Yes. It’s not the Choice axiom which is problematic, but the infinity itself. So it doesn’t mater if ZF or ZFC.
Why do I believe this? It’s known for some time now, that you can’t have an uniform probability distribution over the set of all naturals. That would be an express road to paradoxes.
The problem is, that even if you have a probability distribution where P(0)=0.5, P(1)=0.25, P(2)=0.125 and so on … you can then invite a super-task of swapping two random naturals (using this distribution) at the time 0. Then the next swapping at 0.5. Then the next swapping at 0.75 … and so on.
The question is, what is the probability that 0 will remain in its place? It can’t be more than 0, after the completion of the super-task after just a second. On the other hand, for every other number, that probability of being on the leftmost position is also zero.
We apparently can construct an uniform distribution over the naturals. Which is bad.
The limit of your distributions is not a distribution so there’s no problem.
If there’s any sort of inconsistency in ZF or PA or any other major system currently in use, it will be much harder to find than this. At a meta level, if there were this basic a problem, don’t you think it would have already been noticed?
If there’s any sort of inconsistency in ZF or PA or any other major system currently in use, it will be much harder to find than this.
Indeed, since you can prove ZFC consistent with the aid of an inaccessible cardinal. And you can prove the consistency of an inaccessible cardinal with a Mahlo cardinal, and so on.
I’m not sure that’s strong evidence for the thesis in question. If ZFC had a low-lying inconsistency, ZFC+an inaccessible cardinal would still prove ZFC consistent, but it would be itself an inconsistent system that was effectively lying to you. Same remarks apply to any large cardinal axiom.
What can one expect after this super-task is done to see?
This question presupposes that the task will ever be done. Since, if I understand correctly, you are doing an infinite number of swaps, you will never be done.
You could similarly define a super-task (whatever that is) of adding 1 to a number. Start with 0, at time 0 add 1, add one more at time 0.5, and again at 0.75. What is the value when you are done? Clearly you are counting to infinity, so even though you started with a natural number, you don’t end up with one. That is because you don’t “end up” at all.
What you are doing in many ways amounts to the 18th and early 19th century arguments over whether 1-1+1-1+1-1… converged and if so to what. First formalize what you mean, and then get an answer. And a rough intuition of what should formally work that leads to a problem is not at all the same thing as an inconsistency in either PA or ZFC.
This question presupposes that the task will ever be done
Sure. It’s called super-tasks.
From mathematics we know that not all sequences converge. So the sequence of distributions that you gave, or my example of the sequence 0,1,2,3,4,… both don’t converge. Calling them a supertask doesn’t change that fact.
What mathematicians often do in such cases is to define a new object to denote the hypothetical value at the end of sequence. This is how you end up with real numbers, distributions (generalized functions), etc. To be fully formal you would have to keep track of the sequence itself, which for real numbers gives you Cauchy sequences for instance. In most cases these objects behave a lot like the elements of the sequence, so real numbers are a lot like rational numbers. But not always, and sometimes there is some weirdness.
From the wikipedia link:
In philosophy, a supertask is a countably infinite sequence of operations that occur sequentially within a finite interval of time.
This refers to something called “time”. Most of mathematics, ZFC included, has no notion of time. Now, you could take a variable, and call it time. And you can say that a given countably infinite sequences “takes place” in finite “time”. But that is just you putting semantics on this sequence and this variable.
So the sequence of distributions that you gave, or my example of the sequence 0,1,2,3,4,… both don’t converge. Calling them a supertask doesn’t change that fact.
Your question of “after finishing the supertask, what is the probability that 0 stays in place” doesn’t yet parse as a question in ZFC, because you haven’t specified what is meant by “after finishing the supertask”. You need to formalize this notion before we can say anything about it.
If you’re saying that there is no formalization you know of that makes sense in ZFC, then that’s fine, but that’s not necessarily a strike against ZFC unless you have a competitive alternative you’re offering. The problem could just be that it’s an ill-defined concept to begin with, or you just haven’t found a good formalization. Just because your brain says “that sounds like it make sense”, doesn’t mean it actually makes sense.
To show that ZFC is inconsistent, you would need to display a formal contradiction deduced from the ZFC axioms. “I can’t write down a formalization of this natural sounding concept” isn’t a formal contradiction; the failure is at the modeling step, not inside the logical calculus.
