A collection of research questions on infinity

Trent Dougherty, Rob Koons and I, with the help of one or two friends, made what we think is an interesting collection of research questions on infinity, relevant to philosophy as well as to mathematics, physics and theology.

Conceptual

• How can one classify and define infinities? (E.g., actual, potential, extensive, intensive, ordinal, absolute, unbounded.)
• What does it mean to say that a being is infinite?
• Can something like Cantor’s notion of the absolute infinite be made sense of?  Did Cantor’s theory have theological grounds?
• Are there better general ways of comparing the sizes of infinite sets than subset relations, cardinality, measure, dimension (Hausdorff, etc.) and Baire category?

Divine infinity

• In what way could God contain all mathematical infinities?
• Can an ontological argument based on the idea of an unlimited being be sustained?
• Could the notion of an infinite proof be used to explain the necessity of divine existence?
• Can one being know all orders and kinds of infinity?

Logic, epistemology and rationality

• What does the Löwenheim-Skolem Theorem say about our ability to meaningfully talk about infinities?
• Can one extend probabilistic reasoning to handle about a sample chosen from an infinite number (of moderate or high cardinality) of samples “on par” with one another?
• Do renormalization techniques have probabilistic and epistemological implications?
• Can one define a measure for the components of a multiverse, such as one resulting from string theory, in a way that yields epistemologically responsible predictions?
• How should one rationally reason when there are infinite utilities in view? Infinitesimal probabilities?
• How can one classify the paradoxes of infinity, what can one learn from them, and how can one resolve them? 
• Does one need a logic that handles infinite sentences or propositions and what are the prospects for one?
• Can one engage in plural quantification over proper-class-many objects?  Over greater infinities?
• Is infinity simpler than finite non-zero numbers for the purposes of measuring the complexity of a theory?  What sort of infinity?
• What infinite regresses are vicious?
• Are non-recursive infinite theories, such as “list theories” of mind, scientifically and philosophically acceptable?
• Can Leibniz’s notion of an infinite proof, or the notion of an infinite argument, be made sense of and made useful?
• Is it possible for finite and purely natural beings to come to the concept of infinity?

What infinities there are in physics and metaphysics

• Is a simultaneous infinite number of objects possible? Actual?
• Is an infinite past possible? Actual?
• Can space be infinite? Is it? 
• What alternatives are there to Archimedean spacetimes based on the real numbers?
• Is there an upper cardinality bound on the number of objects, or of physical objects, that could exist?  Could there be proper class many objects or physical objects?
• Are space and time infinitely subdivided or at least infinitely subdivisible? Are they necessarily so? Are they continuous? Necessarily so? 
• Are ‘supertasks’ possible? Non-well-founded processes and non-well-capped ones?
• Can causation proceed through an infinite number of causal intermediaries?
• Could there be infinite intensities: infinite energy, force, density, pleasure, pain?
• Could matter be infinitely complex? Could ‘gunk’ exist? Could matter be infinitely heterogeneous?

Theological non-divine infinities

• Does complete human flourishing require eternal life?
• Does eternal life for humans have to be temporally everlasting?  Does it have to temporally everlasting with respect to external time?
• Could the years of an eternal life have an order type greater than ω?  Would that be better than an  ω-type infinity of eternal life?  
• Is it possible for there to be an infinite sin?  If so, would it deserve an infinite punishment?
• Does an everlasting punishment have to be an infinite punishment?