In my experience with children, one of the easiest-to-grasp concepts of infinity is provided by the transfinite ordinals, since it can be viewed as a continuation of the usual counting manner of children, but proceeding into the transfinite:
1,2,3,⋯,ω,ω+1,ω+2,⋯,ω+ω=ω⋅2,ω⋅2+1,⋯,ω⋅3,⋯,ω2,ω2+1,⋯,ω2+ω,⋯⋯
Presumably this person has no experience with 6 year olds? This explanation is horrendous haha
No it isn't. If you ask a child what comes after infinity, "Infinity + 1" is pretty much the default answer. Any kid who knows multiplication knows "Infinity + Infinity" is the same as "Infinity Times Two". The answer of "Infinity TIMES Infinity" is also popular for kids to say when they know a number bigger than their friend (who just proclaimed infinity is the largest number).
I thought that most of us learn at an early age, as a result of this kind of exchange, that "infinity" is not "the biggest number" or even a number at all, as far as the ordinary notion of "number" goes.
> Any kid who knows multiplication knows "Infinity + Infinity" is the same as "Infinity Times Two".
Or is it "Two Times Infinity"? (Hint: It isn't, because "Two Times Infinity" = "Infinity", while "Infinity Times Two" = "Infinity + Infinity". Not sure every kid knows that.)
I explained basically this to my 4 year old nephew recently. He wanted to count to infinity. I asked him what is the biggest problem with counting to infinity? It's too slow. I said ok let's take bigger steps. We counted by 2's then 10's then hundreds and millions and then zillions and other ridiculous superlative numbers. It doesn't really matter because everything is still too slow. So then we said ok lets make up a number ω that is half way there, One ω, Two ω, done. He's happy. Then I told him to add one more and sent him back to play fetch with the dog.
I taught my kid that the way to think of infinity is that it's like hugs, there's always one more, unlike candy, which is limited and can be counted, infinity cannot be counted.
> this person has experience teaching children mathematics
Just as a FYI, there are plenty of countries in Europe where many 6 year-olds are still in kindergarten not at school, as a result they most likely have not have properly started learning numbers or reading and writing.
Also my first thought. I assume he's writing this to other people who know what transfinite ordinals are (I don't understand the explanation) and would frame it differently with an actual kid. Even in context it's a hilarious quote though, I think it's possible this was on purpose
I think the big assumption that kids can't get "complicated" ideas is faulty.
Sure, they lack rigor, and often will just get the sketch of the idea.
And it's a lot more work to think about how to put things in the terms that a kid will understand given their knowledge so far.
But this idea? "Infinity plus one?!@" --- this is a conversation elementary school kids have on their own. Pulling it a little closer to a sane footing in ordinal analysis is not hard. Half of six year olds can handle it.
On the other hand, there's not a lot of obvious utility to teaching a six year old this particular concept early. On the gripping hand, there is a cost to keeping kids in a bubble where you don't talk about any big ideas (of whatever sort-- mathematical, philosophical, historical, linguistic) at all, or excessively dilute them to the point where they're meaningless.
Richard Feynman would be making disapproving noises.
Explain everything like you're talking to a fifth grader. If you can't, you don't understand your problem fully.
He spend much of his professorship agonizing about how to fit all of physics into a freshman lecture. When he couldn't, he knew we needed to think more about that area.
Transfinite ordinals also known as hyperreals should really be taught in school as they make many parts of math easier: algebraic definition of derivatives (including algebraic derivative of step functions without dirac 'density') and yes: natural addition and multiplication.
> Transfinite ordinals also known as hyperreals should really be taught in school as they make many parts of math easier: algebraic definition of derivatives
Q: What proportion of children study maths long enough to understand derivatives?
They are not explaining to a 6 years old, they explains to somebody who will in their turn explain it to a 6 years old, which is a different task and has to be optimized in a different way.
