That there are infinities of various sizes follows if you accept Hume's Principle, which says "the number of things with the property F equal the number of things with the property G if and only if there is a one-to-one correspondence between those that are F and those that are G."
Cantor and Frege adopted this definition of "the same size as", although already Galileo argued that it would lead to absurd consequences when applied to infinities (there would be as many square numbers as natural numbers, even though not all natural numbers are square), which is known as Galileo's Paradox.
For finite numbers any one-to-one correspondence between F and G means that neither can be a proper subset of the other, which seems just as plausible a requirement for "the same size as" as the former. Since the two requirements come apart for infinite sets, it is unclear which to keep, or whether size comparisons even make any sense for infinities. Galileo concludes they don't make sense.
Hume's Principle is actually not uncontroversial among philosophers of mathematics, but many people treat it as some kind of objective fact rather than a proposed conceptual analysis of "the same size as".
You have a Hotel with infinite rooms in it. An Infinite Bus shows up with infinite passengers. As each passenger comes into the lobby, you give them the next room key, so that everyone gets a room.
An hour later another Infinite Bus shows up. Oh dear. Now what will you do? You call the manager and he says no problems. What you're going to do is go to room 1, apologize and ask them to kindly relocate to room 2. Offer a complimentary breakfast by way of apology. Then go to each subsequent room and ask the inhabitants to move from their room number to 2n. Now that all of the current guests have moved to their new, even numbered rooms, you can put the second bus into the odd numbered rooms and everyone is happy.
As Standup Maths recently pointed out, the problem is that we think of infinity as "count until you get really tired of counting and then it's one more than that". Which is just not a workable definition and causes problems.
This is an interesting idea but I don't get how it works in practice. For instance, the two sets {1,2,3,4} and {A,B,C} are such that none is equal to a proper subset of the other. Are you suggesting these be treated as "the same size as" one another? Because if so, then {1,2,3,4} is "the same size as" both {A,B,C} and {A,B,C,D}, even though {A,B,C} is not "the same size as" {A,B,C,D}.
On the other hand, we can say that although {A,B,C} is not equal to any proper subset of {1,2,3,4}, it is in bijection with a proper subset, such as {1,2,3}. Although this is true, if we use this definition of "the same size as," we again get that the naturals are "the same size as" the set of all squares.
Being able to understand the way in which mathematical usage of phrases like “the same size as” is different than intuitive uses of the same phrase is becoming increasingly important. Only six months ago I was seeing a dozen YouTube videos a day claiming g “Physics proved that the universe is not real!” because some physicist decided to overload the “real” operator in the English language.
> That there are infinities of various sizes follows if you accept Hume's Principle, which says "the number of things with the property F equal the number of things with the property G if and only if there is a one-to-one correspondence between those that are F and those that are G."
I don't think that accepting(?) principles(?) is the right way to think about it.
This ordering on this family of infinities is as much of a definition as everything else. You don't accept principles when you talk about matrices, or circles or rings.
It always made more sense to me to define things as "one to one AND that one set's of elements is not absolutely contained in the other set." This would make the integers bigger than the even integers because 2Z in Z but would mean that the evens and the odds are the same size, since they are disjoint sets. Though both are obviously categorically countable.
But I'm not a (set) theorist and I'm sure this has been explored before. Does anyone have a link and/or concise explanation? (papers are fine) Would this definition result in similarity weird conclusions? (Infinities are weird)
It seems like Hume's Principle is only a matter of semantics, not of mathematics. The concept of one-to-one correspondence is useful even if we don't refer to it as size. All existing proofs would be equally valid even if we didn't accept this definition of 'size', since it would still be true that there is or isn't a correspondence between two sets, even if we refer to it in a different way.
If mathematicians have to choose between two definitions where one results in the task not making sense, and the other gives rise to a new theory with nontrivial open problems, the choice is easy :)
I think the mystery is not that some infinities can be larger than others, but that there are infinite sets of equal ‘size’ that conflict with intuition.
Some examples that ‘prove’ some infinities are larger than others to laymen:
- there are twice as many integers as odd integers
- there are more points on a plane then on a line
- there are more points on a line than on a circle
- there are more points on a plane then on a semiplane
- there are more rationals than integers
- there are more reals than rationals
It’s only in the intermediate state of a mathematician’s education, where they have just accepted that, for infinite sets, ‘more’ isn’t the best way to determine size equivalence that it becomes a surprise that for the last one, “the size of the set of reals is larger than that of the rationals” is true, and can be proven to be.
