Topics Logic

Question Ok, I'm going to go at Godel backwards. I'm going to start from the fact that the universe exists (whatever others may think to the contrary). I'm assuming that the universe is ruled by law. It also seems to me that the universe can't contain any self-contradictions, or it wouldn't exist in the first place. So, its laws are consistent. For a similar reason, they must be complete; if some key part was missing, the universe wouldn't exist. This line of reasoning seems to lead me to: the laws of the universe are both consistent and complete. I know that Godel was talking about formal systems, but it just seems to me that the laws of the universe are the formal system. So, there is at least one example of a formal system that is both consistent and complete, whether or not we can articulate it. Or have I completely missed Godel's idea here? Thanks, JT

Accepted:
June 28, 2010

Comments

A formal system (of the kind to which Gödel's incompleteness theorem applies) is a consistent axiomatized theory which contains a modicum of arithmetic and is such that it is mechanically decidable whether a given sentence is or isn't an axiom.

Why should we think that the "laws of the universe" can be encapsulated in a formal theory in that sense? Why suppose that all the laws can be wrapped up into a single formal system? It isn't at all obvious why that should be so: maybe the laws of the universe are so rich that they always elude being pinned down by a single formal axiomatic system (such that it is mechanically decidable what's an axiom).

Indeed, we might say that Gödel's incompleteness theorem shows that, on a broad enough understanding of "laws of the universe", the laws can't be so pinned down. For any given formal theory, there will be arithmetical truths that particular formal system can't prove -- so the arithmetical laws of the universe, for a start, run always run beyond what can be pinned down in a particular formal system.