That paradox is good in that it cuts to the matter very cleanly.
To my mind it “numbers larger than my number” and “uniform integer” don’t need to be same. There are n smaller numbers and ω-n bigger numbers. (ω-n)/ω is going to be near 1 (infinidesimally so) but not quite up to 1. Maybe crucially (ω-2n)/ω is smaller than (ω-n)/ω ie if I hit a high number my chances are better. I get that standard approach somehow gets into the way of this and I would like to know which axiom I have a bone to pick with.
There is a (from my perspective a problem) that events of 0 probablity can happen and events with probability 1 can fail to happen. The associated verbal language is “almost surely” and “almost never”. Showing me that a thing can almost surely happen doesn’t guarantee it. To my mind this is because some zeroes have rounding to the nearest real and some don’t.
That paradox is good in that it cuts to the matter very cleanly.
To my mind it “numbers larger than my number” and “uniform integer” don’t need to be same. There are n smaller numbers and ω-n bigger numbers. (ω-n)/ω is going to be near 1 (infinidesimally so) but not quite up to 1. Maybe crucially (ω-2n)/ω is smaller than (ω-n)/ω ie if I hit a high number my chances are better. I get that standard approach somehow gets into the way of this and I would like to know which axiom I have a bone to pick with.
There is a (from my perspective a problem) that events of 0 probablity can happen and events with probability 1 can fail to happen. The associated verbal language is “almost surely” and “almost never”. Showing me that a thing can almost surely happen doesn’t guarantee it. To my mind this is because some zeroes have rounding to the nearest real and some don’t.
There are two bones you might pick. One is that probabilities are real numbers. The other is that probabilities are countably additive.