This is a sequence of natural numbers. This sequence does not converge, which means that the limit as n goes to infinite of S(n) is not a natural number (nor a real number for that matter).
You could try to write it as a function of time, S’(t) such that S’(1-0.5^n) = S(n). That is, S’(0)=0, S’(0.5)=1, S’(0.75)=2, etc. A possible formula is S’(t) = -log_2(1-t). You could then ask what is S’(1). The answer is that this is the same as the limit S(infinity), or as log(0), which are both not defined. So in fact S’ is not a function from numbers between 0 and 1 inclusive to natural or real numbers, since the domain excludes 1.
You can similarly define a sequence of distributions over the natural numbers by
T(0) = {i -> 0.5 * 0.5^i}
T(n+1) = the same as T(n) except two values swapped
This is the example that you gave above. The sequence T(n) doesn’t converge (I haven’t checked, but the discussion above suggests that it doesn’t), meaning that the limit “lim_{n->inf} T(n)” is not defined.
Thomas, please read and understand query’s response above. In attempting to dismantle a concept you don’t like, you’ve lost precision. Formalize your questions and concerns rigorously and then see if a seeming contradiction is still there.
Phrasing it as a “super-task” relies on intuitions that are not easily formalized in either PA or ZFC. Think instead in terms of a limit, where your nth distribution and let n go to infinity. This avoids the intuitive issues. Then just ask what mean by the limit. You are taking what amounts to a pointwise limit. At this point, what matters then is that it does not follow that a pointwise limit of probability distributions is itself a probability distribution.
If you prefer a different example that doesn’t obfuscate as much what is going on we can do it just as well with the reals. Consider the situation where the nth distribution is uniform on the interval from n to n+1. And look at the limit of that (or if you insist move back to having it speed up over time to make it a supertask). Visually what is happening each step is a little 1 by 1 square moving one to the right. Now note that the limit of these distributions is zero everywhere, and not in the nice sense of zero at any specific point but integrates to a finite quantity, but genuinely zero.
This is essentially the same situation, so nothing in your situation has to do with specific aspects of countable sets.
Wildberger’s complaints are well known, and frankly not taking very seriously. The most positive thing one can say about it is that some of the ideas in his rational trignometry do have some interesting math behind them, but that’s it. Pretty much no mathematican who has listened to what he has to say have taken any of it seriously.
Sure, I know he is not taken very seriously. That is his own point, too.
In the time of Carl Sagan, in the year 1986 or so, I became an anti Saganist. I realized that his million civilization in our galaxy alone is an utter bullshit. Most likely only one exists.
Every single astro-biologist or biologist would have said to a dissident like myself—you don’t understand evolution, sire, it’s mandatory!
20 years later, on this site, Rare Earth is a dominant position. Or at least—no aliens position.
On the National Geographic channel and elsewhere, you still listen “how previously unexpected number of Earth like planets will be detected”.
I am not afraid of mathematicians more than of astrobiologists. Largely unimpressed.
I’m not sure what your point is here. Yes, experts sometimes have a consensus that turns out to be wrong. If one is lucky one can even turn out to be right when the experts are wrong if one takes sufficiently many contrarian positions (although the idea that many millions of civilizations in our galaxy was a universal among both biologists and astro-biologists is definitely questionable), but in this case, the experts have really thought about these ideas a lot, and haven’t gotten anywhere.
If you prefer an example other than Wildberger, when Edward Nelson claimed to have a contradiction in PA, many serious mathematicians looked at what he had done. It isn’t like there’s some special mathematical mob which goes around suppressing these things. I literally had a lunch-time conversation a few days ago with some other mathematician where the primary topic was essentially if there is an inconsistency in ZFC where would we expect to find it and how much of math would likely be salvageable? In fact, that conversation was one of the things that lead me along to the initial question in this subthread.
I am not afraid of mathematicians more than of astrobiologists. Largely unimpressed.
Neither of these groups are groups you should be afraid of and I’m a little confused as why you think fear should be relevant.
I’m not sure why you think that. This may depend strongly on what you mean by an in infinitary method. Is induction infinitary? Is transfinite induction infinitary?
Physics is only good, when you expel all the infinities out of it.
I’m not sure what you mean by this, and in so far as I can understand it doesn’t seem to be true. Physicists use the real numbers all the time which are an infinite set. They use integration and differentiation which involves limits. So what do you mean?
