Godel's Incompleteness Theorems are a pair of deep and disconcerting results in Mathematics. The main thrust of the results is any logical system will have one of two flaws: incompleteness or inconsistency. Both of these cases have disturbing implications for Math. If our construction of Math is incomplete, then there are statements we can make which are neither true nor false given our starting assumptions. If Math is inconsistent, then all statements are both true and false. The hope is that we have selected axioms such that Math is incomplete as that is the lesser of the two evils. It is worth noting that there is still a lot we can do in this case. The implication of the theorem is there is always a boundary on what we can prove, no matter how many axioms we have.
Why am I discussing a Mathematical results that is rarely used in a blog about games? If you think about it the rules of a game are like the axioms of a logical system. This means that Godel's Incompleteness Theorems apply to every game in existence! This means that there are either situations we can describe which the rules don't cover or the rules contradict themselves somewhere. Examining help forums for games, it is clear that many games suffer from one of these problems. Though this is usually chocked up to insufficient playtesting. So is Godel telling us that there is no amount of playtesting that can resolve this issue and we should just always expect such problems from our games?
Looking carefully at what incompleteness is, we see that this actually not a problem for most games. A game suffering from incompleteness just has some situation which the rules cannot decide. There is nothing that states that this situation can occur during gameplay or even that the components of the game that would cause issues even exist. The best game to explore this idea with is Magic: the Gathering. Magic being incomplete implies that there is a card or combination of cards which the rules cannot handle. This is like the double Opalescence and Humility problem before the introduction of layers. The thing about Godel is the cards that cause a rules issue might not even have been created yet. There could be a card or cards that we can write up that are perfectly valid Magic cards, but have something undecidable in the rules. So Magic escapes the Godel's problems by simply not having all possible cards existing.
This leads us to the main release valve in games for Godel's results, games are (usually) finite. Just like Magic avoids the problem by only having a tiny portion of all the infinite possible cards actually existing, other games can avoid being incomplete by bounding what can happen in game. So any game with a bounded scope can avoid the paradox of Godel's Theorems by being playtested to the point that the inevitable undecidable situation exists outside the boundary of the game.
Hence any single game can avoid rules confusion. Though there are two special cases worth thinking about more: expansions and RPGs. Expansions should be just like the base game, with enough playtesting one can push the inevitable incompleteness outsidethe scope of the game. This is usually done by adding rules to resolve situations. Under Godel's Theorem, adding new axioms does not resolve incompleteness as even with an additional axiom, there will still be some other case not covered. It is literally impossible to cover all the cases in an infinite game without contradicting yourself. We see that expansions always add rules. Sometimes they even completely rework the base rules because the new rules came in direct conflict with how new situations were handled.
This rewriting of the basic axioms is more common than one would expect. It happens with paradigm shifts in the sciences, like classical Newtonian physics to General Relativity. Or the many attempts at axiomatizing set theory in the 20th century. This has also happened in Magic with the big 6th Edition rules change. Through this we see than even expansions can avoid Godel, though they might require a complete rewrite of the rules to do so.
Tabletop RPGs are the most complex games to design. Unlike essentially all other games, they are infinite in scope. Any situation the players can imagine is possible and people are constantly pushing the limit all the time. In an environment where so many people are applying their collective creativity, it is inevitable that someone will try something that is impossible to decide with the rules. Unlike most other games, RPGs have an ultimate authority who is technically above the rules, the Game Master. It is because there is a person empowered to unilaterally resolve situations that these games function. When a situation the rules deem undecidable arises, the Game Master has the power to resolve it.
Hence, one should expect people to have game worlds run by different Game Masters function differently. One last implication of Godel's Theorems is that any undecidable statement can be determined to be either true or false and still form a consistent system. For example, see the continuum hypothesis. This means that in any RPG there will always be legitimate rules disagreements. Moreover, neither person will be wrong!
This ended up being a bit longer and more philosophical than I intended at the beginning. To sum up, Godel's Incompleteness Theorems do apply to games. Though due to their finite nature, it cannot be used as an excuse for games that needs more playtesting. The one exception are RPGs, where their unbounded nature means incompleteness will always be encountered. Luckily, there is a Game Master empowered to resolve problems as they occur.
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