Could this actually hold?
Please disprove the following:
- For a system to be deterministic, its underlying rules must be consistent.
- For a system to be deterministic, its underlying rules must be complete.
- No system of rules can be both complete and consistent per Gödel’s Incompleteness Theorems.
- Thus no system can be deterministic.
FYI: This proof dawned on me while researching Gödel’s Incompleteness Theorems – but I realize that points 1 & 2 above may perhaps be invalidated by presenting cases for deterministic systems with incomplete or inconsistent rule sets. Can such cases exist?
Update (2011-01-30): After some good comments, I offer this:
- Thesis: The universe contains all there is and all there ever will be, it is a complete and closed system and causally deterministic (Laplace’s demon)
- According to Gödel’s Incompleteness Theorems, such a system would have to contain paradoxes (inconsistencies), potentially rendering the system indeterministic.
- To prove the thesis of a causally deterministic universe, one would have to prove (why) the universe would never encounter any such paradoxes breaking the determinism – and prove why the universe itself would never encounter Turing’s Halting Problem when deciding any effect ever in the universe.