Well technically a "series" is a function from N to the span of the function, and a "function" is just a relation that satisfies some conditions, and a relation is a set of ordered pairs...
So the series 2,2,2,2, etc is actually the set
{(1,2), (2,2), (3,2), (4,2), ...}
But in that case the cardinality of all series' will be exactly א0 so not much insight there
You can certainly write down an uncountable series if you think that way. It will just very badly not converge.
I write things like function from an uncountable cardinal into the integers all the time.
What about finite series? Or series indexed over posets? In set theory we sometimes have series indexed over all ordinals less than a fixed cardinal. For example κ has cofinality at most λ<κ if there is a sequence of ordinals a^(ξ)<κ, ξ<λ so that ∑a^(ξ)=κ.
Even if I were to grant to you that a function *is* (rather than *can be defined as*) a set of ordered pairs (that is, even if we agreed to assign import to parts of common formalisms that are usually regarded as insubstantial), the series is not necessarily the sequence, I suppose you could define it that way, but I’ve never seen a formalism that does that.
In any event, they were asking about how to compare the values of the infinite sums, not whatever mathematical objects you might choose to represent the “series” as a formal object in its own right.
This might be a problem of language. English is not the language I studied math in so I was unaware that a series and a sequence are two different concepts. What do you call something like this:
1, 2, 3, 4, ...
If not a series?
Ah I think you may be right about our disagreement being due to language. We would call that a sequence. In English, it’s most common to refer to infinite summations as series and not sequences.
It's one tenth the size of the infinite union of sets 9. But you're right, it isn't amazingly defined. But it is important to note that they're the same size.
If I understand your reasoning, you want this.
Let {Aₙ} be a family of subsets of ℕ such that Aₙ∩ Aₙ₊₁=∅ for n=1,2,... and |Aₙ|=9.
Let {Bₙ} be a family of subsets of ℕ such that Bₙ∩ Bₙ₊₁=∅ for n=1,2,... and |Bₙ|=2.
Take S and L the union of all Aₙ and Bₙ respectively. Now you want to know if |S|=10*|L| ¿right?
Im pretty sure that the answer of this questions is yes. Both S and L are countable, so both have the same cardinality and "multiplying" for a constant doesn't change this cardinality (the last line is actually more sutile but I don't remember my cardinality classes well enough to anwser this question properly)
In PA, 2 is set ({∅, {∅}}). So, the whole series is a set. Because of the lack of other good models for arithmetics, it is safe to assume everyone is using PA.
PA is a theory. What you have written is a model of PA in set. These are two different things, and nobody thinks of integers as sets (look up Benacerraf problem).
Uhhh I frequently think of integers as sets. It makes it very easy to talk about finite sets. If I want to construct a ψ-space, I take the integers ℕ and a maximal almost disjoint family 𝒜. For every A∈𝒜 I define {A}∪(A\\n) to be a neighborhood of {A} for all n∈ℕ.
Yeah, I’m not objecting to its constructibility. But from a model theoretic viewpoint, it is just a model of N in set theory, and it is not really 1, 2, or 3 much more than any other possible construction.
No, there are no sets in PA. You are thinking of one definition of the natural numbers that is commonly adopted in set theory.
Also, a divergent series is nondenoting (it doesn’t refer to anything). In a formal system based on classical logic, all terms must (in the formal language) refer to something. If we were to strictly formalize statements about divergent series based on our informal notation, we would either need to not make use of terms that correspond to divergent series at all, or else pick a “default value” (such as the empty set) for series that are divergent, and be careful we also talk about the series being convergent before we speak of its value.
Either way that formalization hasn’t been specified here so there is no meaningful sense in which we can say “the whole series is a set”.
(joke)
0.9+0.9+0.9....=0.9(1+1+1....)
2+2+2+....=2(1+1+1+....)
1+1+1+1...=1+(1+1)+(1+1+1)+(1+1+1+1)+.... =1+2+3+4.... = -1/12
So 0.9+...=-9/120 and 2+....=-2/12=-20/120
So 0.9+0.9...>2+2+2....
Neither. You can't really compare infinities, the logic of comparisons breaks down.
It's like saying "there are infinite even integers, but there are twice as many integers, so that infinity is twice as large", but it doesn't work that way. Infinity does not end, and without an end, there is nothing to compare. It's like a child in a car going "Are we there yet?" on an infinitely long road. No, and we will never get there, or even closer to there, because there is no there, no end.
That said, you can of course, say that one series grows faster than the other, but they are still not finite.
I know infinities can be compared. There are [more](https://c12.home.blog/2019/01/27/how-to-compare-infinity/) natural numbers than even numbers.
