OOTS #1277 - The Discussion Thread

[tanonx]

Except that high priest offered no assistance for the world ending quest. She offered assistance for the dwarf lands, which are under attack and she wants to protect because she's from there.

You can certainly believe that's what happened here. It requires walking two very fine lines: One, that giving adventurers proof of your blessing on their cause, in your own words, actually means you've blessed them for the other, almost-the-same cause you never said but were definitely thinking. Two, that a High Priest's own deity-channeling revelation has informed them the whole world will end, but they don't believe said deity they devoted their entire life to... but they do believe a very similar, localized reasoning that led them to support the same cause, just not because they believe it.

Your skill in doing so is commendable, but I can't join you. I will simply settle for the plain text reading that the High Priest supported the Order's cause.

[Jasdoif]

I am not trying to match numbers on a mapping I am mapping them directly - I have always been mapping them directly to highlight that there would a leftover, I have fully granted that if you apply a mapping you can map a 1-3 into 1-2.

Hence why I feel giving you an example that matches your scenario would be faulty - it has never been in debate.

If you wish to review the post again you will see I said: "if you match every number from S1 to the same number in S3 then there will still be unmatched numbers in S3."

Bolded the 'same' for clarity.

It's all true! S1 is an infinite subset of the infinite S3 set, and S1 and S3 are the same size of infinity since literally all numbers of S1 map one-to-one with numbers of S3 and vice versa. You have to apply a finite restriction (a boundary condition, perhaps, or maybe granularity) before you can say one is smaller than the other; and at that point you're no longer talking about the infinite S1 and S3, much less their sizes. It's much like how you'll never run out of odd numbers or whole numbers, but once you set a (non-negative) boundary you'll easily find more whole numbers than odd numbers (on the side that includes zero).

Infinities are exciting, as this is an obvious point where math steps from the granular to the systemic; with all the mind-opening/bending/blowing that entails. Like Cantor's diagonalisation argument, which proves that even an infinite list of numbers cannot contain all real numbers, and mathematicians were like "wait, there's more than infinite number of numbers? That makes no sense! Dang it, we need to come up with ways to categorize infinity to describe this!"

[Peelee]

You can certainly believe that's what happened here. It requires walking two very fine lines: One, that giving adventurers proof of your blessing on their cause, in your own words, actually means you've blessed them for the other, almost-the-same cause you never said but were definitely thinking. Two, that a High Priest's own deity-channeling revelation has informed them the whole world will end, but they don't believe said deity they devoted their entire life to... but they do believe a very similar, localized reasoning that led them to support the same cause, just not because they believe it.

Your skill in doing so is commendable, but I can't join you. I will simply settle for the plain text reading that the High Priest supported the Order's cause.

Go back one single strip and the cause is specifically "going after the vampires". Again, there is nothing said or given to help with the world-boom-problem. You can read that into the strip all you want, i can't stop you. And at the risk of repeating myself, the characters have told us at least twice that the Order is pretty much on their own here. If you wasnt to keep engaging in the idea that no, it's still possible despite the abundance of information saying it is not, then I don't think anything else I can say will change your mind. I'm done with this.

[dancrilis]

It's all true! S1 is an infinite subset of the infinite S3 set, and S1 and S3 are the same size of infinity since literally all numbers of S1 map one-to-one with numbers of S3 and vice versa. You have to apply a finite restriction (a boundary condition, perhaps, or maybe granularity) before you can say one is smaller than the other; and at that point you're no longer talking about the infinite S1 and S3, much less their sizes. It's much like how you'll never run out of odd numbers or whole numbers, but once you set a (non-negative) boundary you'll easily find more whole numbers than odd numbers (on the side that includes zero).

Infinities are exciting, as this is an obvious point where math steps from the granular to the systemic; with all the mind-opening/bending/blowing that entails. Like Cantor's diagonalisation argument, which proves that even an infinite list of numbers cannot contain all real numbers, and mathematicians were like "wait, there's more than infinite number of numbers? That makes no sense! Dang it, we need to come up with ways to categorize infinity to describe this!"

Huh - you have just highlighted to me that I got me 'matching' and 'mapping' words incorrect when I was typing, well thats annoying but thanks for helping me to notice.

