It won’t change “fundamentally” i.e. in any important way, but it will slightly change the meaning of the probability decreasing on average, because for example, the probability of all hypotheses of complexity 17 might be 0, and if this is the case then there is no greater complexity which will give a lower probability for this particular value. But in any case either the terms will sum to 1 after a finite number of values, in which case the probability will drop permanently to 0 after that, or it will not, in which case the theorem will continue to hold as applied to all the hypotheses that have some positive probability.
Just because there are infinitely many options doesn’t mean that it’s reasonable to assign a probability of 1/infinity. These options are claims about reality that you make with a language; saying that their probability is such a value is much the same as if I said that the probability that you are right in this disagreement is 1/infinity. If someone can say this in the context of my argument, he can say it any context, and this would not be standard probability theory.
Is the claim “you can have probability distributions with infinitely many options and no such distribution can be uniform” a premise or conclusion of your argument?
I don’t agree with the above but even if I did I would assert that it is trivial to reverse the order of ranking such that the arbitrary limitations on the probability distribution lead you to completely the opposite conclusion.
You can’t reverse the order, because there is a least element but no greatest element. So they can’t be swapped round.
The time taken to reverse the ordering of an ordered list depends on the size of the list. Yes, if the size is infinite you cannot reverse it. Or print it out. Or sum it. Or find the median. Infinity messes things up. It certainly doesn’t lead us to assuming:
“Things can be considered in order of complexity but not in order of simplicity (1/compexity) and therefore the most complex things must be lower probability than simple things.”
I say this is just demonstrating “things you shouldn’t do with infinity”, not proving Occams Razor.
the most complex things must be lower probability than simple things
That is not a conclusion. By the definitions given, there is no “most complex thing”; rather, for any given element you pick out that you might think is the most complex thing, there are infinitely many that are more complex.
The claim you mention is a conclusion from standard probability theory.
The order can’t be reversed, just as you can’t reverse the order of the natural numbers to make them go from high to low. Yes, you could start at a google and count down to 1, but then after that you would be back at counting up again. In the same way every possible ordering of hypotheses will sooner or later start ordering them from simpler to more complex.
It won’t change “fundamentally” i.e. in any important way, but it will slightly change the meaning of the probability decreasing on average, because for example, the probability of all hypotheses of complexity 17 might be 0, and if this is the case then there is no greater complexity which will give a lower probability for this particular value. But in any case either the terms will sum to 1 after a finite number of values, in which case the probability will drop permanently to 0 after that, or it will not, in which case the theorem will continue to hold as applied to all the hypotheses that have some positive probability.
Just because there are infinitely many options doesn’t mean that it’s reasonable to assign a probability of 1/infinity. These options are claims about reality that you make with a language; saying that their probability is such a value is much the same as if I said that the probability that you are right in this disagreement is 1/infinity. If someone can say this in the context of my argument, he can say it any context, and this would not be standard probability theory.
Is the claim “you can have probability distributions with infinitely many options and no such distribution can be uniform” a premise or conclusion of your argument?
I don’t agree with the above but even if I did I would assert that it is trivial to reverse the order of ranking such that the arbitrary limitations on the probability distribution lead you to completely the opposite conclusion.
You can’t reverse the order, because there is a least element but no greatest element. So they can’t be swapped round.
The time taken to reverse the ordering of an ordered list depends on the size of the list. Yes, if the size is infinite you cannot reverse it. Or print it out. Or sum it. Or find the median. Infinity messes things up. It certainly doesn’t lead us to assuming:
“Things can be considered in order of complexity but not in order of simplicity (1/compexity) and therefore the most complex things must be lower probability than simple things.”
I say this is just demonstrating “things you shouldn’t do with infinity”, not proving Occams Razor.
That is not a conclusion. By the definitions given, there is no “most complex thing”; rather, for any given element you pick out that you might think is the most complex thing, there are infinitely many that are more complex.
Infinity is not magic.
The claim you mention is a conclusion from standard probability theory.
The order can’t be reversed, just as you can’t reverse the order of the natural numbers to make them go from high to low. Yes, you could start at a google and count down to 1, but then after that you would be back at counting up again. In the same way every possible ordering of hypotheses will sooner or later start ordering them from simpler to more complex.