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.

In this chapter, we defend an explanationist version of proper functionalism. After explaining proper functionalism’s initial appeal, we note two major objections to proper functionalism: creatures with no design plan who appear to have knowledge (Swampman) and creatures with malfunctions that increase reliability. We then note how proper functionalism needs to be clarified because there are cases of what we call warrant-compatible malfunction. We then formulate our own view: explanationist proper functionalism, which explains the warrant-compatible malfunction cases and helps to (...) block the above objections. We also advance a positive argument for explanationist proper functionalism. (shrink)

According to commonsense, some collections of objects compose wholes, and others do not. However, philosophers have found serious difficulties with attempts to preserve this thesis, and especially with attempts to preserve the existence of just those composite objects recognized by commonsense. In this paper, I defend a classical solution to this problem: "it is the mind that maketh each thing to be one" (Berkeley, Siris, sect. 356). According to this view, which I call 'mereological idealism,' it is when a plurality (...) is united in thought under a concept that a unified whole comes to exist. After explaining the view in more detail, I show how it escapes three standard arguments against commonsense views of composition. (shrink)

Probes the nature of linguistic or symbolic action as it relates to specific novels, plays, and poems.

Alexander Pruss and I have proposed an analysis of omnipotence which makes no use of the problematic terms 'power' and 'ability'. However, this raises an obvious worry: if our analysis is not related to the notion of power, then how can it count as an analysis of omnipotence, the property of being all-powerful, at all? In this paper, I show how omnipotence can be understood as the possession of infinite power (general, universal, or unlimited power) rather than the possession of (...) particular powers. I then argue that the ordinary notions of particular powers or abilities can be understood as applying vague limitations to the notion of infinite power. The vagueness of these limitations is the source of the well-known difficulties in the analysis of ability. (shrink)