In the comments of the answer the author says they have a 4 and a 9 year old:
"Bill, despite your emphatic comments, I know for a fact that counting into the ordinals is something that children can easily learn. I have two young children (ages 4 and 9), who are happy to discuss ℵα
for small ordinals α---although my daughter's pronunciation sounds more like Olive0, Olive1---and my son can count up to small countably infinite ordinals. The pattern below ωω is not difficult to grasp. Below ω2, it is rather like counting to 100, since the numbers have the form ω⋅n+k, essentially two digits"
Ordinals are hard to grasp for people that know the standard school curriculum, know about countability and uncountable sets, cardinality, and the basic properties and arithmetic of cardinality.
I don't know why would it be hard for people that haven't been familiarized with a similar but different concept?
He has children (not sure about age right now) and discusses mathematics often with them. His tweets have had many interesting examples.
I do not think he means he would use symbols to explain to children, but that the notion of counting natural numbers that children have easily generalises to counting transfinite numbers.
The way I would explain it to a 6 year old would be like this:
Infinity isn't a number really, it's a concept, like the word many or the word few. If someone says they have many of something, you don't think is that odd or even you just know they have a lot of it. Infinity is kind of like that, it explains the idea of things going on forever, not an exact quantity of things like the number 10 or 11.
The way I would explain it to a 6 year old would be like this:
There are natural numbers, like 0,1,2 and so on. Natural numbers can be odd or even. There is no such natural number as infinity. Therefore the question if 'infinity' is odd or even is meaningless. It does not even type-check.
In math people like well-formed questions, and generally don't like ill-formed questions.
The fallback metaphor I use in these situations or similar ones, "What's outside of the universe" for example, is the old, "What's North of the North Pole?" Then you explain that we can create questions and statements in our languages which don't have logical, mathematical or physical validity. Although we can often describe scientific and technical concepts in common languages, that's just a translation, the real language is math.
> In math people like well-formed questions, and generally don't like ill-formed questions.
This is not so simple, though. Ill formed questions can be interesting as a motivation to formalise them (ie make them well-formed) in generalising/abstracting concepts into new concepts. Eg how even/odd has been generalised to transfinite numbers.
The only thing that is a little off here to me is that I don't think there is mathematical notation for "many" or "few". And yet infinity does have mathematical notation and is used in some equations, no?
It's an analogy, meant to show the similarities between two things in a limited way, to illustrate an idea. They do not have to be exactly the same in every way.
The answer that says "Here is a simple example that has some hope of being comprehensible to a 6-year-old." and then begins "Consider the ring of polynomial functions with integer coefficients, ..." gets upvoted tens of times.
Even the answer that uses "numerocity", "refined cardinality", and "logarithm" as the explanation to a 6-year-old gets upvoted.
The answer, https://math.stackexchange.com/a/49065/13638, that says as the answer-to-a-6-year-old the same thing that several commenters have actually posted here (e.g. https://news.ycombinator.com/item?id=35790064 for one of many), on Hacker News in just the past hour or so, and that explains in terms that a 6-year-old has at least a chance of having encountered, gets 5 votes in 12 years and the submitter is banned from the site.
The answer is using big words but the concept is simple. Like talking about fractions as a quotient ring over a field.
A six year old can absolutely grasp that even means "being able to be split into two equally sized piles" where equally sized means each thing in the left pile can be matched to something in the right. 6 apples is even because you can split them into 3 and 3.
Then for infinity you separate them into the even and odd numbers, boom. Infinity is even.
Saying "infinity isn't a number", to me, is so much worse an answer because it's not satisfying. Because both you and the 6 year old know that isn't right. The 6 year old is grasping at a bigger concept but doesn't have the words.
So there seems like a glaring hole in the answer, but maybe I'm missing something. Because:
> It is easy to prove from this definition by transfinite recursion that the ordinals come in an alternating even/odd pattern, and that every limit ordinal (and hence every infinite cardinal) is even.
Sure, if we use the natural numbers and start at 1, then we can group:
[1, 2], [3, 4], [5, 6], ...
and prove infinity is even.
But we could also just as easily group:
1, [2, 3], [4, 5], [6, 7], ...
and prove infinity is odd.
It's the same if we try to split into two equal subsets, because we can split into:
[1, 3, 5, ...]
