property. We prove here that Menas' property is not equivalent to the partition property. We also show that if α is the least cardinal greater than κ such that P κ α bears a measure without the partition property, then α is inaccessible and Π 2 1 -indescribable.

If θ is any singular cardinal of cofinality ω 1 , we produce a forcing extension in which MA holds below θ but fails at θ. The failure is due to a partial order which splits a gap of size θ in P(ω).

Keywords: Elementary Submodel; Real Line; Order-Isomorphic.

We continue the investigation of Gregory trees and the Cantor Tree Property carried out by Hart and Kunen. We produce models of MA with the Continuum arbitrarily large in which there are Gregory trees, and in which there are no Gregory trees.

We consider a well-known partial order of Prikry for producing a collapsing function of minimal degree. Assuming MA + ≠ CH, every new real constructs the collapsing map.

We consider a well-known partial order of Prikry for producing a collapsing function of minimal degree. Assuming $MA + \neq CH$, every new real constructs the collapsing map.

The separation, uniformization, and other properties of the Borel and projective hierarchies over hyperfinite sets are investigated and compared to the corresponding properties in classical descriptive set theory. The techniques used in this investigation also provide some results about countably determined sets and functions, as well as an improvement of an earlier theorem of Kunen and Miller.

Physicians generally recommend that patients resuscitated with naloxone after opioid overdose stay in the emergency department for a period of observation in order to prevent harm from delayed sequelae of opioid toxicity. Patients frequently refuse this period of observation despiteenefit to risk. Healthcare providers are thus confronted with the challenge of how best to protect the patient’s interests while also respecting autonomy, including assessing whether the patient is making an autonomous choice to refuse care. Previous studies have shown that physicians (...) have widely divergent approaches to navigating these conflicts. This paper reviews what is known about the effects of opioid use disorder on decision-making, and argues that some subset of these refusals are non-autonomous choices, even when patients appear to have decision making capacity. This conclusion has several implications for how physicians assess and respond to patients refusing medical recommendations after naloxone resuscitation. (shrink)

It is widely agreed among broadly Anselmian theists that God is in some sense the 'delimiter of possibilities.' In other words, the scope of possibility is explained by the manner in which the universe emanates from God. However, existing accounts of God's role here—in terms of freedom, choice, or power—face serious difficulties. The present paper provides a new account of God's role as the delimiter of possibilities in terms of the different manner in which the non-actuality of non-actual states of (...) affairs is explained. Intuitively and metaphorically, we can think of the menu of states of affairs being 'whittled down' in successive stages of a process of divine decision-making, with the different species of impossibility corresponding to when and how the states of affairs are ruled out. I argue that this approach provides genuine contingency while also preserving the Principle of Sufficient Reason. (shrink)