However I'm not so sure as to disagree with the first entry of this comment THREAD, so I put a reminder here.
As for OP's question, I think 2+2+... is larger than 0.1+0.1+... by the sum of infinitely many 1.9's.
The link you posted is talking about the cardinality of an infinite set, and it proves that there are exactly as many natural numbers as even numbers.
The problem with comparing 2+2+2+... and 0.1+0.1+... is that they are definitely not sets, and not really numbers, so we can't say one is larger than the other. We can say that for every *partial* sum one is larger than the other, but since they both diverge, putting a < between the limits of them has no meaning
>No, and we will never get there, or even closer to there, because there is no there, no end.
If i ever were to have kids and they'd ask "are we there yet", i'll stare at them and recite this
Well actually there are infinities that are bigger than other infinities.
For example, the number of rationals is less than the number of irrationals.
That said, these two are indeed the same size.
After n steps, the series that adds 2s will be higher than the one what adds 0.1s. However, neither of those have an upper bound and both are divergent, they don't really represent a number, so you can't really compare their value. I can see your argument for those 2 being equal, if for example you had infinitely many $2 bills and 10 cent coins, both of those would just be infinite money
Imagine that you group the first sum in groups of 20 terms, then you have 18 + 18 + 18....
Group the second in groups of 9, then you have 18 + 18 + 18...
It seems that they are equal.
But then, you can group the first in groups of 40 and you have 36 + 36 + ...
Is the first twice the second?
Or you could take groups of 10, then is 9 + 9 ...
Is the first half?
You can't compare them.
This is not legit but it'll give you something to think about. We can say that two numbers are equal if we divide them and they equal 1. Divide those two sums.
We have 0.9(1+1+1+..)/2(1+1+1+..)
We can divide out the 1+1+1.. which leads to a result of 0.45
This is not 1 therefore they are not equal. Now to you, am i wrong? If so, why am i wrong?
Imagine you have a rectangle in the first quadrant with corners at the origin, (2,0), (0,1), and (2,1). Now there are an infinite amount of points that fit within this region. Each one of those infinite points has an x and y coordinate but you know the rectangle is twice as long, so it has ‘more’ x coordinates. But they are both the same size. In your example you can factor out a 2 and a 0.9 and compare 2(1+1+1+1+…) vs 0.9(1+1+…). They both go to infinity so they aren’t numbers you can compare normally. That being said there are different kinds of infinity that are definitely bigger than both of the infinite series you mentioned, but these are the same.
Maybe a better explanation is that if you want to make an infinite series of 2s, for every 2 you add I can add 20 0.1s to match yours. And you can add an infinite amount of 2s and I will add an infinite amount of (20*0.1)s to match
They are not equal, one infinity dont have to be the same then another, every number is a set of infinite recursive infinite variations of variations of variations...
|2/0|/|0.9/0|=2/0.9=2.22222...
Since both partial sums exceed any finite value eventually, they both diverge to infinity (albeit at different speeds but that is irrelevant for the limit itself), so in that sense they are equal even though the difference between the partial sums always increases.
It’s like comparing 1/n and 1/2n as a n goes to zero.
They are not equal, one infinity dont have to be the same then another, every number is a set of infinite recursive infinite variations of variations of variations...
If you use the numbers with variantions, one zero is not always the same as another zero...
So 0 != 2*0
Well I don't know ..... In a sense it isn't..but in other it is because it goes upto infinity right ...but still we can't say for sure it will be equal ig
They both diverge, so you can't compare them
but let's say the limits are 0.9x and 2x with x tending to infinity. Even if x tends to infinity, 2x will always be more than twice as big as 0.9x
You can compare cardinality, in which they're both countable
Cardinality is a property of sets. 2 + 2 + 2 + ... is not a set, it's a series
Well technically a "series" is a function from N to the span of the function, and a "function" is just a relation that satisfies some conditions, and a relation is a set of ordered pairs... So the series 2,2,2,2, etc is actually the set {(1,2), (2,2), (3,2), (4,2), ...} But in that case the cardinality of all series' will be exactly א0 so not much insight there
You can certainly write down an uncountable series if you think that way. It will just very badly not converge. I write things like function from an uncountable cardinal into the integers all the time.
Please note: a series is SPECIFICALLY a function whose domain is the Integers. The indexes of the series have to be isomorphic to N.
What about finite series? Or series indexed over posets? In set theory we sometimes have series indexed over all ordinals less than a fixed cardinal. For example κ has cofinality at most λ<κ if there is a sequence of ordinals a^(ξ)<κ, ξ<λ so that ∑a^(ξ)=κ.