[2, 4, 6, ...]
and say it's even. Or we can divide:
1
[2, 4, 6, ...]
[3, 5, 7, ...]
and prove it's odd because we have two equal subsets plus one left over.
So I'm missing the reason for why the second versions aren't just as valid.
(Of course, I'm more inclined to agree with many commenters here that it's just a category error, and asking whether infinity is even/odd is as useful as asking whether democracy is blonde or brunette.)
The definition given was 'if there is another ordinal 𝛽 such that 2⋅𝛽=𝛼' [1], but the intuition is better explained by the post below:
> A set 𝑆 has even cardinality if it can be written as the disjoint union of two subsets 𝐴,𝐵 which have the same cardinality. [2]
In other words, a set is even if it can be paired up, by finding one grouping where it pairs. Finding alternative groupings that do not pair does not matter.
OK, so I guess I'm just understanding that mathematicians arbitrarily decided to prioritize "even" over "odd"?
Because as I stated in another comment, you could just as easily say odd cardinality exists if you can find two subsets with the same cardinality and there's one element left over, and otherwise we call it even.
So at the end of the day, what you're saying is that ultimately infinity would be even just because mathematicians arbitrarily defined 'even' that way -- not because there's any intuitive logic behind it, any deeper justification, or any necessary consistency with parity for finite sets.
It might be a better explanation but those two are very much not equivalent.
Actually the fact that splitting it into pairs is the same as splitting into two equal sets of equal cardinality is itself non-trivial. The reason why shows up when you try to get the two definitions closer together.
Splitting an ordinal into pairs is essentially splitting it into ordered pairs (a_i, b_i) such that the map i to a_i is monotonic and for no i<j the pair (a_i, b_i) overlaps with (a_j, b_j) in the sense that a_j <= b_j.
Splitting a set into pairs is splitting it into sets {a_i, b_i} such that for no i != j the two sets {a_i, b_i} and {a_j, b_j} overlap.
These two are note the same, you can split pretty much any infinite set into two disjoint sets of equal cardinality.
It's hard to get the definitions general enough to get one definition for both ordinals and sets. Mostly because products of ordinals are a bit weird. For sets (and most other types of mathematical objects) it doesn't matter which way around you pair things up, but for the ordinals you end up with a completely different object if you do it the other way around and this is apparently the more interesting definition of the two.
I would weaken the definition of even/odd to say that a set is even if /there exists/ a way to pair things off, and odd if /there is no way/ to pair things off (ie, not even). So the countable numbers would be even.
But that seems redundant with countable/uncountable sets, because then every countable infinite set would be even (e.g. rational numbers), and every uncountable infinite set would be odd (e.g. real numbers).
It's also not clear to me what justification there would be for a "preference" for the "even" category that way -- it seems arbitrary. Why not be odd if there exists a way to pair things off such that one is left over, and even if there isn't such a way?
But pairity is just a question of sorting the countable numbers into sets of size two, and the more general form even of that is sorting into sets of size N. It's just as easy to say that the countable numbers are odd if there exists a way to sort them into sets of size three. So I'd argue the countable numbers are odd.
And you then you could retort with sets of size four, and I could use five, and then we can argue about whether we'll end up at the limit with more odd sets or even sets, and now we're arguing in circles. Reductio ad absurdum.
> we can...prove infinity is even....and prove infinity is odd...
> maybe I'm missing something
The answer said:
> the usual definition is that an ordinal number 𝛼 is even if... Otherwise, it is odd.
In other words, if a number could be proved to be even, it is even. If not, it is odd.
Using their definition, there is no such thing as "proving a number is odd". You'd have to do it by failing to prove it's evenness. In the case of infinity, because we can successfully prove evenness, it's even and not odd.
Omega is the lowest countable infinity. There's no parity within a countable infinite as you describe.
It's only even or odd with respect to other infinities which the cardinal numbers can count based on the presence of a bijection or not. It's a kind of relative parity.
> There's no parity within a countable infinite as you describe.
That directly contradicts the quoted text I included from the original answer though, as far as I understand. It directly asserted that "every infinite cardinal is even".
> It's a kind of relative parity.