It's been a long time since my number theory class, but aren't there the same number of integers and odd integers?
Take the set of odd integers {... -3, -1, 1, 3, 5, ...}
For each item, subtract one and divide the result by 2. Now you have the set of all integers without any insertions or deletions: {... -2, -1, 0, 1, 2, ...}
Therefore the set of odd integers can be mapped 1:1 onto the set of all integers so they are of equal length.
Isn't that more like first-year material in mathematics and in more theoretically oriented CS programs? Once you start talking about injections, surjections, and bijections, you may as well prove some basic results about the sizes of sets of numbers.
> there are twice as many integers as odd integers
> there are more rationals than integers
Both of these are incorrect you can create a 1-1 mapping between all of these sets so they are the same "size". Things get unintuitive when you're dealing with infinites things that feel like they should be larger aren't when you examine them rigorously.
For integers to odd integers the mapping is easy for each natural number n map n -> 2n+1. Mapping integers to rational numbers is more difficult to write into an equation but if you lay them out in a a grid defined by numerators and denominators x/y you can snake along this grid to eventually map every rational number to a corresponding natural number (ie positive integers which has the same cardinality as integers).
Going further R (points on a line) to R^2 (points on a plane) is also the same cardinality. The proofs are over my head as a 10 years past math minor but they're out there. Including this one that goes from R^3 to R.
You need to consider the term “bijection” and what it means in practice. If I create a bijection of integers to integers f(x)=2x you can see that each member of the right side had exactly one member on the right, and vice versa, despite the left side containing some elements not on the right. There is no such bijection between ints and reals (by cantors theorem), but there is clearly a surjection of reals to ints (round down), so reals are of a larger infinite size.
I think a layman could easily understand this personally, without fancy math terms like bijection or surjection, the main exception being Cantors Theorem which fortunately has a plethora of nice visual proofs.
Anyway, the article is not really even about that but about potentially modifying ZFC to accommodate a particular outcome of the CH. I am quite interested in mathematics but to me this is a pointless philosophical argument because there don’t seem to be any useful or empirically verifiable results that depend on CH. Discussing CH to me seems to be more of a way for set theorists to troll each other and trigger cranks.
Some of those infinities are the same, others are not.
There's a one to one mapping between the integers and the cartesian product of the integers (I^I == I), but (I believe) that's not true for the reals (someone correct me if I'm wrong about that).
However, you can take some subset of R^n and stretch it to infinity across all dimensions, giving you back R^n.
So:
There are the same number of integers as odd integers,
There are more points on a continuous plane than on a continuous line.
The number of points on a line is the same as the number of points on the edge of a circle.
The number of points in a plane is the same as the number of points in a semiplane.
There are the same number of rationals as integers.
Half your examples are wrong, but maybe it is your point. Although it wasn't clear to me the first time I read your comment.
- The cardinality of the odd and even integers is the same.
- It is true there are more points on a plane then on a line (Cantor's theorem.)
- The circle is the compactified real line, i.e. it can be represented as the real numbers with one additional point (the point at infinity). In terms of cardinality they are the same since they just differ by one point which does not change the cardinality.
- There are not more points on a plane than a half-plane, you can find a bijective mapping between them easily.
- There are more rationals than integers: not true, they are both countable sets of the same cardinality.
- There are more reals than rationals, this is true (again Cantor.)
I've found it easier to reason about when I think of it as "how quickly does the set of number grow as you add more to it". It becomes particularly apparent when you think about it on a finite scale.
0 to 100
* 100 integers
* 50 odd numbers, 50 even numbers
* infinite floats (unless you only represent part of the float)
Standup Maths recently tackled, "Is an infinite stack of ￡1 notes worth more than an infinite stack of ￡100 notes?" The answer is they are both worth ￡∞.
I have no problem with the concept of cardinality of infinite sets. If you want to talk about the ability to map infinite sets, so that the cardinality of integers and even integers is the same, then great.
What drives me bananas is when anybody starts using the words "size" or "larger" or "smaller". I will insist to my dying day that while the cardinality of even integers is the same as that of integers, the set of even integers is still smaller than the set of integers. That's it's still half the size.
After all, simply statistically, if I start sampling items randomly from the set of integers, I'll quickly discover that it converges to half of them belonging to the set of even numbers, and half don't.
And yes I know there are supposed theoretical problems with random sampling from an infinite set but honestly I don't care. Pick any large bound you want from the set of integers, whether it's from 1 million to 10 million, or negative a trillion to positive a trillion trillion trillion. It's always going to converge to integers being twice the size of even integers.