I’m not sure what you mean by this, and in so far as I can understand it doesn’t seem to be true. Physicists use the real numbers all the time which are an infinite set.
I’m not sure what you mean by this, and in so far as I can understand it doesn’t seem to be true. Physicists use the real numbers all the time which are an infinite set.
The problem there is that certain specific models of physics end up giving infinite values for measurable quantities—this is a known problem and has been an area of active research since early work with renormalization in the 1930s. This is not at all an attempt to banish infinity in any general sense.
Now, when there in no God, the Infinity is its substitute, most people would love to exist. But it’s just another blunder.
This is not at all an attempt to banish infinity in any general sense.
Of course it is. Nothing infinite has been spotted so far.
This is rhetoric without content.
Is it? Is this same “rhetoric” against aliens also without a content? If I say that people want aliens, because they have lost angels, is this really without a content?
Not only that there is no infinite God, even infinite sets are probably just a miracle.
This is not at all an attempt to banish infinity in any general sense.
Of course it is. Nothing infinite has been spotted so far.
I’m not sure how your sentence is a response to my sentence.
This is rhetoric without content.
Is it? Is this same “rhetoric” against aliens also without a content? If I say that people want aliens, because they have lost angels, is this really without a content?
Not only that there is no infinite God, even infinite sets are probably just a miracle.
Generally, yes, the content level is pretty low. It essentially amounts to Bulverism, where one is focusing on claimed intents and motives rather than focusing on the substantive issue of whether there’s an inconsistency in PA or ZFC that can arise due to issues with supertasks or other ideas related to infinity.
It may well be that specific people or groups are adopted aliens in a way that is essentially replacing deities. The Raelians and other New Age groups certainly fall into that categoyr. But it is a mistake to therefore claim that in general, people believe in aliens as a replacement for belief in a deity. And it is an even more serious mistake to make such claims about infinite sets. If you see physicists praying to infinite sets, or claiming that infinite sets are responsible for the creation of the universe or humanity, or claim that infinite sets will somehow save us, or claim that infinite sets have an agency to them, or claim that infinite sets have a special mystery and majesty to them that merits worship, or if they start wars with or excommunicate people who don’t believe in infinite sets or believe in a different type of infinite set, then there would be an argument.
I don’t give a damn about infinity. If it is doable, why not? But is it? That’s the only question.
Then, a supertask mixes the infinite set of naturals and we are witnessing “the irresistible force acting on an unmovable object”. What the Hell will happen? Will we have finite numbers on the first 1000 places? We should, but bigger, no matter which will be.
The “irresistible force” is just an empty word. And so is “unmovable object”. And so is “infinity” and so is “supertask”.
Empty words. So every theory which encompasses them is flawed. More than likely.
There is an argument there, but it certainly is not one based on ZFC, since no axiom of set theory says anything about time or what can be accomplished in time.
Physics is built on top of mathematics, and almost all of mathematics can be built on top of ZFC (there are other choices). But there is as much time in ZFC as there are words in a single pixel on your screen.
I don’t give a damn about infinity. If it is doable, why not? But is it? That’s the only question.
I’m not sure what you mean by this, especially given your earlier focus on whether infinity exists and whether using it in physics is akin to religion. I’m also not sure what “it” is in your sentence, but it seems to be the supertask in question. I’m not sure in that context what you mean by “doable.”
Then, a supertask mixes the infinite set of naturals and we are witnessing “the irresistible force acting on an unmovable object”. What the Hell will happen? Will we have finite numbers on the first 1000 places? We should, but bigger, no matter which will be.
The “irresistible force” is just an empty word. And so is “unmovable object”. And so is “infinity” and so is “supertask”.
I’m not at all sure what this means. Can you please stop using analogies can make a specific example of how to formalize this contradiction in ZFC?
This seems to be essentially the same argument and it seems like the exact same problem: an assumption that an intuitive limit must exist. Limits don’t always exist when you want them to, and we have a lot of theorems about when a point-wise limit makes sense. None of them apply here.
I don’t think this conversation is being very productive so this is likely my final reply.
Just answer me a simple question.
? How do the first 1000 naturals look like, after mixing supertask described above has finished its job,
You may say that this supertask is impossible.
You may say that there is no set of all naturals.
