The discussion centers on the concept of probability in relation to continuous random variables, specifically whether the probability of an event occurring at an exact moment in time can ever be zero. Participants explore the implications of continuous time, mathematical modeling, and the nature of events in a probabilistic framework.
Participants express a range of views, with no clear consensus on whether the probability of an event occurring at an exact moment can be considered zero in practical terms. The discussion remains unresolved, with competing interpretations of mathematical principles and their implications for real-world events.
Participants acknowledge limitations in their understanding of the relationship between mathematical models and physical reality, as well as the nuances of defining events in continuous time.
quadraphonics said:Indeed, the cost of including infinity in a number system is that certain basic expressions must remain undefined, when they include infinity. But so what? They're still defined as usual for every finite number, and the expressions where infinity is sensible (like 1/\infty=0) are still well defined.
There is also a finite number that does not work in many basic algebraic manipulations, but is nevertheless included in many standard number systems.
Archosaur said:Can you give me an example of a finite number that I can't multiply both sides of an equation by?
Archosaur said:I would agree that the limit as n -> infinity of 1/n = zero
But I'm having a hard time with this 1/infinity = 0 thing...
quadraphonics said:\infty, as it is used in the extended reals, is essentially a shorthard for that type of limit expression.