I mean if we can deal with ratios in calculus down to infinitesimal sizes using limits, we can sure as heck go the other way, the limit as the bounds go to infinity and the proportion still continues to hold perfectly.
Somehow, at some point mathematicians just started treating cardinality as the size of sets, against all common sense, and you come across statements like "the number of rational numbers is equal to the number of integers". Nonsense. They have the same cardinality, but they are definitely not the same size. It's simply mathematical gaslighting and yes, I will die on this hill! :)
> Pick any large bound you want from the set of integers, whether it's from 1 million to 10 million, or negative a trillion to positive a trillion trillion trillion. It's always going to converge to integers being twice the size of even integers.
But how can there be more integers than the are even integers when for each and every even integer there exists a corresponding integer? And how can there be more rationals, when we can find an integer for every rational number?
You are thinking in terms of very large sets, but literally infinite sets are simply not the same as very large finite sets. And they are not even the same as the limits of cardinality N sets as N grows to infinity.
For example, if we take the set of all integers less than N and the set of all even integers less than N, we can show that no mapping can exist between the two, even as N grows to infinity - so, one is indeed larger than the other. And yet, a mapping appears if N is literally infinite.
Yes, if you tautologically select a finite subset of an infinite set, it tautologically has a comparable size which doesn’t equal the same constraints to select a finite subset of another infinite set. It would be a little bit more accurate to describe these comparisons as “faster” vs “slower” (ie you’ll reach the end of the conceptual subset of integers “before” you reach the end of the conceptual subset of even integers), but that belies the tautology.
> Pick any large bound you want from the set of integers, whether it's from 1 million to 10 million, or negative a trillion to positive a trillion trillion trillion. It's always going to converge to integers being twice the size of even integers.
I think you have to understand this: Even the largest of numbers you can fathom (ie: 10e5000, as in gazillions of trillions) is still closer to 0 than to infinity.
We know the set is smaller (or half) for a determined infinity. What we don't know, is how big the size is going to be when you "go" to infinity.
I've also been convinced that any comparative might have issues. Let's take the set of all primes.
On the one hand, I can trivially demonstrate they are less numerous than the set of integers
On the other hand, Cantor's exercises for showing larger cardinality can be exercised equally on the set of prime numbers
At least as far as I've been able to see. This shows a contradiction that can be easily accommodated for with multi-dimensional or an otherwise richer descriptor set. Any refutation that starts like "assume there exist a function f that generates all the primes as a sequence of integers" I reject. The whole point is that I'm not assuming that - maybe a proof that no such function can exist for the set of primes is an undiscovered property.
If you decide to be extremely disagreeable and stubbornly throw out any speculative, at least I'm left with only with the contradiction. They seem to behave way closer to the irrationals then anything else.
I think a more useful exercise is whether the peano successor function can be defined. For the irrationals, I think the answer is "not a chance". For the primes the intuition is "maybe" but when you start stating why, nearly every argument you give (statistically you can do it - with enough information you can do it, it is a discrete number and so on) you can defend for the irrationals with the same reasons.
>> if we can deal with ratios in calculus due to infinitesimal sizes using limits
Cauchy proved this in the 19th century using analysis. You did not cite a proof that we can somehow use analysis to do something vaguely similar for infinities.’
>> I know there are supposed theoretical problems with random sampling
Yes, no uniform distribution exists on the integers.
>> But I don’t care
Clearly. It drives me bananas when people knowingly post comments in mathematical discussion that have no rigor and instead are appeals to emotion.
> What drives me bananas is when anybody starts using the words "size" or "larger" or "smaller". I will insist to my dying day that while the cardinality of even integers is the same as that of integers, the set of even integers is still smaller than the set of integers. That's it's still half the size.
Note that size of a set has nothing to do with individual set members and subset relationship. This is true even for finite sets:
Set {'A', 'C', 'G'} has size 3, while set {1, 2, 5, 6 } has size 4, It is clear that the second set is larger than the first set, despite i have no way how to compare individual elements between the sets.
But what does even mean that set {'A', 'C', 'G'} has size 3? We get such result by counting the elements, which is just process of defining one-to-one mapping to {0, 1, 2} (or traditionally {1, 2, 3}, it does not matter), canonical representation of size 3.
From this point of view, the concept of infinite cardinality is just a natural extension of this.
One basic misconception people often have is that the subset relation implies smaller cardinality (i.e. "size"). E.g., all odd integers are integers, so there are more integers that odd integers. But this is not a very useful notion of size. For example, how would we compare the set of multiples of two with the set of multiples of three? We want a notion of size independent of the underlying "objects" in our sets. This is why we define cardinality in terms of mappings between sets.