The resulting pointwise limit exists, and it gives each positive integer a probability of zero. This is fine because the pointwise limit of a distribution on a countable set is not necessarily itself a distribution. Please take a basic real analysis course.
Why do you believe that? And do you also believe that ZF is inconsistent?
Yes. It’s not the Choice axiom which is problematic, but the infinity itself. So it doesn’t mater if ZF or ZFC.
Why do I believe this? It’s known for some time now, that you can’t have an uniform probability distribution over the set of all naturals. That would be an express road to paradoxes.
The problem is, that even if you have a probability distribution where P(0)=0.5, P(1)=0.25, P(2)=0.125 and so on … you can then invite a super-task of swapping two random naturals (using this distribution) at the time 0. Then the next swapping at 0.5. Then the next swapping at 0.75 … and so on.
The question is, what is the probability that 0 will remain in its place? It can’t be more than 0, after the completion of the super-task after just a second. On the other hand, for every other number, that probability of being on the leftmost position is also zero.
We apparently can construct an uniform distribution over the naturals. Which is bad.
The limit of your distributions is not a distribution so there’s no problem.
If there’s any sort of inconsistency in ZF or PA or any other major system currently in use, it will be much harder to find than this. At a meta level, if there were this basic a problem, don’t you think it would have already been noticed?
Indeed, since you can prove ZFC consistent with the aid of an inaccessible cardinal. And you can prove the consistency of an inaccessible cardinal with a Mahlo cardinal, and so on.
I’m not sure that’s strong evidence for the thesis in question. If ZFC had a low-lying inconsistency, ZFC+an inaccessible cardinal would still prove ZFC consistent, but it would be itself an inconsistent system that was effectively lying to you. Same remarks apply to any large cardinal axiom.
What can one expect after this super-task is done to see?
Nothing?
It has been noticed, but never resolved properly. A consensus among top mathematicians, that everything is/must be okay prevails.
One dissident.
https://www.youtube.com/watch?t=27&v=4DNlEq0ZrTo
This question presupposes that the task will ever be done. Since, if I understand correctly, you are doing an infinite number of swaps, you will never be done.
You could similarly define a super-task (whatever that is) of adding 1 to a number. Start with 0, at time 0 add 1, add one more at time 0.5, and again at 0.75. What is the value when you are done? Clearly you are counting to infinity, so even though you started with a natural number, you don’t end up with one. That is because you don’t “end up” at all.
Sure. It’s called super-tasks.
https://en.wikipedia.org/wiki/Supertask
“a supertask is a countably infinite sequence of operations that occur sequentially within a finite interval of time.”
You can’t avoid supertasks, when you endorse infinity.
Therefore, I don’t.
What you are doing in many ways amounts to the 18th and early 19th century arguments over whether 1-1+1-1+1-1… converged and if so to what. First formalize what you mean, and then get an answer. And a rough intuition of what should formally work that leads to a problem is not at all the same thing as an inconsistency in either PA or ZFC.
There are no axioms of ZFC that imply that such a task can be completed.
From mathematics we know that not all sequences converge. So the sequence of distributions that you gave, or my example of the sequence 0,1,2,3,4,… both don’t converge. Calling them a supertask doesn’t change that fact.
What mathematicians often do in such cases is to define a new object to denote the hypothetical value at the end of sequence. This is how you end up with real numbers, distributions (generalized functions), etc. To be fully formal you would have to keep track of the sequence itself, which for real numbers gives you Cauchy sequences for instance. In most cases these objects behave a lot like the elements of the sequence, so real numbers are a lot like rational numbers. But not always, and sometimes there is some weirdness.
From the wikipedia link:
This refers to something called “time”. Most of mathematics, ZFC included, has no notion of time. Now, you could take a variable, and call it time. And you can say that a given countably infinite sequences “takes place” in finite “time”. But that is just you putting semantics on this sequence and this variable.
I don’t understand you.
Your question of “after finishing the supertask, what is the probability that 0 stays in place” doesn’t yet parse as a question in ZFC, because you haven’t specified what is meant by “after finishing the supertask”. You need to formalize this notion before we can say anything about it.
If you’re saying that there is no formalization you know of that makes sense in ZFC, then that’s fine, but that’s not necessarily a strike against ZFC unless you have a competitive alternative you’re offering. The problem could just be that it’s an ill-defined concept to begin with, or you just haven’t found a good formalization. Just because your brain says “that sounds like it make sense”, doesn’t mean it actually makes sense.