What is relative parity? The original question was whether infinity is even or odd... I don't know what you mean by relative parity.
>To explain the idea to a child, I would focus on the principal idea: whether finite or infinite, a number is even when it can be divided into pairs. For finite sets, this is the same as the ability to divide the set into two sets of equal size, since one may consider the first element of each pair and the second element of each pair.
The answer this quote came from is amazingly obtuse, but it does make me think that infinity must be even since infinity can be divided into 2 pairs, each of which is of equal size since both are infinity.
In mathematics, you can define things in different ways to get different answers. Ways of defining things tend to be highlighted as true (in at least some context) if they are interesting and useful, and ignored if not. I don't think the definition based on "dividing into pairs" is particularly interesting or useful in the context of the child's understanding of numbers, because it's too vague to be useful, and it doesn't lead to any insights.
The definition based on transfinite ordinals explained in the same answer does seem interesting, and I wouldn't be surprised if it were useful. I think this is a case of simplification gone wrong, where everything interesting was lost in the translation to more accessible terminology.
A more honest thing to say to a child would be that the way even and odd are defined only make sense for finite numbers. It's true for the definition they know, and it introduces them to the important insight that logical rules that are created for one kind of thing might not work when applied to something else. I think this would be more accessible and stimulating for a six-year-old than giving them a half-baked verbal imitation of a result from transfinite mathematics.
They'll be thrilled later if they study math and discover that there are definitions of "infinity" and "even" that yield an answer to their childhood question.
An even more honest thing to say is that infinity when used as a number is a hack introduced by mathematicians to make notation and reasoning simple in some cases, but that it can be dangerous in other cases, like any other hack. If you want to use infinity in a safe way, then use limits around your expressions.
(And this quickly resolves the case of this article, since lim x->inf x-2*floor(x/2) does not exist).
The concept of "infinity - 1" doesn't exist. Subtraction isn't defined for ordinals. Furthermore even if you try to define it, it doesn't work for limit ordinals.
If you are thinking about the difference between
[0,1,2,3,…]
and
0, [1,2,3,4,…]
Then I regret to inform you the former is omega and the latter is 1+omega which is the same as omega. In other words attempting to subtract one from infinity by removing from the front results in infinity.
In the general case, the comparability of cardinals relies on the axiom of choice. In other words, they are comparable, but they require a slightly unintuitive foundation to establish that they are always comparable.
> Unlike the case of even integers, one cannot go on to characterize even ordinals as ordinal numbers of the form β2 = β + β. Ordinal multiplication is not commutative, so in general 2β ≠ β2. In fact, the even ordinal ω + 4 cannot be expressed as β + β, and the ordinal number
For a six year old, I'd tell that infinite is not a number so it's not even or odd. If s/he even get's a Ph.D. in math, s/he will understand.
Moreover, I remember when I was a graduate T.A. that one day before lunch I went to a class to learn about the https://en.wikipedia.org/wiki/Alexandroff_extension in the morning. (The idea is that you add one ∞ to a set of numbers to get a compact set. And in the new set ∞ is (almost) a number as good as the other numbers.) After lunch, I went to teach limits to first years students, and with a total straight face I told them that ∞ is not a number.
> After lunch, I went to teach limits to first years students, and with a total straight face I told them that ∞ is not a number.
When you apply Alexandroff extension to add the point at infinity to, say, the real numbers, what you're left with is not a set of numbers (i.e. a field) anymore. So it makes sense to say that ∞ is not a number. Moreover, the way ∞ is used in analysis is different from Alexandroff compactification, in that you usually use two infinities (±∞) as a shorthand for quantification over increasing or decreasing sequences of real numbers (this can be formalized using extended real numbers [0] or other gadgets but doing so has no advantages in a first-year analysis class, and might in fact make matters worse).
It was a long time ago, something like an optional course in Advanced Functional Analysis. It was about the algebras of functions with and without unity, and how to complete the ones without unity using the compactification (i.e. including a ∞) and a few variants.
> two infinities (±∞)
It depends. In the real numbers it depends, but in most cases I agree that it's better to use two. In complex analysis it's much better to have only one infinity. And there are more weird case like the projective plane where you have one infinity in each direction.