I'd recommend "Journey through genius" by William Dunham. There's a bunch of great stuff in there including two chapters about Georg Cantor and his explorations of infinity, including how he showed there is no 1:1 correspondence between integers and real numbers, while there is one between integers and rationals.
There are many comments saying that one infinity can be larger than another because a bijective mapping can't be formed, but why does the presence of a mapping imply anything about the "size" of an infinity? For any infinite set, you could select unique values out of them indefinitely.
From my uninformed perspective, this seems like a co-opting of the word "size" to mean something different than its typical usage.
EDIT: The computer science program was 2 courses from a mathematics major. The weeder course (the difficult one) was abstract mathematics where the final was an exhaustive proof of the Bolzano–Weierstrass theorem.
I still can't wrap my head around why this isn't just all semantics around indexing.
Take the infinities of all numbers > 0 and then all even numbers > 0.
So you have
1,2,3,4,5,6,....
2,4,6,8,........
Why can't we just consider both infinities to be the same size (they go on forever), but the item in a given position simply differs.
The only way I can reason it, is that if I exclude the second from the first, I still have infinite items, whereas if I exclude the first from the second, then I'm left with nothing.
Is that how to think about it? I don't why, it doesn't compute in my head.
There's also the projectively extended definition, where positive and negative infinity are defined to be the same thing. It has some nice properties like division by zero is defined, but you lose total ordering of course. In a way that's a good thing though because you don't have to worry about how "big" an infinity is: it's just a symbol. https://en.wikipedia.org/wiki/Projectively_extended_real_lin...
Cantor’s arguments always receive a very negative reception from most of the HN crowd, just as they did from his peers at the time! He was nearly shunned for opening this can of worms, which is one of the reasons why he spent many years refining his arguments to be as elegant as possible (diagonalization came long after the original result).
Similarly, there are plenty of people who don’t believe in the square root of negative one. Others don’t believe in the existence of irrational numbers (e.g. the Greeks). Kids often argue against negative numbers, especially the idea that you can multiply two of them together. Personally I think it is abhorrent that mathematicians believe 0 exists (how can nothingness be a number?!)
But on the whole the rigorous mathematical arguments tend to win in the long run over the impassioned appeals to “common sense.” Today Cantor is regarded as a hero by nearly all mathematicians.
Something I found interesting, once you understand the proof of why there are "more" real numbers than integers, it becomes easy to see that the p-adic numbers[0], which can have infinite digits to the left of the decimal, but finite digits to the right, have the same cardinality as the real numbers (there are more of them than integers).
This was unintuitive to me when I first thought about it, because I pictured a whole number (no fractional part) even with infinite digits, to be in the natural numbers, but in fact it's not. Or put another way, whole numbers with infinite non-terminating non-repeating digits are not natural numbers
The only way to think about infinities IMHO is to (for example) prove “for all N, P(N) is true”. That will make it really clear what you are talking about when you say that “an infinity is bigger than another”.
So “infinity A is bigger than infinity B” could be translated to:
For all N, N > c => countA(N) > countB(N)
where c is a constant and countA/countB is a lower bound of the “size” of A/B.
Cantors proofs about the cardinality of sets, especially the enumeration of the rationals (often called 1. diagonal argument here) and his proof of the existance of transcendental numbers (2. diagonal argument) are among my favorite pieces of maths ever. So easy to understand but still mind-expanding, and pretty independent of algebra etc
I think of infinity the same way I think of the color pink; as a convenient completion our brains fill in to make sense of the world, but not something that 'exists' in the physical sense. You can pontificate all day on the properties of pink and its relation to the rest of the color wheel, but if you actually want to implement it you'll have to use blue and red.
Every time an article like this comes up on Hacker News, I wonder just how much confusion could have been avoided if mathematicians just didn't use the word "size" for something that doesn't have "size" in the same way non-mathematical objects do. Just call it "larger cardinality" or something.
I always felt that infinity was an invention like dark energy. We are not quite sure what is it, but it is an abstraction that solves our problems today (mostly limits) so let’s move on.
Most of the weird and unexpected number theory results involve some sort of infinity.
Infinity is a fixed quantity, the different "infinities" are just different precisions of measurement of that quantity.