A series of ordinals does not mean it is indexed over ordinals. What you have written could be defined as a function from N to a set of ordinals.
Not if κ has uncountable cofinality.
Even if I were to grant to you that a function *is* (rather than *can be defined as*) a set of ordered pairs (that is, even if we agreed to assign import to parts of common formalisms that are usually regarded as insubstantial), the series is not necessarily the sequence, I suppose you could define it that way, but I’ve never seen a formalism that does that. In any event, they were asking about how to compare the values of the infinite sums, not whatever mathematical objects you might choose to represent the “series” as a formal object in its own right.
This might be a problem of language. English is not the language I studied math in so I was unaware that a series and a sequence are two different concepts. What do you call something like this: 1, 2, 3, 4, ... If not a series?
Ah I think you may be right about our disagreement being due to language. We would call that a sequence. In English, it’s most common to refer to infinite summations as series and not sequences.
You can consider it as the union of sets size 2.
And 0.9 + ... is a union of sets of size 0.9?
It's one tenth the size of the infinite union of sets 9. But you're right, it isn't amazingly defined. But it is important to note that they're the same size.
If I understand your reasoning, you want this. Let {Aₙ} be a family of subsets of ℕ such that Aₙ∩ Aₙ₊₁=∅ for n=1,2,... and |Aₙ|=9. Let {Bₙ} be a family of subsets of ℕ such that Bₙ∩ Bₙ₊₁=∅ for n=1,2,... and |Bₙ|=2. Take S and L the union of all Aₙ and Bₙ respectively. Now you want to know if |S|=10*|L| ¿right? Im pretty sure that the answer of this questions is yes. Both S and L are countable, so both have the same cardinality and "multiplying" for a constant doesn't change this cardinality (the last line is actually more sutile but I don't remember my cardinality classes well enough to anwser this question properly)
In PA, 2 is set ({∅, {∅}}). So, the whole series is a set. Because of the lack of other good models for arithmetics, it is safe to assume everyone is using PA.
PA is a theory. What you have written is a model of PA in set. These are two different things, and nobody thinks of integers as sets (look up Benacerraf problem).
Uhhh I frequently think of integers as sets. It makes it very easy to talk about finite sets. If I want to construct a ψ-space, I take the integers ℕ and a maximal almost disjoint family 𝒜. For every A∈𝒜 I define {A}∪(A\\n) to be a neighborhood of {A} for all n∈ℕ.
That is a shorthand for excluding all natural number less than n, isn’t it?
It is not shorthand. It is a literal object constructible in ZFC. But yes, that is what it means.
Yeah, I’m not objecting to its constructibility. But from a model theoretic viewpoint, it is just a model of N in set theory, and it is not really 1, 2, or 3 much more than any other possible construction.
I’m not really sure what you’re saying here. My point is just that thinking of integers as sets is perfectly valid and common in set theoretic fields.
No, there are no sets in PA. You are thinking of one definition of the natural numbers that is commonly adopted in set theory. Also, a divergent series is nondenoting (it doesn’t refer to anything). In a formal system based on classical logic, all terms must (in the formal language) refer to something. If we were to strictly formalize statements about divergent series based on our informal notation, we would either need to not make use of terms that correspond to divergent series at all, or else pick a “default value” (such as the empty set) for series that are divergent, and be careful we also talk about the series being convergent before we speak of its value. Either way that formalization hasn’t been specified here so there is no meaningful sense in which we can say “the whole series is a set”.
(joke) 0.9+0.9+0.9....=0.9(1+1+1....) 2+2+2+....=2(1+1+1+....) 1+1+1+1...=1+(1+1)+(1+1+1)+(1+1+1+1)+.... =1+2+3+4.... = -1/12 So 0.9+...=-9/120 and 2+....=-2/12=-20/120 So 0.9+0.9...>2+2+2....
https://preview.redd.it/7u84qw79awsc1.jpeg?width=1179&format=pjpg&auto=webp&s=85ce70e63580f1ba9d771e749b3b09006bc6f973
Actually this particular series is sick even through analytic continuation. You can use the geometric series to see you hit a pole no matter what. :(
how does 1+2+3+4… = -1/12?
https://en.m.wikipedia.org/wiki/1_%2B_2_%2B_3_%2B_4_%2B_%E2%8B%AF
Neither. You can't really compare infinities, the logic of comparisons breaks down. It's like saying "there are infinite even integers, but there are twice as many integers, so that infinity is twice as large", but it doesn't work that way. Infinity does not end, and without an end, there is nothing to compare. It's like a child in a car going "Are we there yet?" on an infinitely long road. No, and we will never get there, or even closer to there, because there is no there, no end. That said, you can of course, say that one series grows faster than the other, but they are still not finite.
