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This post is a very brief introduction to some of the basic concepts of set theory. Set theory is a branch of mathematical-logic, that has wide applications across disciplines. Its not just used in the obvious way of studying the foundations of mathematics by mathematicians but also in physics, social science and even in philosophey. I'm only going to cover in this post what's needed for predicate logic (although you can do propositional logic without set theory) but it has for many other applications. A set is a collection of elements, or members; the notation for a set is specified by listing its components. So the set of even numbers can be represented as $E: \left \{ 2,4,6,8 ... \right \}$ $E: \left \{ x: x > 0 \wedge even\right \}$ Either of these notations is valid. Further, elements of a set can only be in that set once. $E: \left \{ 2,2,2,4,4,6,8 ... \right \} = E: \left \{ 2,4,6,8 ... \right \}$ The notation used to indicate that something is a

William Lane Craig and the Hartle-Hawking No Boundary Proposal

Standard hot Big Bang cosmology represents the universe as beginning from a singular dense point, with no prior description or explanation of classical spacetime. Quantum cosmology is different in that it replaces the initial singularity with a description in accord with some law the "quantum mechanical wave function of the universe", different approaches to quantum cosmology differ in their appeal either to describe the origin of the material content of the universe e.g., Tyron 1973, Linde 1983a, Krauss 2012 or the origin of spacetime itself e.g., Vilenkin 1982, Linde 1983b, Hartle-Hawking 1983, Vilenkin 1984. These last few proposals by Vilenkin, Hartle-Hawking and others are solutions to the Wheeler-DeWitt equation come under a category of proposals called "quantum gravity cosmologies" which make cosmic applications of an approach to quantum gravity called "closed dynamic triangulation" or CDT (also known as Euclidean quantum gravity). I've n

Can inflation be eternal into the past?

Back in 2003 a paper appeared on the arXiv titled " Inflationary spacetimes are not past complete " that was published by Arvind Borde, Alan Guth and Alexander Vilenkin which has had considerable amounts of attention online. The theorem is rather uninteresting but simple and doesn't require a very complicated understanding of math. So I thought I'd explain the result here. It's purpose is to demonstrate that inflationary models are geodesically incomplete into the past which they take as "synonymous to a beginning" but Vilenkin stresses that the theorem can be extended to non inflationary models so long as the condition of the theorem that the average rate of expansion is never below zero is met. These models too are incomplete into the past. Consider the metric for an FRW universe with an exponential expansion Where the scale factor is Since the eternal inflation model is a "steady state cosmology" the mass density and the Hub

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