> So it makes sense to say that ∞ is not a number.
I agree, it's not longer a field and the operation lose many properties if you try to extend them. So I said "(almost) a number". Anyway, the weird part is that in some cases you can write f(∞) in an advanced math course, but you can never write f(∞) in a fist year math course.
> The problem with transfinite is that you lose commutatively.
Depends on which transfinite algebra you're working with. If you restrict "number" to mean "element of an ordered field" (thus excluding things like the "complex numbers" but matching the usual intuition of how numbers should behave) then you can't include Cantor's ordinals but you can include the Surreal Numbers. Those include infinite ordinals and (due to being a field) have commutative addition and multiplication operations.
I'd say that the problem with transfinites is that you lose intuitive understanding of what's going on, and one of those intuitions is commutativity.
People seem to assume that they know a couple of tricks about infinity (adding, multiplying) and don't stop to think that there should be a much more rigorous definition. Which, they shouldn't -- the average person will never _actually_ care about transfinites.
Imagine the + in C++ when you have to add two complex numbers. They are just a struct with x and y, and some magic to make all operations work as intended.
The use of the + in this example is more like the concatenation of strings, like "Hello " + "World!" is "Hello World!". But in this case, the content of the string doesn't matter so "Hello " == "World!" and there are some magical strings that are infinite.
The idea is that anyone can overload the symbol + and sum whatever they want. It's not necessary to use + with numbers. Obviously, most silly overloads are ignored, and nobody use them. In this case it's a popular overload so it is teach in an advanced math course and has it's own Wikipedia article.
The concept of "number" has a lot of definitions in mathematics. I agree with this [0] more in depth explanation that calling infinity strictly not a number is not useful (though it certainly is not e.g. a natural number). But more importantly, the concept of evenness readily generalizes to ordinals, so as long as we specify that we are in (or move into) that context, then the question is well formed and interesting.
I agree. It's an interesting intellectual exercise, but I am not sure if we would miss out on anything if we just had a symbol(s) for specific really large discrete numbers.
Sometimes I wonder if there's a better math language waiting to be invented that eschews the non-discrete.
Readit News
Teacher: “Well, there’s 100 million and one”
Child: “I was pretty close then!”
Or is it "Two Times Infinity"? (Hint: It isn't, because "Two Times Infinity" = "Infinity", while "Infinity Times Two" = "Infinity + Infinity". Not sure every kid knows that.)
(Full disclosure: have three children and plenty of STEM in the family)
I'm not sure that's the _default_ answer, of course one might easily get that answer if at least one parent has a STEM background.
Schools don't teach about infinity to young children. A pity, really.
In case anyone is curious, this person has experience teaching children mathematics. For example, on his blog, we have
http://jdh.hamkins.org/math-for-six-year-olds/http://jdh.hamkins.org/math-for-seven-year-olds-graph-colori...http://jdh.hamkins.org/math-for-eight-year-olds/http://jdh.hamkins.org/math-for-nine-year-olds-fold-punch-cu...
The most recent post in his category "Math for Kids" is in fact teaching how to count ordinals up to omega-squared: http://jdh.hamkins.org/counting-to-infinity-poster/
Just as a FYI, there are plenty of countries in Europe where many 6 year-olds are still in kindergarten not at school, as a result they most likely have not have properly started learning numbers or reading and writing.
https://www.statista.com/chart/13378/when-do-children-start-...
Sure, they lack rigor, and often will just get the sketch of the idea.
And it's a lot more work to think about how to put things in the terms that a kid will understand given their knowledge so far.
But this idea? "Infinity plus one?!@" --- this is a conversation elementary school kids have on their own. Pulling it a little closer to a sane footing in ordinal analysis is not hard. Half of six year olds can handle it.
On the other hand, there's not a lot of obvious utility to teaching a six year old this particular concept early. On the gripping hand, there is a cost to keeping kids in a bubble where you don't talk about any big ideas (of whatever sort-- mathematical, philosophical, historical, linguistic) at all, or excessively dilute them to the point where they're meaningless.