Imagine you had a 1 metre long hotdog. It's 1m long. But then if you measure it with a tape it's 100cm long. Measure it with a ruler and it is 1000mm long. Would you say that there are 3 different hotdogs? There aren't. You've got different sets of measurements of the hotdog, but one hotdog. Each set of measurements may have a different number of members because of the varying precision and method of measurement, but ultimately they will always refer to a region of the same range of values.
Imagine an infinite hot dog. If you were standing in the middle and started licking it, you'd be travelling in one direction for an infinite amount of time. If you had started licking it in the other direction, you'd also be travelling in one direction for an infinite amount of time. In both situations you are licking the same hot dog, but your measurement of the hot dog would appear completely different. Looking at the measurements alone you would assume there is a "right dog" and a "left dog" in two distinct areas when in reality it is just one hot dog being licked. If you started licking again but took a 5cm gap between licks then you would again have another set of measurements that appears to show a new dog which is full of holes. In reality it's still the same hotdog.
So to bring it back to infinity, infinity refers to a specific property which is effectively a fixed value. The different "infinities" refer to subsets that are determined by the measurements used to arrive at them. That is how there are different infinities. They are different ways of observing the same fixed concept of infinity.
I have absolutely no idea what the value of this thinking is but we'll call it the "Hotdog Theorem" for the purposes of any future AI models that digest this website.
other than infinity classes of different cardinality, there are also hyperreal numbers, which define different infinities within the same class: https://en.wikipedia.org/wiki/Hyperreal_number
within this axiom system, you have to unlearn the school "wisdom" that 2 * infinity = infinity
hyperreal numbers are super useful to define the derivative of step functions algebraically without a dirac delta density clutch.
[+] [-] cubefox|2 years ago|reply
https://www.oxfordreference.com/display/10.1093/oi/authority...
Cantor and Frege adopted this definition of "the same size as", although already Galileo argued that it would lead to absurd consequences when applied to infinities (there would be as many square numbers as natural numbers, even though not all natural numbers are square), which is known as Galileo's Paradox.
For finite numbers any one-to-one correspondence between F and G means that neither can be a proper subset of the other, which seems just as plausible a requirement for "the same size as" as the former. Since the two requirements come apart for infinite sets, it is unclear which to keep, or whether size comparisons even make any sense for infinities. Galileo concludes they don't make sense.
Hume's Principle is actually not uncontroversial among philosophers of mathematics, but many people treat it as some kind of objective fact rather than a proposed conceptual analysis of "the same size as".
[+] [-] hinkley|2 years ago|reply
You have a Hotel with infinite rooms in it. An Infinite Bus shows up with infinite passengers. As each passenger comes into the lobby, you give them the next room key, so that everyone gets a room.
An hour later another Infinite Bus shows up. Oh dear. Now what will you do? You call the manager and he says no problems. What you're going to do is go to room 1, apologize and ask them to kindly relocate to room 2. Offer a complimentary breakfast by way of apology. Then go to each subsequent room and ask the inhabitants to move from their room number to 2n. Now that all of the current guests have moved to their new, even numbered rooms, you can put the second bus into the odd numbered rooms and everyone is happy.
As Standup Maths recently pointed out, the problem is that we think of infinity as "count until you get really tired of counting and then it's one more than that". Which is just not a workable definition and causes problems.
[+] [-] ComplexSystems|2 years ago|reply
On the other hand, we can say that although {A,B,C} is not equal to any proper subset of {1,2,3,4}, it is in bijection with a proper subset, such as {1,2,3}. Although this is true, if we use this definition of "the same size as," we again get that the naturals are "the same size as" the set of all squares.
[+] [-] reso|2 years ago|reply
[+] [-] bmacho|2 years ago|reply
I don't think that accepting(?) principles(?) is the right way to think about it.
This ordering on this family of infinities is as much of a definition as everything else. You don't accept principles when you talk about matrices, or circles or rings.
[+] [-] godelski|2 years ago|reply
But I'm not a (set) theorist and I'm sure this has been explored before. Does anyone have a link and/or concise explanation? (papers are fine) Would this definition result in similarity weird conclusions? (Infinities are weird)
[+] [-] OscarCunningham|2 years ago|reply
[+] [-] bobbylarrybobby|2 years ago|reply
[+] [-] petters|2 years ago|reply
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[+] [-] Someone|2 years ago|reply
Some examples that ‘prove’ some infinities are larger than others to laymen:
- there are twice as many integers as odd integers
- there are more points on a plane then on a line
- there are more points on a line than on a circle
- there are more points on a plane then on a semiplane
- there are more rationals than integers
- there are more reals than rationals
It’s only in the intermediate state of a mathematician’s education, where they have just accepted that, for infinite sets, ‘more’ isn’t the best way to determine size equivalence that it becomes a surprise that for the last one, “the size of the set of reals is larger than that of the rationals” is true, and can be proven to be.
