Is one out of infinity different from zero?

[lIllIlIIIl]

In another question I posted on here, I asked about a hypothetical roulette wheel with an infinite amount of spaces. Each number has a one-in-infinity chance of being selected, yet each time the wheel is spun, one number wins those odds. My initial question was how this was possible, and I had that answered. Now my question is, is one out of infinity different from zero?

In my initial question, it was discussed how p<0 is different from something being impossible. Now, one-in-infinity is interesting to me because one is infinitely far from its potential maximum in this scenario, which I think makes it infinitely small. Is something that is infinitely small different from nothingness? If so, why and how, and if not, does that mean that every number is secretly zero?

I'm a high school freshman in Algebra 1, so my understanding of numbers is at an annoying point where I have questions about them but need to take at least a year of stats to understand the answers. Sorry if this is a stupid question, it just confuses me how being infinitely far from a maximum is different from being nothing. I hope this makes sense.

 

[pinball1970]

In another question I posted on here, I asked about a hypothetical roulette wheel with an infinite amount of spaces. Each number has a one-in-infinity chance of being selected, yet each time the wheel is spun, one number wins those odds. My initial question was how this was possible, and I had that answered. Now my question is, is one out of infinity different from zero?

In my initial question, it was discussed how p<0 is different from something being impossible. Now, one-in-infinity is interesting to me because one is infinitely far from its potential maximum in this scenario, which I think makes it infinitely small. Is something that is infinitely small different from nothingness? If so, why and how, and if not, does that mean that every number is secretly zero?

I'm a high school freshman in Algebra 1, so my understanding of numbers is at an annoying point where I have questions about them but need to take at least a year of stats to understand the answers. Sorry if this is a stupid question, it just confuses me how being infinitely far from a maximum is different from being nothing. I hope this makes sense.

Infinity is not a number but this will be better explained by @jbriggs444 and @PeroK among others

 

[PeroK]

In another question I posted on here, I asked about a hypothetical roulette wheel with an infinite amount of spaces. Each number has a one-in-infinity chance of being selected, yet each time the wheel is spun, one number wins those odds. My initial question was how this was possible, and I had that answered. Now my question is, is one out of infinity different from zero?

To begin with $\frac 1 \infty$ is undefined. As @pinball1970 says, infinity is not a number which can be used in the arithmetic options.

In my initial question, it was discussed how p<0 is different from something being impossible. Now, one-in-infinity is interesting to me because one is infinitely far from its potential maximum in this scenario, which I think makes it infinitely small. Is something that is infinitely small different from nothingness? If so, why and how, and if not, does that mean that every number is secretly zero?

This is not mathematics. Mathematics is precise about how things are defined.

I'm a high school freshman in Algebra 1, so my understanding of numbers is at an annoying point where I have questions about them but need to take at least a year of stats to understand the answers. Sorry if this is a stupid question, it just confuses me how being infinitely far from a maximum is different from being nothing. I hope this makes sense.

There's no royal road to mathemtics, as someone once said.

 

[jedishrfu]

While not directly answering your question, working with infinity can produce ambiguous or nonsense answers and must thought about very carefully.

Veritaseum's video on Hilbert's hotel shows when and where it can break down and you are stuck between countable and uncountable infinities.



and mathologer runs into similar oddities in his visual logarithms video:

 

[BWV]

If you phrase it properly, your intuition is correct - for a continuous random variable with an uncountably infinite range, the probability that it takes any particular value is zero, it can only be defined on an interval:

https://en.wikipedia.org/wiki/Random_variable

Similarly the limit of a function like 1/x is zero as x approaches infinity

but that does not mean that you can treat ∞ like a number, if 1/∞ = 0 then multiplying both sides by ∞ would get you 1=0

 

[lIllIlIIIl]

So is my problem that I'm treating infinity like I would any other number rather than treating it like what it is, which is confusing?

 

[PeroK]

So is my problem that I'm treating infinity like I would any other number rather than treating it like what it is, which is confusing?

Yes. You cannot do arithmetic operations on "infinity".

 

[jbriggs444]

To begin with $\frac 1 \infty$ is undefined. As @pinball1970 says, infinity is not a number which can be used in the arithmetic options.

There is a slightly finer point that can be made.

There is no uniform probability distribution over the set of positive integers. Or over any countably infinite set for that matter.

 

[weirdoguy]

I think, that you can also read about actual infinity and potential infinity.

 

[WWGD]

Yes. You cannot do arithmetic operations on "infinity".

You actually can. There are both Cardinal and Ordinal Arithmetic, though I assume you had a different intent in mind. Here infinity would be an equivalent class, represented by an element. Two sets are equivalent as Cardinals iff(def.) there is a bijection between them.

 

[weirdoguy]

There are both Cardinal and Ordinal Arithmetic

Which are "actual infinities" whereas infinity denoted by $\infty$ and connected to limits is "potential".

 

[pbuk]

I think, that you can also read about actual infinity and potential infinity.

Which are "actual infinities" whereas infinity denoted by $\infty$ and connected to limits is "potential".

The concepts of actual and potential infinities do not have any place in modern mathematics.