Explain everything like you're talking to a fifth grader. If you can't, you don't understand your problem fully.
He spend much of his professorship agonizing about how to fit all of physics into a freshman lecture. When he couldn't, he knew we needed to think more about that area.
https://en.wikipedia.org/wiki/Hyperreal_number
Q: What proportion of children study maths long enough to understand derivatives?
Things people say on HN :)
"Bill, despite your emphatic comments, I know for a fact that counting into the ordinals is something that children can easily learn. I have two young children (ages 4 and 9), who are happy to discuss ℵα for small ordinals α---although my daughter's pronunciation sounds more like Olive0, Olive1---and my son can count up to small countably infinite ordinals. The pattern below ωω is not difficult to grasp. Below ω2, it is rather like counting to 100, since the numbers have the form ω⋅n+k, essentially two digits"
why misquote someone and claim their idea is hard to understand?
I don't know why would it be hard for people that haven't been familiarized with a similar but different concept?
I do not think he means he would use symbols to explain to children, but that the notion of counting natural numbers that children have easily generalises to counting transfinite numbers.
Infinity isn't a number really, it's a concept, like the word many or the word few. If someone says they have many of something, you don't think is that odd or even you just know they have a lot of it. Infinity is kind of like that, it explains the idea of things going on forever, not an exact quantity of things like the number 10 or 11.
For any collection with a certain number of things (such as n potatoes), I can always name a collection with more things, such as n+1 potatoes.
"Infinity" is a word that we use for games like that.
PS: they seem to _know_ that endless*endless > endless but do not dare to admit it
There are natural numbers, like 0,1,2 and so on. Natural numbers can be odd or even. There is no such natural number as infinity. Therefore the question if 'infinity' is odd or even is meaningless. It does not even type-check.
In math people like well-formed questions, and generally don't like ill-formed questions.
This is not so simple, though. Ill formed questions can be interesting as a motivation to formalise them (ie make them well-formed) in generalising/abstracting concepts into new concepts. Eg how even/odd has been generalised to transfinite numbers.
Sensei: Grasshopper, lecture over today.
The answer that says "Here is a simple example that has some hope of being comprehensible to a 6-year-old." and then begins "Consider the ring of polynomial functions with integer coefficients, ..." gets upvoted tens of times.
Even the answer that uses "numerocity", "refined cardinality", and "logarithm" as the explanation to a 6-year-old gets upvoted.
The answer, https://math.stackexchange.com/a/49065/13638, that says as the answer-to-a-6-year-old the same thing that several commenters have actually posted here (e.g. https://news.ycombinator.com/item?id=35790064 for one of many), on Hacker News in just the past hour or so, and that explains in terms that a 6-year-old has at least a chance of having encountered, gets 5 votes in 12 years and the submitter is banned from the site.
A six year old can absolutely grasp that even means "being able to be split into two equally sized piles" where equally sized means each thing in the left pile can be matched to something in the right. 6 apples is even because you can split them into 3 and 3.
Then for infinity you separate them into the even and odd numbers, boom. Infinity is even.
Saying "infinity isn't a number", to me, is so much worse an answer because it's not satisfying. Because both you and the 6 year old know that isn't right. The 6 year old is grasping at a bigger concept but doesn't have the words.
> It is easy to prove from this definition by transfinite recursion that the ordinals come in an alternating even/odd pattern, and that every limit ordinal (and hence every infinite cardinal) is even.
Sure, if we use the natural numbers and start at 1, then we can group:
and prove infinity is even.But we could also just as easily group:
and prove infinity is odd.It's the same if we try to split into two equal subsets, because we can split into:
and say it's even. Or we can divide: and prove it's odd because we have two equal subsets plus one left over.So I'm missing the reason for why the second versions aren't just as valid.
(Of course, I'm more inclined to agree with many commenters here that it's just a category error, and asking whether infinity is even/odd is as useful as asking whether democracy is blonde or brunette.)