To show that ZFC is inconsistent, you would need to display a formal contradiction deduced from the ZFC axioms. “I can’t write down a formalization of this natural sounding concept” isn’t a formal contradiction; the failure is at the modeling step, not inside the logical calculus.
Define the sequence S by
This is a sequence of natural numbers. This sequence does not converge, which means that the limit as n goes to infinite of S(n) is not a natural number (nor a real number for that matter).
You could try to write it as a function of time, S’(t) such that S’(1-0.5^n) = S(n). That is, S’(0)=0, S’(0.5)=1, S’(0.75)=2, etc. A possible formula is S’(t) = -log_2(1-t). You could then ask what is S’(1). The answer is that this is the same as the limit S(infinity), or as log(0), which are both not defined. So in fact S’ is not a function from numbers between 0 and 1 inclusive to natural or real numbers, since the domain excludes 1.
You can similarly define a sequence of distributions over the natural numbers by
This is the example that you gave above. The sequence T(n) doesn’t converge (I haven’t checked, but the discussion above suggests that it doesn’t), meaning that the limit “lim_{n->inf} T(n)” is not defined.
Thomas, please read and understand query’s response above. In attempting to dismantle a concept you don’t like, you’ve lost precision. Formalize your questions and concerns rigorously and then see if a seeming contradiction is still there.
Phrasing it as a “super-task” relies on intuitions that are not easily formalized in either PA or ZFC. Think instead in terms of a limit, where your nth distribution and let n go to infinity. This avoids the intuitive issues. Then just ask what mean by the limit. You are taking what amounts to a pointwise limit. At this point, what matters then is that it does not follow that a pointwise limit of probability distributions is itself a probability distribution.
If you prefer a different example that doesn’t obfuscate as much what is going on we can do it just as well with the reals. Consider the situation where the nth distribution is uniform on the interval from n to n+1. And look at the limit of that (or if you insist move back to having it speed up over time to make it a supertask). Visually what is happening each step is a little 1 by 1 square moving one to the right. Now note that the limit of these distributions is zero everywhere, and not in the nice sense of zero at any specific point but integrates to a finite quantity, but genuinely zero.
This is essentially the same situation, so nothing in your situation has to do with specific aspects of countable sets.
Wildberger’s complaints are well known, and frankly not taking very seriously. The most positive thing one can say about it is that some of the ideas in his rational trignometry do have some interesting math behind them, but that’s it. Pretty much no mathematican who has listened to what he has to say have taken any of it seriously.
Sure, I know he is not taken very seriously. That is his own point, too.
In the time of Carl Sagan, in the year 1986 or so, I became an anti Saganist. I realized that his million civilization in our galaxy alone is an utter bullshit. Most likely only one exists.
Every single astro-biologist or biologist would have said to a dissident like myself—you don’t understand evolution, sire, it’s mandatory!
20 years later, on this site, Rare Earth is a dominant position. Or at least—no aliens position.
On the National Geographic channel and elsewhere, you still listen “how previously unexpected number of Earth like planets will be detected”.
I am not afraid of mathematicians more than of astrobiologists. Largely unimpressed.
I’m not sure what your point is here. Yes, experts sometimes have a consensus that turns out to be wrong. If one is lucky one can even turn out to be right when the experts are wrong if one takes sufficiently many contrarian positions (although the idea that many millions of civilizations in our galaxy was a universal among both biologists and astro-biologists is definitely questionable), but in this case, the experts have really thought about these ideas a lot, and haven’t gotten anywhere.
If you prefer an example other than Wildberger, when Edward Nelson claimed to have a contradiction in PA, many serious mathematicians looked at what he had done. It isn’t like there’s some special mathematical mob which goes around suppressing these things. I literally had a lunch-time conversation a few days ago with some other mathematician where the primary topic was essentially if there is an inconsistency in ZFC where would we expect to find it and how much of math would likely be salvageable? In fact, that conversation was one of the things that lead me along to the initial question in this subthread.
Neither of these groups are groups you should be afraid of and I’m a little confused as why you think fear should be relevant.
I doubt that any proof in FAI will use infinitary methods.