 

[weirdoguy]

Maybe, but I found them useful to stop thinking about $\infty$ as a number. Or maybe it was just the usage of word "potential" vs. "actual" that made it "clicked". Funny enough, I learned about them during lectures on calculus.

 

[jbriggs444]

Which are "actual infinities" whereas infinity denoted by $\infty$ and connected to limits is "potential".

To my way of thinking, the $\infty$ which appears in limits ($\lim_{x \to \infty}$) and integrals ($\int_0^\infty$) is not a potential infinity.

Instead, to the extent that we pretend that $\infty$ denotes an entity at all in such contexts, it is topological in nature. It is the positive element added in the two point compactification of the real numbers.

 

[Algr]

I've had this whole discussion before, and found the whole subject frustrating and a bit unnerving. The best thing to do is remember that this can never matter in the real world.

A roulette wheel, even a hypothetical one, can't have an infinite amount of spaces. At some point the spaces will be smaller than an atom, or a planck length, and at that point the equation 1/2 = 0 will be true in context. So, an infinite number of spaces can't happen no matter what you do, and any questions beyond that will yield nonsense.

 

[WWGD]

I've had this whole discussion before, and found the whole subject frustrating and a bit unnerving. The best thing to do is remember that this can never matter in the real world.

A roulette wheel, even a hypothetical one, can't have an infinite amount of spaces. At some point the spaces will be smaller than an atom, or a planck length, and at that point the equation 1/2 = 0 will be true in context. So, an infinite number of spaces can't happen no matter what you do, and any questions beyond that will yield nonsense.

Mathematics is not restricted by real-life constraints. That's for Physics, Mathematical Physics.

 

[pbuk]

You actually can. There are both Cardinal and Ordinal Arithmetic, though I assume you had a different intent in mind. Here infinity would be an equivalent class, represented by an element. Two sets are equivalent as Cardinals iff(def.) there is a bijection between them.

But we are dealing with probability theory here, and the axioms of probability do not admit cardinal or ordinal arithmetic, infinities, infinitessimals or anything other than $\mathbb R$. Please don't introduce things that will further confuse the OP.

Mathematics is not restricted by real-life constraints. That's for Physics, Mathematical Physics.

But mathematics is constrained by axioms, in this case the Kolomogrov Axioms which do not admit infinities (or infinitessimals). This means that we can have a mathematical model of a finite roulette wheel, but we cannot have a mathematical model of an infinite roulette wheel.

 

[WWGD]

Fair-enough. Maybe I did not consider the context enough.

 

[PeterDonis]

is one out of infinity different from zero?

You can't even formulate this question in a well-defined manner without having a number system in which "infinity" is well defined. Neither the integers nor the rationals nor the reals have this property. You would have to work with something like the hyperreals:

https://en.wikipedia.org/wiki/Hyperreal_number

one-in-infinity is interesting to me because one is infinitely far from its potential maximum

Not in the integers, rationals, or reals. In those number systems there is no "potential maximum" since infinity is not well-defined, and the number one is a finite distance from every other number.

which I think makes it infinitely small

No, one is not infinitely small in any number system. Not even in the hyperreals.

Is something that is infinitely small different from nothingness?

Again, you can't even formulate this question unless you are working with a number system that includes "infinitely small" quantities like the hyperreals. (In the hyperreals, the answer to your question is yes: there are infinitely small quantities that are different from zero.)

does that mean that every number is secretly zero?

No.

it just confuses me how being infinitely far from a maximum is different from being nothing.

The issues you are having are because you are using the term "infinity" without really understanding what it does or does not mean, or whether it actually corresponds to something you can reason about. This is a common pitfall caused by trying to use ordinary language to do precise reasoning, for which it is a very poor tool.

 

[PeterDonis]

mathematics is constrained by axioms, in this case the Kolomogrov Axioms which do not admit infinities (or infinitessimals)

Not all number systems have this constraint. But ones that don't, like the hyperreals, are much less well known.

 

[WWGD]

You can't even formulate this question in a well-defined manner without having a number system in which "infinity" is well defined. Neither the integers nor the rationals nor the reals have this property. You would have to work with something like the hyperreals:

https://en.wikipedia.org/wiki/Hyperreal_numberNot in the integers, rationals, or reals. In those number systems there is no "potential maximum" since infinity is not well-defined, and the number one is a finite distance from every other number.No, one is not infinitely small in any number system. Not even in the hyperreals.Again, you can't even formulate this question unless you are working with a number system that includes "infinitely small" quantities like the hyperreals. (In the hyperreals, the answer to your question is yes: there are infinitely small quantities that are different from zero.)No.The issues you are having are because you are using the term "infinity" without really understanding what it does or does not mean, or whether it actually corresponds to something you can reason about. This is a common pitfall caused by trying to use ordinary language to do precise reasoning, for which it is a very poor tool.

Such numbers exist, IIRC, within the Surreal numbers.

 

[Dale]

So is my problem that I'm treating infinity like I would any other number rather than treating it like what it is, which is confusing?

Yes, that is a problem. Infinity is not a real number. However, there are other number systems, like the hyperreals, where there are legitimate infinite numbers.