> A set 𝑆 has even cardinality if it can be written as the disjoint union of two subsets 𝐴,𝐵 which have the same cardinality. [2]
In other words, a set is even if it can be paired up, by finding one grouping where it pairs. Finding alternative groupings that do not pair does not matter.
[1] https://math.stackexchange.com/a/49046
[2] https://math.stackexchange.com/a/49045
Because as I stated in another comment, you could just as easily say odd cardinality exists if you can find two subsets with the same cardinality and there's one element left over, and otherwise we call it even.
So at the end of the day, what you're saying is that ultimately infinity would be even just because mathematicians arbitrarily defined 'even' that way -- not because there's any intuitive logic behind it, any deeper justification, or any necessary consistency with parity for finite sets.
Actually the fact that splitting it into pairs is the same as splitting into two equal sets of equal cardinality is itself non-trivial. The reason why shows up when you try to get the two definitions closer together.
Splitting an ordinal into pairs is essentially splitting it into ordered pairs (a_i, b_i) such that the map i to a_i is monotonic and for no i<j the pair (a_i, b_i) overlaps with (a_j, b_j) in the sense that a_j <= b_j.
Splitting a set into pairs is splitting it into sets {a_i, b_i} such that for no i != j the two sets {a_i, b_i} and {a_j, b_j} overlap.
These two are note the same, you can split pretty much any infinite set into two disjoint sets of equal cardinality.
It's hard to get the definitions general enough to get one definition for both ordinals and sets. Mostly because products of ordinals are a bit weird. For sets (and most other types of mathematical objects) it doesn't matter which way around you pair things up, but for the ordinals you end up with a completely different object if you do it the other way around and this is apparently the more interesting definition of the two.
It's also not clear to me what justification there would be for a "preference" for the "even" category that way -- it seems arbitrary. Why not be odd if there exists a way to pair things off such that one is left over, and even if there isn't such a way?
Deleted Comment
And you then you could retort with sets of size four, and I could use five, and then we can argue about whether we'll end up at the limit with more odd sets or even sets, and now we're arguing in circles. Reductio ad absurdum.
> maybe I'm missing something
The answer said:
> the usual definition is that an ordinal number 𝛼 is even if... Otherwise, it is odd.
In other words, if a number could be proved to be even, it is even. If not, it is odd.
Using their definition, there is no such thing as "proving a number is odd". You'd have to do it by failing to prove it's evenness. In the case of infinity, because we can successfully prove evenness, it's even and not odd.
It's only even or odd with respect to other infinities which the cardinal numbers can count based on the presence of a bijection or not. It's a kind of relative parity.
That directly contradicts the quoted text I included from the original answer though, as far as I understand. It directly asserted that "every infinite cardinal is even".
> It's a kind of relative parity.
What is relative parity? The original question was whether infinity is even or odd... I don't know what you mean by relative parity.
I would have thought the reverse, but ¯\_(ツ)_/¯
And if you implement, e.g.:
Then IsEven(Infinity) == false and IsOdd(Infinity) == false, as expected.The answer this quote came from is amazingly obtuse, but it does make me think that infinity must be even since infinity can be divided into 2 pairs, each of which is of equal size since both are infinity.
The definition based on transfinite ordinals explained in the same answer does seem interesting, and I wouldn't be surprised if it were useful. I think this is a case of simplification gone wrong, where everything interesting was lost in the translation to more accessible terminology.
A more honest thing to say to a child would be that the way even and odd are defined only make sense for finite numbers. It's true for the definition they know, and it introduces them to the important insight that logical rules that are created for one kind of thing might not work when applied to something else. I think this would be more accessible and stimulating for a six-year-old than giving them a half-baked verbal imitation of a result from transfinite mathematics.
They'll be thrilled later if they study math and discover that there are definitions of "infinity" and "even" that yield an answer to their childhood question.
(And this quickly resolves the case of this article, since lim x->inf x-2*floor(x/2) does not exist).
This is true,
but the same is true of (infinity - 1)
Therefor infinity must also be odd.
If you are thinking about the difference between
and Then I regret to inform you the former is omega and the latter is 1+omega which is the same as omega. In other words attempting to subtract one from infinity by removing from the front results in infinity.Deleted Comment
Edit: perhaps you meant one is in the middle?
