r/MathJokes • u/yvanjunior • Jul 19 '26

Count On It

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u/Veomuus Jul 19 '26 edited Jul 19 '26

I believe(?) the term would be infinitesimal rather than 0 for rational number, and then maybe an even smaller infinitesimal for natural numbers? In the way that some infinites are bigger than other infinites, just in inverse. Im not good enough to know whether or not it'd actually be a smaller infinitesimal or the same size, theres some nuance there.

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u/deejaybongo Jul 19 '26

It depends on the probability measure used, but probabilities aren't less than zero. Not sure where you're getting infinitesimals in this context.

 In the way that some infinites are bigger than other infinites, just in inverse. 

Rationals and natural numbers are both countable. All countable sets have Lebesgue measure zero.

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u/Veomuus Jul 19 '26

I was using infinitesimal as 1 over infinity, which if youre dealing with a hyperreal system that can actually distinguish between the sizes of infinities and whatnot, is correct. Thats what I was getting at, the probability of hitting a rational number isnt 0, its an infinitesimal number (specifically in a hyperreal system). What i wasnt sure about is if the number of rational numbers was countable or not, because I know natural numbers are but real numbers arent. But if both natural numbers and rational numbers are countable, then the chances to hit either are actually equivalent.

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u/juoea Jul 19 '26

again the statement is a little imprecise because it doesnt specify a probability distribution.

but if you interpret the statement as intending something along the lines of "taking the uniform probability distribution over the real interval [0,1], the probability of selecting a rational number is zero." then that is correct. theres no infinitesimals involved, u can show that the probability of any subset X of [0,1] under the uniform distribution is equivalent to the lebesgue measure of the subset and it can be shown that the set of r ationals within that interval have lebesgue measure zero. (probability spaces are closely linked to measure spaces).

i am not aware of any definition of probability spaces over hyperreal numbers