Now my question is, is one out of infinity different from zero?

If $\omega$ is a positive infinite hyperreal then $\epsilon=1/\omega$ is an infinitesimal hyperreal, and $0<\epsilon$. $\omega$ is larger than any real number and $\epsilon$ is smaller than any positive real number but larger than 0.

One subtlety is that “infinity” isn’t a single hyperreal number. There are many infinite hyperreals. For example, $\omega/2 < \omega -1 <\omega < \omega +1 < 2\omega$ are all infinite hyperreal numbers.

it just confuses me how being infinitely far from a maximum is different from being nothing

There is no maximum. As shown above, for any positive infinite hyperreal you can always make a larger number just by adding 1.

Is something that is infinitely small different from nothingness? If so, why and how, and if not,

The hyperreals obey the transfer principle. Any first order statement that is true about the reals is also true about the hyperreals.

If $x$ is any positive real number then $0<1/x$. So by the transfer principle, if $X$ is any positive hyperreal number then $0<1/X$. Since $\omega$ is a positive hyperreal number then $0<1/\omega=\epsilon$

 

[BWV]

But mathematics is constrained by axioms, in this case the Kolomogrov Axioms which do not admit infinities (or infinitessimals). This means that we can have a mathematical model of a finite roulette wheel, but we cannot have a mathematical model of an infinite roulette wheel.

What about an uncountably infinite one? Map (0,1] in R to a circle and offer generous odds to anyone wanting to bet on the wheel stopping on rational number

 

[PeterDonis]

Map (0,1) in R to a circle

You can't; (0, 1) is an open set but a circle is a closed set.

 

[WWGD]

You can't; (0, 1) is an open set but a circle is a closed set.

You can map it, but it won't be a homeomorphism. The two have the same cardinality, uncountable infinity, so a bijection exists, just not likely a continuous one.
. You can map (0,1] to the circle using $e^{i2\pi *t}$,
Can't think of which to use for (0,1).

 

[pbuk]

What about an uncountably infinite one? Map (0,1] in R to a circle

Roulette wheels are numbered with integers.

and offer generous odds to anyone wanting to bet on the wheel stopping on rational number

How would you prove to them that they had lost?

Again this is no help to the OP.

 

[jbriggs444]

What about an uncountably infinite one? Map (0,1] in R to a circle and offer generous odds to anyone wanting to bet on the wheel stopping on rational number

This one works. I believe that the key is countable additivity.

So you lay all of these real numbers in (0,1] out in a circular arrangement. You come up with a probability distribution such that the probability associated with each singleton set (such as {0.1}, {0.5} or {$\frac{1}{e}$}) is zero. This probability distribution meets our requirements for uniformity. The probability of getting any one result is the same as getting any other.

But does such a distribution obey the Kolmogorov axioms?

The key is the third axiom. If you have a bunch of mutually exclusive alternatives and ask about the probability of getting one of them, that is required to be the sum of the individual probabilities. So, naively thinking, we need to have the sum of all of those zero probabilities to add up to 1.

The thing is, the third axiom is only good for countable unions. Here we have an uncountable union. So there is, in fact, no requirement that all those zero probabilities add to 1. Or that their "sum" even be well defined.

 

[BWV]

This one works. I believe that the key is countable additivity.

So you lay all of these real numbers in (0,1] out in a circular arrangement. You come up with a probability distribution such that the probability associated with each singleton set (such as {0.1}, {0.5} or {$\frac{1}{e}$}) is zero. This probability distribution meets our requirements for uniformity. The probability of getting any one result is the same as getting any other.

But does such a distribution obey the Kolmogorov axioms?

The key is the third axiom. If you have a bunch of mutually exclusive alternatives and ask about the probability of getting one of them, that is required to be the sum of the individual probabilities. So, naively thinking, we need to have the sum of all of those zero probabilities to add up to 1.

The thing is, the third axiom is only good for countable unions. Here we have an uncountable union. So there is, in fact, no requirement that all those zero probabilities add to 1. Or that their "sum" even be well defined.

However if you take a subset that contains just the rational numbers it has measure zero - so the probability of selecting one rational number from the countably infinite set is zero per the OP (because the entire set has measure zero)?

 

[jbriggs444]

However if you take a subset that contains just the rational numbers it has measure zero - so the probability of selecting one rational number from the countably infinite set is zero per the OP (because the entire set has measure zero)?

The sum of countably many zeroes is zero, so yes, the probability of selecting a rational number from the distribution that we are discussing is necessarily zero.

However, this is not a consequence of the set having zero measure. One can have a set of measure zero without having that set be countable and without its probability (by some chosen distribution other than the Lebesgue measure) being zero.

The Cantor ternary set is the classic example of an uncountable set of measure zero.

 

[PeterDonis]

You can map it, but it won't be a homeomorphism.

Yes, agreed, I wasn't precise enough.

 

[WWGD]

Why not just use a RNG on the interval?

 

[PeterDonis]

Why not just use a RNG on the interval?

A RNG can't generate a random number out of an infinite set. Nor, if you are talking about the uncountable interval (0, 1), can it generate infinite decimals. A RNG can only randomly choose an element from a finite set.