Flowing the standard notation, where the usual infinite in the integer or the real line is "ω = ∞ = 1,2,3,..."
ω+1 = ω+1 , i.e. "the next thing after infinity"
1+ω = ω , i.e. "the same infinity as before"
2ω = ω , i.e. "the same infinity as before", so it's even
1+2ω = ω , i.e. "the same infinity as before", so it looks odd, but don't fall in that trap
ω2 = ω2 , i.e. "two infinities chained together", that is weird
Two more weird example from https://en.wikipedia.org/wiki/Even_and_odd_ordinals
> Unlike the case of even integers, one cannot go on to characterize even ordinals as ordinal numbers of the form β2 = β + β. Ordinal multiplication is not commutative, so in general 2β ≠ β2. In fact, the even ordinal ω + 4 cannot be expressed as β + β, and the ordinal number
> (ω + 3)2 = (ω + 3) + (ω + 3) = ω + (3 + ω) + 3 = ω + ω + 3 = ω2 + 3
> is not even.
For a six year old, I'd tell that infinite is not a number so it's not even or odd. If s/he even get's a Ph.D. in math, s/he will understand.
Moreover, I remember when I was a graduate T.A. that one day before lunch I went to a class to learn about the https://en.wikipedia.org/wiki/Alexandroff_extension in the morning. (The idea is that you add one ∞ to a set of numbers to get a compact set. And in the new set ∞ is (almost) a number as good as the other numbers.) After lunch, I went to teach limits to first years students, and with a total straight face I told them that ∞ is not a number.
When you apply Alexandroff extension to add the point at infinity to, say, the real numbers, what you're left with is not a set of numbers (i.e. a field) anymore. So it makes sense to say that ∞ is not a number. Moreover, the way ∞ is used in analysis is different from Alexandroff compactification, in that you usually use two infinities (±∞) as a shorthand for quantification over increasing or decreasing sequences of real numbers (this can be formalized using extended real numbers [0] or other gadgets but doing so has no advantages in a first-year analysis class, and might in fact make matters worse).
[0] https://en.wikipedia.org/wiki/Extended_real_number_line
> two infinities (±∞)
It depends. In the real numbers it depends, but in most cases I agree that it's better to use two. In complex analysis it's much better to have only one infinity. And there are more weird case like the projective plane where you have one infinity in each direction.
> So it makes sense to say that ∞ is not a number.
I agree, it's not longer a field and the operation lose many properties if you try to extend them. So I said "(almost) a number". Anyway, the weird part is that in some cases you can write f(∞) in an advanced math course, but you can never write f(∞) in a fist year math course.
Depends on which transfinite algebra you're working with. If you restrict "number" to mean "element of an ordered field" (thus excluding things like the "complex numbers" but matching the usual intuition of how numbers should behave) then you can't include Cantor's ordinals but you can include the Surreal Numbers. Those include infinite ordinals and (due to being a field) have commutative addition and multiplication operations.
People seem to assume that they know a couple of tricks about infinity (adding, multiplying) and don't stop to think that there should be a much more rigorous definition. Which, they shouldn't -- the average person will never _actually_ care about transfinites.
I find this concept perplexing. To me this implies that "infinity" has a value. How can you add 1 to a thing that by definition has no value?
Imagine the + in C++ when you have to add two complex numbers. They are just a struct with x and y, and some magic to make all operations work as intended.
The use of the + in this example is more like the concatenation of strings, like "Hello " + "World!" is "Hello World!". But in this case, the content of the string doesn't matter so "Hello " == "World!" and there are some magical strings that are infinite.
The idea is that anyone can overload the symbol + and sum whatever they want. It's not necessary to use + with numbers. Obviously, most silly overloads are ignored, and nobody use them. In this case it's a popular overload so it is teach in an advanced math course and has it's own Wikipedia article.
[0]: https://math.stackexchange.com/a/36298
Dead Comment
Sometimes I wonder if there's a better math language waiting to be invented that eschews the non-discrete.
https://en.wikipedia.org/wiki/Ordinal_number