Toby Ord

We present a new method for expressing Chaitin’s random real, Ω, through Diophantine equations. Where Chaitin’s method causes a particular quantity to express the bits of Ω by fluctuating between finite and infinite values, in our method this quantity is always finite and the bits of Ω are expressed in its fluctuations between odd and even values, allowing for some interesting developments. We then use exponential Diophantine equations to simplify this result and finally show how both methods can also be used to create polynomials which express the bits of Ω in the number of positive values they assume.

Consequentialism is often charged with being self-defeating, for if a person attempts to apply it, she may quite predictably produce worse outcomes than if she applied some other moral theory. Many consequentialists have replied that this criticism rests on a false assumption, confusing consequentialism’s criterion of the rightness of an act with its position on decision procedures. Consequentialism, on this view, does not dictate that we should be always calculating which of the available acts leads to the most good, but instead advises us to decide what to do in whichever manner it is that will lead to the best outcome. Whilst it is typically afforded only a small note in any text on consequentialism, this reply has deep implications for the practical application of consequentialism, perhaps entailing that a consequentialist should eschew calculation altogether.