The simplest possible axiom, axiom 0.

There is an axiom (I will enounce it immediately) which I will call “Axiom 0″, with the following nice properties:

1. It is the simplest possible.
2. It is the richest possible: we can derive from it a very wide spectrum of consequences; let’s name these consequences “the Metaverse”.
3. Our universe is a [logically] consistent subset of the Metaverse.
4. Any consistent universe is a subset of the Metaverse.
5. Axiom 0 itself is outside of any of the consistent universes it creates.

What is this axiom?

1. An axiom establishes logical relations between some objects. In order to have an axiom which makes sense, it has to refer to something. The simplest axioms are those which do not rely on things outside themselves. Therefore, they have to be self-referential.
2. There are two self-referential propositions which can be considered the simplest possible:
Axiom 1: “Axiom 1 is true.”
and
Axiom 0: “Axiom 0 is not true.”
From these, Axiom 1 is trivial and has no new consequences. On the other hand, Axiom 0 is the richest possible, because any statement can be derived from it, as it is known from Logic.
3-4. Any consistent universe, or “possible world”, is therefore a subset of the consequences of Axiom 0. Including our universe.
5. But Axiom 0 itself is inconsistent, and has to remain outside any universe from the Metaverse. The Metaverse itself is inconsistent.

The Metaverse contains all possible worlds, so all these can be related to Tegmark’s “Mathematical Universe Hypothesis”.

Among these universes, there is a special one (at least to us): the one which we observe and wonder where did its laws come from.