How infinity can be used in mathematics

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Discussion Overview

The discussion revolves around the concept of infinity in mathematics, particularly in relation to probability and calculus. Participants explore the implications of infinitely small chances, infinitely large areas, and the mathematical treatment of expressions involving infinity.

Discussion Character

  • Exploratory
  • Technical explanation
  • Conceptual clarification
  • Debate/contested

Main Points Raised

  • One participant questions how many times an event would occur given an infinitely small chance, an infinitely large area, and an infinitely small time frame.
  • Another participant emphasizes the need for careful definitions of "infinitely small" and "infinitely large," suggesting that these terms can lead to different interpretations.
  • There is a discussion about whether 1/infinity can be considered a limit of 0, with one participant asserting that it cannot be directly equated to zero.
  • Infinity/infinity is debated, with some arguing it is undefined while others suggest it depends on the context of how the values approach infinity.
  • Clarifications are made regarding the use of the term "infinite," with one participant stressing that it refers to a concept without end rather than just a large number.
  • Concerns are raised about the misuse of infinity in mathematical expressions, particularly in relation to probability and measure theory.
  • One participant challenges the idea of having a smallest non-zero time period, arguing that time is not quantized in the usual model.
  • Examples are provided to illustrate the necessity of specific conditions when discussing probabilities in infinite contexts.

Areas of Agreement / Disagreement

Participants express differing views on the treatment of infinity in mathematical contexts, particularly regarding its implications for probability and calculus. No consensus is reached on how to properly handle expressions involving infinity.

Contextual Notes

Limitations in the discussion include the ambiguity of terms like "infinitely small" and "infinitely large," as well as the need for more specific scenarios to clarify the mathematical treatment of probabilities in infinite contexts.

musky_ox
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Here is the question that was plaguing me:
If there was an infinitely small chance of something happening, but an infinitely large test area, and and infinitely small amount of time for it to happen in, how many times would it happen?

Now, i am not sure how infinity can be used in mathematics. I am just starting calculus.

Here are some questions:
Can you say that 1/infinity is just a limit of 0?

What would you say for infinity/infinity? Would this simply be undefined? If you said that both were equally infinite, logically would the answer would be infinity still, or 1? :eek:

However, if you had infinity/infinity squared...? When i think of this, i picture a really small time, a really large space, and a really small chance of the happening. It seems that it should logically happen once, but not mathematically.

Please give me your thoughts on this, i could be totally wrong with this since I am not sure if i can use infinity like this. :bugeye: Not sure whether this should have been in the physics forum because its kind of abstract, but i couldn't post there for some reason.
 
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musky_ox said:
Here is the question that was plaguing me:
If there was an infinitely small chance of something happening, but an infinitely large test area, and and infinitely small amount of time for it to happen in, how many times would it happen?

You have to be a bit more careful with infinity. When you say "infinitely small" does that mean zero or small but finite? It makes a difference! Likewise for "infinitely large area" - finite or infinite?

musky_ox said:
Now, i am not sure how infinity can be used in mathematics. I am just starting calculus.

I think the word you're looking for is infinitesimal. In calculus it means "arbitrarily small." For example, you could divide a domain into aribitrarily small pieces and end up with an aribtrarily large number of pieces.

musky_ox said:
Here are some questions:
Can you say that 1/infinity is just a limit of 0?

No, but you can say that [itex]\frac {1}{n}[/itex] gets arbitrarily close to zero as n becomes arbitrarily large.

musky_ox said:
What would you say for infinity/infinity? Would this simply be undefined? If you said that both were equally infinite, logically would the answer would be infinity still, or 1? :eek:

See my previous comment!

musky_ox said:
However, if you had infinity/infinity squared...? When i think of this, i picture a really small time, a really large space, and a really small chance of the happening. It seems that it should logically happen once, but not mathematically.

infinity/infinity simply makes no sense mathematically. In calculus, the result depends on the details of how both the numerator and denominater become aribtrarily large.
 
Okay i will clarify that... i mean infinite as in no end... not just a large number. What i was trying to say was:

You have a never ending space in all directions, and an the chance of an event ocurring is 0.0000...1 (1/infinity). You have the smallest possible time period; there is a time, but it is the closest thing possible to 0. When you first think of the situation, you think it would happen infinite times, but if i said say that that they were both equally infinite, would they cancel out? I am not sure if there is some way to communicate this. Lots of people look at it and say "well the top of the equation is infinite, so it is automatically larger than any value in the denominator and thus the answer must be infinite." However, the chance of it happening is also infinitly small.
 
You cannot draw any conclusions without specifics as I indicated in my previous post. You're implying that the "probability per unit area" changes in some manner with the total area and then you're letting the total area tend toward infinity. Do you have a specific problem in mind?
 
Its a purely abstract idea, id like to know what you get from it.

If there was an infinitely small chance of something happening, but an infinitely large test area in which it could happen in, and it was given an infinitely small amount of time to happen in, how many times would it happen theoretically?
 
musky_ox said:
the chance of an event ocurring is 0.0000...1 (1/infinity).

No, no and no again, that makes no sense in the real numbers. (An infinite number of 0s then a 1 presumably.)

Nor is there the smallest non-zero time period (time is not qunatized in the usual model).

You are completely misusing the word infinity. There are several well defined situations when one can talk about infinite objects in many senses, especially in probability: that is what measure theory does, but this is not one of them.

Let us demonstrate by example why you need to give more information:

Let us suppose you are "picking a point at random from the (positive) real numbers" which is what you're basically attempting to describe. What is the probability that number lies in [0,1]? or [1,2], or [2,3],... if it were the same non-zero probablitity in each, say, e, then e+e+e... must equal 1, but that's non-sense, hence you need some better description of how the probabilities are distributed.
 

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