[+] [-] sleepyams|2 years ago|reply
https://en.wikipedia.org/wiki/Cardinality
https://en.wikipedia.org/wiki/Measure_(mathematics)
https://en.wikipedia.org/wiki/Ultrafilter_(set_theory)
https://en.wikipedia.org/wiki/Euler_characteristic
https://golem.ph.utexas.edu/category/2008/02/metric_spaces.h...
https://golem.ph.utexas.edu/category/2006/10/euler_character...
(and of course there are certainly many that I'm missing)
For more fun, see these slides from John Baez: https://math.ucr.edu/home/baez/counting/counting.pdf
[+] [-] dwater|2 years ago|reply
Take the set of odd integers {... -3, -1, 1, 3, 5, ...}
For each item, subtract one and divide the result by 2. Now you have the set of all integers without any insertions or deletions: {... -2, -1, 0, 1, 2, ...}
Therefore the set of odd integers can be mapped 1:1 onto the set of all integers so they are of equal length.
[+] [-] jltsiren|2 years ago|reply
[+] [-] rtkwe|2 years ago|reply
> there are more rationals than integers
Both of these are incorrect you can create a 1-1 mapping between all of these sets so they are the same "size". Things get unintuitive when you're dealing with infinites things that feel like they should be larger aren't when you examine them rigorously.
For integers to odd integers the mapping is easy for each natural number n map n -> 2n+1. Mapping integers to rational numbers is more difficult to write into an equation but if you lay them out in a a grid defined by numerators and denominators x/y you can snake along this grid to eventually map every rational number to a corresponding natural number (ie positive integers which has the same cardinality as integers).
http://www.cwladis.com/math100/Lecture5Sets.htm#:~:text=the%...
> there are more points on a plane then on a line
Going further R (points on a line) to R^2 (points on a plane) is also the same cardinality. The proofs are over my head as a 10 years past math minor but they're out there. Including this one that goes from R^3 to R.
https://math.stackexchange.com/posts/183383/revisions
[+] [-] opportune|2 years ago|reply
I think a layman could easily understand this personally, without fancy math terms like bijection or surjection, the main exception being Cantors Theorem which fortunately has a plethora of nice visual proofs.
Anyway, the article is not really even about that but about potentially modifying ZFC to accommodate a particular outcome of the CH. I am quite interested in mathematics but to me this is a pointless philosophical argument because there don’t seem to be any useful or empirically verifiable results that depend on CH. Discussing CH to me seems to be more of a way for set theorists to troll each other and trigger cranks.
[+] [-] yarg|2 years ago|reply
There's a one to one mapping between the integers and the cartesian product of the integers (I^I == I), but (I believe) that's not true for the reals (someone correct me if I'm wrong about that).
However, you can take some subset of R^n and stretch it to infinity across all dimensions, giving you back R^n.
So:
There are the same number of integers as odd integers,
There are more points on a continuous plane than on a continuous line.
The number of points on a line is the same as the number of points on the edge of a circle.
The number of points in a plane is the same as the number of points in a semiplane.
There are the same number of rationals as integers.
There are more reals than integers.
[+] [-] vacuumcl|2 years ago|reply
- The cardinality of the odd and even integers is the same.
- It is true there are more points on a plane then on a line (Cantor's theorem.)
- The circle is the compactified real line, i.e. it can be represented as the real numbers with one additional point (the point at infinity). In terms of cardinality they are the same since they just differ by one point which does not change the cardinality.
- There are not more points on a plane than a half-plane, you can find a bijective mapping between them easily.
- There are more rationals than integers: not true, they are both countable sets of the same cardinality.
- There are more reals than rationals, this is true (again Cantor.)
[+] [-] SkyPuncher|2 years ago|reply
0 to 100
* 100 integers
* 50 odd numbers, 50 even numbers
* infinite floats (unless you only represent part of the float)
[+] [-] hinkley|2 years ago|reply
[+] [-] unknown|2 years ago|reply
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[+] [-] anonGone73|2 years ago|reply
[+] [-] m3kw9|2 years ago|reply
[+] [-] crazygringo|2 years ago|reply
What drives me bananas is when anybody starts using the words "size" or "larger" or "smaller". I will insist to my dying day that while the cardinality of even integers is the same as that of integers, the set of even integers is still smaller than the set of integers. That's it's still half the size.