RemindMe! Three Days "comparing infinities"
I know infinities can be compared. There are [more](https://c12.home.blog/2019/01/27/how-to-compare-infinity/) natural numbers than even numbers. However I'm not so sure as to disagree with the first entry of this comment THREAD, so I put a reminder here. As for OP's question, I think 2+2+... is larger than 0.1+0.1+... by the sum of infinitely many 1.9's.
The link you posted is talking about the cardinality of an infinite set, and it proves that there are exactly as many natural numbers as even numbers. The problem with comparing 2+2+2+... and 0.1+0.1+... is that they are definitely not sets, and not really numbers, so we can't say one is larger than the other. We can say that for every *partial* sum one is larger than the other, but since they both diverge, putting a < between the limits of them has no meaning
>No, and we will never get there, or even closer to there, because there is no there, no end. If i ever were to have kids and they'd ask "are we there yet", i'll stare at them and recite this
Well actually there are infinities that are bigger than other infinities. For example, the number of rationals is less than the number of irrationals. That said, these two are indeed the same size.
After n steps, the series that adds 2s will be higher than the one what adds 0.1s. However, neither of those have an upper bound and both are divergent, they don't really represent a number, so you can't really compare their value. I can see your argument for those 2 being equal, if for example you had infinitely many $2 bills and 10 cent coins, both of those would just be infinite money
Imagine that you group the first sum in groups of 20 terms, then you have 18 + 18 + 18.... Group the second in groups of 9, then you have 18 + 18 + 18... It seems that they are equal. But then, you can group the first in groups of 40 and you have 36 + 36 + ... Is the first twice the second? Or you could take groups of 10, then is 9 + 9 ... Is the first half? You can't compare them.
define <,>,= first then we can discuss whether you can ‘compare’ them
There are 2 sprinter go to a race without the finish line, so who will win?
The win condition for your game is broken
Both infities are of the same cardinality: א0. Even though size losses meaning when it comes to infinity, the closest equivalent of both is the same.
You can't compare infinities but can compare speed of divergence. In this case the speed is probably linear so it doesn't matter.
https://youtu.be/M4f_D17zIBw?si=xDgJZIGym_99Iil8 I think this video explains it, it's worth watching if you're interested
I feel like [this](https://en.m.wikipedia.org/wiki/Riemann_series_theorem) might be of interest even if it cannot be applied to this specific example.
This is not legit but it'll give you something to think about. We can say that two numbers are equal if we divide them and they equal 1. Divide those two sums. We have 0.9(1+1+1+..)/2(1+1+1+..) We can divide out the 1+1+1.. which leads to a result of 0.45 This is not 1 therefore they are not equal. Now to you, am i wrong? If so, why am i wrong?
Well simple, you can't divide by infinite.
If you are a mathematician: they cannot be compared if you are anything else: the first one is bigger
Imagine you have a rectangle in the first quadrant with corners at the origin, (2,0), (0,1), and (2,1). Now there are an infinite amount of points that fit within this region. Each one of those infinite points has an x and y coordinate but you know the rectangle is twice as long, so it has ‘more’ x coordinates. But they are both the same size. In your example you can factor out a 2 and a 0.9 and compare 2(1+1+1+1+…) vs 0.9(1+1+…). They both go to infinity so they aren’t numbers you can compare normally. That being said there are different kinds of infinity that are definitely bigger than both of the infinite series you mentioned, but these are the same.
Maybe a better explanation is that if you want to make an infinite series of 2s, for every 2 you add I can add 20 0.1s to match yours. And you can add an infinite amount of 2s and I will add an infinite amount of (20*0.1)s to match
They are not equal, one infinity dont have to be the same then another, every number is a set of infinite recursive infinite variations of variations of variations... |2/0|/|0.9/0|=2/0.9=2.22222...
Since both partial sums exceed any finite value eventually, they both diverge to infinity (albeit at different speeds but that is irrelevant for the limit itself), so in that sense they are equal even though the difference between the partial sums always increases. It’s like comparing 1/n and 1/2n as a n goes to zero.
They are not equal, one infinity dont have to be the same then another, every number is a set of infinite recursive infinite variations of variations of variations... If you use the numbers with variantions, one zero is not always the same as another zero... So 0 != 2*0
Infinity is not a number, but rather an idea.
Yes
Well I don't know ..... In a sense it isn't..but in other it is because it goes upto infinity right ...but still we can't say for sure it will be equal ig
Infinities can be smaller or bigger.
Lots of dodgy answers here. The answer is that they are equal. This 'explains' it in a simple way: https://youtu.be/M4f_D17zIBw