I’m not sure why you think that. This may depend strongly on what you mean by an in infinitary method. Is induction infinitary? Is transfinite induction infinitary?
Physics is only good, when you expel all the infinities out of it.
Even more so for a subset of physics, such as FAI or molecular dynamics or something.
Well, some of us think that this should be applied to the mathematics itself.
I’m not sure what you mean by this, and in so far as I can understand it doesn’t seem to be true. Physicists use the real numbers all the time which are an infinite set. They use integration and differentiation which involves limits. So what do you mean?
https://physics.aps.org/articles/v2/70
http://blogs.discovermagazine.com/crux/2015/02/20/infinity-ruining-physics/#.Vh0LnHqqpBc
Now, when there in no God, the Infinity is its substitute, most people would love to exist. But it’s just another blunder.
The problem there is that certain specific models of physics end up giving infinite values for measurable quantities—this is a known problem and has been an area of active research since early work with renormalization in the 1930s. This is not at all an attempt to banish infinity in any general sense.
This is rhetoric without content.
Of course it is. Nothing infinite has been spotted so far.
Is it? Is this same “rhetoric” against aliens also without a content? If I say that people want aliens, because they have lost angels, is this really without a content?
Not only that there is no infinite God, even infinite sets are probably just a miracle.
I’m not sure how your sentence is a response to my sentence.
Generally, yes, the content level is pretty low. It essentially amounts to Bulverism, where one is focusing on claimed intents and motives rather than focusing on the substantive issue of whether there’s an inconsistency in PA or ZFC that can arise due to issues with supertasks or other ideas related to infinity.
It may well be that specific people or groups are adopted aliens in a way that is essentially replacing deities. The Raelians and other New Age groups certainly fall into that categoyr. But it is a mistake to therefore claim that in general, people believe in aliens as a replacement for belief in a deity. And it is an even more serious mistake to make such claims about infinite sets. If you see physicists praying to infinite sets, or claiming that infinite sets are responsible for the creation of the universe or humanity, or claim that infinite sets will somehow save us, or claim that infinite sets have an agency to them, or claim that infinite sets have a special mystery and majesty to them that merits worship, or if they start wars with or excommunicate people who don’t believe in infinite sets or believe in a different type of infinite set, then there would be an argument.
I don’t give a damn about infinity. If it is doable, why not? But is it? That’s the only question.
Then, a supertask mixes the infinite set of naturals and we are witnessing “the irresistible force acting on an unmovable object”. What the Hell will happen? Will we have finite numbers on the first 1000 places? We should, but bigger, no matter which will be.
The “irresistible force” is just an empty word. And so is “unmovable object”. And so is “infinity” and so is “supertask”.
Empty words. So every theory which encompasses them is flawed. More than likely.
And yes, supertask can be established in ZFC.
The topic is also exercised here:
http://mathforum.org/kb/thread.jspa?forumID=13&threadID=2278300&messageID=7498035
There is an argument there, but it certainly is not one based on ZFC, since no axiom of set theory says anything about time or what can be accomplished in time.
So you say, ZFC has nothing to do with time? Time in physics is uncovered in ZFC?
Physics is built on top of mathematics, and almost all of mathematics can be built on top of ZFC (there are other choices). But there is as much time in ZFC as there are words in a single pixel on your screen.
I’m not sure what you mean by this, especially given your earlier focus on whether infinity exists and whether using it in physics is akin to religion. I’m also not sure what “it” is in your sentence, but it seems to be the supertask in question. I’m not sure in that context what you mean by “doable.”
I’m not at all sure what this means. Can you please stop using analogies can make a specific example of how to formalize this contradiction in ZFC?
This seems to be essentially the same argument and it seems like the exact same problem: an assumption that an intuitive limit must exist. Limits don’t always exist when you want them to, and we have a lot of theorems about when a point-wise limit makes sense. None of them apply here.
Just answer me a simple question.
How do the first 1000 naturals look like, after mixing supertask described above has finished its job,
You may say that this supertask is impossible.
You may say that there is no set of all naturals.
Whatever you think about it. Everything else is pretty redundant.
I don’t think this conversation is being very productive so this is likely my final reply.
The resulting pointwise limit exists, and it gives each positive integer a probability of zero. This is fine because the pointwise limit of a distribution on a countable set is not necessarily itself a distribution. Please take a basic real analysis course.