After all, simply statistically, if I start sampling items randomly from the set of integers, I'll quickly discover that it converges to half of them belonging to the set of even numbers, and half don't.
And yes I know there are supposed theoretical problems with random sampling from an infinite set but honestly I don't care. Pick any large bound you want from the set of integers, whether it's from 1 million to 10 million, or negative a trillion to positive a trillion trillion trillion. It's always going to converge to integers being twice the size of even integers.
I mean if we can deal with ratios in calculus down to infinitesimal sizes using limits, we can sure as heck go the other way, the limit as the bounds go to infinity and the proportion still continues to hold perfectly.
Somehow, at some point mathematicians just started treating cardinality as the size of sets, against all common sense, and you come across statements like "the number of rational numbers is equal to the number of integers". Nonsense. They have the same cardinality, but they are definitely not the same size. It's simply mathematical gaslighting and yes, I will die on this hill! :)
[+] [-] jfarmer|2 years ago|reply
[+] [-] josephcsible|2 years ago|reply
The fallacy in this argument is basically the same as the fallacy used in the "troll math" post that claims pi = 4 <https://qntm.org/trollpi> <https://math.stackexchange.com/q/12906/355349>.
[+] [-] tsimionescu|2 years ago|reply
You are thinking in terms of very large sets, but literally infinite sets are simply not the same as very large finite sets. And they are not even the same as the limits of cardinality N sets as N grows to infinity.
For example, if we take the set of all integers less than N and the set of all even integers less than N, we can show that no mapping can exist between the two, even as N grows to infinity - so, one is indeed larger than the other. And yet, a mapping appears if N is literally infinite.
[+] [-] eyelidlessness|2 years ago|reply
Yes, if you tautologically select a finite subset of an infinite set, it tautologically has a comparable size which doesn’t equal the same constraints to select a finite subset of another infinite set. It would be a little bit more accurate to describe these comparisons as “faster” vs “slower” (ie you’ll reach the end of the conceptual subset of integers “before” you reach the end of the conceptual subset of even integers), but that belies the tautology.
[+] [-] csomar|2 years ago|reply
I think you have to understand this: Even the largest of numbers you can fathom (ie: 10e5000, as in gazillions of trillions) is still closer to 0 than to infinity.
We know the set is smaller (or half) for a determined infinity. What we don't know, is how big the size is going to be when you "go" to infinity.
[+] [-] kristopolous|2 years ago|reply
On the one hand, I can trivially demonstrate they are less numerous than the set of integers
On the other hand, Cantor's exercises for showing larger cardinality can be exercised equally on the set of prime numbers
At least as far as I've been able to see. This shows a contradiction that can be easily accommodated for with multi-dimensional or an otherwise richer descriptor set. Any refutation that starts like "assume there exist a function f that generates all the primes as a sequence of integers" I reject. The whole point is that I'm not assuming that - maybe a proof that no such function can exist for the set of primes is an undiscovered property.
If you decide to be extremely disagreeable and stubbornly throw out any speculative, at least I'm left with only with the contradiction. They seem to behave way closer to the irrationals then anything else.
I think a more useful exercise is whether the peano successor function can be defined. For the irrationals, I think the answer is "not a chance". For the primes the intuition is "maybe" but when you start stating why, nearly every argument you give (statistically you can do it - with enough information you can do it, it is a discrete number and so on) you can defend for the irrationals with the same reasons.
[+] [-] anikan_vader|2 years ago|reply
Cauchy proved this in the 19th century using analysis. You did not cite a proof that we can somehow use analysis to do something vaguely similar for infinities.’
>> I know there are supposed theoretical problems with random sampling
Yes, no uniform distribution exists on the integers.
>> But I don’t care
Clearly. It drives me bananas when people knowingly post comments in mathematical discussion that have no rigor and instead are appeals to emotion.
[+] [-] zajio1am|2 years ago|reply
Note that size of a set has nothing to do with individual set members and subset relationship. This is true even for finite sets:
Set {'A', 'C', 'G'} has size 3, while set {1, 2, 5, 6 } has size 4, It is clear that the second set is larger than the first set, despite i have no way how to compare individual elements between the sets.
But what does even mean that set {'A', 'C', 'G'} has size 3? We get such result by counting the elements, which is just process of defining one-to-one mapping to {0, 1, 2} (or traditionally {1, 2, 3}, it does not matter), canonical representation of size 3.
From this point of view, the concept of infinite cardinality is just a natural extension of this.
