Can there be ratios using different sizes of infinities?

  • Thread starter student34
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Main Question or Discussion Point

Pretend that some god has every real number in its head, and it is thinking of only one of them randomly. There seems to be a better chance of it being a real number, minus the naturals, rather than just a natural number. If this example is even allowed to be conceived, what does the ratio look like? If it is ℵ0/C, and the quotient is ≤ ℵ0, wouldn't it mean that there is a 0% chance of picking a rational number?

Also, if this logic is not ridiculous, then pretend that this time the god thought of a rational number instead. Isn't there a greater chance of it being a integer rather than just a natural number? If so, then how can the naturals and the integers have the same cardinality?
 

Answers and Replies

  • #2
pwsnafu
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Pretend that some god has every real number in its head, and it is thinking of only one of them randomly.
This question is meaningless unless you specify how the numbers are being chosen.
If we had a six sided die, I can say "with equal probability choose a number". Each element has probability 1/6. There are six sides, each 1/6, so they add to 1 (i.e. 100%)

I can't do this with an infinite set. If the probabilities are equal, then adding an infinite number will give me infinity, not 1.

If I want to choose (say) "a natural number at random" I need them to be different probabilities. Eg. The probability of choosing n could be 2-n.
 
  • #3
chiro
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Hey student34 and welcome to the forums.

There are a few issues here.

First of all if you are allowed to choose a specific number it means that you can not model this with a continuous distribution, since a continuous distribution allows you to only consider the probability of a non-zero interval (and unions thereof).

So this means you have a discrete distribution for a start.

So in the way you have stated your question, you can't even construct a proper distribution in the above manner, if you are trying to use a continuous distribution. You also need to specify a proper probability for each outcome that is non-zero. If every outcome has the same probability and the number of events tends to infinity, then you get a probability of zero and this reflects one way of interpreting why the probability of a single point in a continuous distribution is zero: they both correspond to the same situation happening.

If you do however want to consider the experiment, you either need to create a continuous distribution where a non-zero interval corresponds to some event (like the naturals, integers, rationals whatever) and the complement corresponds to everything else: if this is properly defined in all aspects (i.e. Kolmogorov Axioms, definition of PDF makes sense in context of the problem) then you can use that to calculate your PDF.
 
  • #4
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Sorry, but I just don't know enough about math to know why my OP doesn't make sense. I probably should have even asked.
 
  • #5
chiro
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Sorry, but I just don't know enough about math to know why my OP doesn't make sense. I probably should have even asked.
Basically it boils down to you specifying a distribution so that you can calculate a probability.

The post I mentioned above says the reason why you can't use a continuous distribution and the reason is that you can't just calculate a probability for one value in a continuous distribution because it is zero. You can calculate a probability for a collection of values like say from 0 to 0.1 or 1 to 1.0000011 or something like that but not for an interval that is essentially 0 length. So it means we need a discrete distribution.

Now you have an infinite number of values and if it really is random its going to be uniform (basically the probability of picking any value is the same as all the other values) which means if you have N choices, then the probability is 1/N. But you have infinitely many numbers and the limit of the probability in this case is 0.

This is the paradox: if you use the above formulation you don't even get a proper probability to begin with and you automatically have problems.

The only way to really settle this is to check that the ratio of cardinalities is non-zero if you wish to say "bunch up all natural numbers" as one event and everything else as the complementary event instead of the way I did above which was to treat every single possible numeric outcome as a single event. Doing what I did shows that the probability is zero and doesn't even make sense, but if you can show that you can obtain a non-zero probability for the ratio of cardinalities, then this means that you can obtain a probability.
 
  • #6
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In application (if this is even possible), it would be so close to 0, they would consider it 0%, but the real issue is, this is a limit & infinite series problem as chiro says. The chances of picking any specific number would approach 0. Check out / review the Infinite Series (with rieman sums). It might help you come to your own conclusion ;D
 
  • #7
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What the OP is asking, although extended to all the integers, may be modelled by a Dirac comb function. However, this distribution is non-normalizible.
 

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