For those who don’t know, Pascal’s wager is an attempt to make a logical argument for believing in god. The idea is that if you agree that heaven has infinite value, it follows that if your belief is that the probability god exists is greater than 0 (and even Richard Dawkins would only rate himself at 6 out of 7 in the strength of his belief that god does not exist) then the value of believing in god is infinite, because any non zero number multiplied by infinity is infinity – if you assign a probability of 0.01 of god existing then the value of believing in him to reach heaven is 0.01 * infinity = infinity. For more information, check out the wikipedia article.

Allowing the possibilities of states that have infinite value (such as heaven) implies that any decisions that can both potentially lead to reaching these infinitely valuable states necessarily have equal value. For example, exercising for 50 minutes and punching one’s self in the face are both of equal value to someone who believes that the probability of reaching heaven after doing either of these things is not exactly 0.

Assigning a future state the value of infinity in decision systems has a very powerful destructive effect. Because a future state of infinite value has a non-zero probability of being reached, all the parent nodes have infinite value too and that will ultimately lead to the current node also having infinite value, meaning that there will be indifference to all current decisions. Thereby making it not much of a decision system anymore because it has no way to discern between any choices.

Those who do believe in states of infinite value but don’t believe that all decisions that don’t rule out the possibility of reaching all values of infinite states have equal value are forced into resolving the contradiction. Either they must accept that essentially all decisions have equal value to them, or they must deny the existence of the possibility of states that have infinite value – it’s not consistent to hold both beliefs. And it’s my belief that denying the existence of states of infinite value is the most intuitive decision that most people already agree with. I don’t have any proof that this is the correct assumption, but I think most people would agree with me that anything that results in almost all decisions having exactly the same value is not valid.

While adding in a new rule for decisions systems is a heavy handed approach to solving what seems to be a small problem in Pascal’s Wager, it actually solves the problems of the destructive forces infinities have on decision systems.

At this point you may be thinking “but wait, if heaven does last infinity long and each year has a value greater than 0, it’s value must be infinite”. In that case I would like to outline how it is possible that even though heaven can last infinitely long, it does not necessarily have infinite value even though each year has a positive value. Consider the series 1/2 + 1/4 + 1/8 + … Does it add up to infinity? It doesn’t, here’s a visual proof

Things that have an infinitely long existence in the temporal realm do not necessarily have infinite value even though each instance of that point of time has a value greater than 0.

Similarly, you can assign a probability distribution of the value of an unknown entity such that it never adds up to infinity. E.g. the probability it is worth 1 is 1/2, the probability it’s worth 2 is 1/4 and so on. This is another way to assign a positive probability to all possible values of an unknown entity without requiring you to be absolutely certain that it is not worth more than a given amount (i.e. not needing to assign a probability that it has a value above x to be exactly 0).

In summary, this post is meant to outline the fact that you must either accept that almost all decisions have the same value or that states that have infinite value do not exist.