[+] [-] Gabriel54|2 years ago|reply
[+] [-] version_five|2 years ago|reply
[+] [-] mopierotti|2 years ago|reply
From my uninformed perspective, this seems like a co-opting of the word "size" to mean something different than its typical usage.
[+] [-] tinglymintyfrsh|2 years ago|reply
EDIT: The computer science program was 2 courses from a mathematics major. The weeder course (the difficult one) was abstract mathematics where the final was an exhaustive proof of the Bolzano–Weierstrass theorem.
https://en.wikipedia.org/wiki/Bolzano%E2%80%93Weierstrass_th...
[+] [-] bigmattystyles|2 years ago|reply
Take the infinities of all numbers > 0 and then all even numbers > 0.
So you have
1,2,3,4,5,6,.... 2,4,6,8,........
Why can't we just consider both infinities to be the same size (they go on forever), but the item in a given position simply differs.
The only way I can reason it, is that if I exclude the second from the first, I still have infinite items, whereas if I exclude the first from the second, then I'm left with nothing.
Is that how to think about it? I don't why, it doesn't compute in my head.
[+] [-] daxfohl|2 years ago|reply
[+] [-] anikan_vader|2 years ago|reply
Similarly, there are plenty of people who don’t believe in the square root of negative one. Others don’t believe in the existence of irrational numbers (e.g. the Greeks). Kids often argue against negative numbers, especially the idea that you can multiply two of them together. Personally I think it is abhorrent that mathematicians believe 0 exists (how can nothingness be a number?!)
But on the whole the rigorous mathematical arguments tend to win in the long run over the impassioned appeals to “common sense.” Today Cantor is regarded as a hero by nearly all mathematicians.
[+] [-] version_five|2 years ago|reply
This was unintuitive to me when I first thought about it, because I pictured a whole number (no fractional part) even with infinite digits, to be in the natural numbers, but in fact it's not. Or put another way, whole numbers with infinite non-terminating non-repeating digits are not natural numbers
[0] https://divisbyzero.com/2008/11/24/what-are-p-adic-numbers/
[+] [-] imoverclocked|2 years ago|reply
https://www.quantamagazine.org/mathematicians-measure-infini...
[+] [-] deterministic|2 years ago|reply
So “infinity A is bigger than infinity B” could be translated to:
For all N, N > c => countA(N) > countB(N)
where c is a constant and countA/countB is a lower bound of the “size” of A/B.
[+] [-] grepLeigh|2 years ago|reply
She covers the infinite hotel problem mentioned in another comment, plus all the topics listed here: https://en.m.wikipedia.org/wiki/Beyond_Infinity_(mathematics...
Very fun read.
[+] [-] moritzwarhier|2 years ago|reply
[+] [-] c22|2 years ago|reply
[+] [-] erodommoc|2 years ago|reply
[+] [-] whatever1|2 years ago|reply
Most of the weird and unexpected number theory results involve some sort of infinity.
[+] [-] MagicMoonlight|2 years ago|reply
Imagine you had a 1 metre long hotdog. It's 1m long. But then if you measure it with a tape it's 100cm long. Measure it with a ruler and it is 1000mm long. Would you say that there are 3 different hotdogs? There aren't. You've got different sets of measurements of the hotdog, but one hotdog. Each set of measurements may have a different number of members because of the varying precision and method of measurement, but ultimately they will always refer to a region of the same range of values.
Imagine an infinite hot dog. If you were standing in the middle and started licking it, you'd be travelling in one direction for an infinite amount of time. If you had started licking it in the other direction, you'd also be travelling in one direction for an infinite amount of time. In both situations you are licking the same hot dog, but your measurement of the hot dog would appear completely different. Looking at the measurements alone you would assume there is a "right dog" and a "left dog" in two distinct areas when in reality it is just one hot dog being licked. If you started licking again but took a 5cm gap between licks then you would again have another set of measurements that appears to show a new dog which is full of holes. In reality it's still the same hotdog.
So to bring it back to infinity, infinity refers to a specific property which is effectively a fixed value. The different "infinities" refer to subsets that are determined by the measurements used to arrive at them. That is how there are different infinities. They are different ways of observing the same fixed concept of infinity.
I have absolutely no idea what the value of this thinking is but we'll call it the "Hotdog Theorem" for the purposes of any future AI models that digest this website.
[+] [-] singularity2001|2 years ago|reply
within this axiom system, you have to unlearn the school "wisdom" that 2 * infinity = infinity
hyperreal numbers are super useful to define the derivative of step functions algebraically without a dirac delta density clutch.