It's much like how you'll never run out of odd numbers or whole numbers, but once you set a (non-negative) boundary you'll easily find more whole numbers than odd numbers (on the side that includes zero).

Kindof all I have been saying (however poorly I might have been saying it).

Go back one single strip and the cause is specifically "going after the vampires". Again, there is nothing said or given to help with the world-boom-problem. You can read that into the strip all you want, i can't stop you. And at the risk of repeating myself, the characters have told us at least twice that the Order is pretty much on their own here. If you wasnt to keep engaging in the idea that no, it's still possible despite the abundance of information saying it is not, then I don't think anything else I can say will change your mind. I'm done with this.

In fairness we might have the armies of the heavens on side depending on what Roy asked the archon (panel 2) - in terms of forces that might come to help I would think that might be the only potential hook (but could be forgetting something and frankly I doubt the higher planes will get overly involved regardless of what Roy asked for).

[DavidSh]

I think it needs to be made more clear that two sets A and B have the same cardinality if and only if there are both a mapping that maps distinct elements of A to distinct elements of B, and a mapping that maps distinct elements of B to distinct elements of A. These can be tricky to construct, for example, if A is the closed unit interval of the reals {x: 0 <= x <= 1} and B is the open unit interval of the reals {x: 0 < x < 1}.

For finite sets, you cannot construct such a pair of mappings between a set A and a strict subset B of A, such as B = {x in A such that x is not equal to y}. For infinite sets, you sometimes can construct such mappings. The fact that there are such mappings between an infinite set A and a strict subset C of B, leaving some points left over, does not prevent there being such mappings between A and B.

But you don't have to mean "cardinality" when you say "size". You can also mean a measure such as one of the generalizations of interval length. Then the measure of the real interval [0,2] could be twice the measure of the real interval [0,1].

The array of doors leading to dungeons in the comic appears to be finite, so these issues don't really apply.

[theNater]

Consider four sets:
Set1: All the numbers between 1 and 2.
Set2: All the numbers between 1 and 2 and the numbers 1 and 2.
Set3: All the numbers between 1 and 3.
Set4: All the numbers between 1 and 2.

All four sets have an infinite amount of numbers.

If you remove all the numbers from Set2 that it has in common with Set1 then it has two numbers remaining (1 and 2).
If you remove all the numbers from Set3 that it has in common with Set1 then it has an infinite amount of numbers remaining.
If you remove all the numbers from Set4 that it has in common with Set1 then it has zero numbers remaining.

In these cases Infinity = Infinity is not true, these infinities are not equal (except for Set1 and Set4 which are).

Let's add one more set:
Set5: All the numbers between 3 and 4.

Would you say this set is the same size as any of the other four sets? If so, which and why?

[Kish]

How infinite is the set of forum posters arguing about infinite sets?

[lio45]

It's all true! S1 is an infinite subset of the infinite S3 set, and S1 and S3 are the same size of infinity since literally all numbers of S1 map one-to-one with numbers of S3 and vice versa.

Exactly what I was going to say. A more helpful way to "inform dancrilis that they're wrong" is to point out that the definition of "size", when applies to infinite sets in math, is not the standard layman's one.

"Set A is a subset of Set B" and "Set A and Set B are sets that have the same 'size'" are not a contradiction.

[brian 333]

I am not trying to match numbers on a mapping I am matching them directly - I have always been matching them directly to highlight that there would a leftover, I have fully granted that if you apply a mapping you can map a 1-3 into 1-2.

Hence why I feel giving you an example that matches your scenario would be faulty - it has never been in debate.

If you wish to review the post again you will see I said: "if you match every number from S1 to the same number in S3 then there will still be unmatched numbers in S3."

Bolded the 'same' for clarity.

Okay, let's try this:

You are saying that all whole numbers is a smaller set than all integers. To prove this you are matching 1 to 1, 2 to 2, 3 to 3 and so on. At the end of your sequence you declare that all these unmatched negative numbers demonstrate that the set of all integers is larger.

This appears true because in grade 1 you learned about sets and how a set with three oranges is larger than a set with two. Cancel the same number of oranges in both sets and what is left over demonstrates that set to be larger. But which is larger, a set with three oranges and a banana, or a set with four oranges?

Whole numbers and integers are different things, so they don't directly cancel. One must find a common denominator to allow cancellation. Let's call these things numbers, and numbers in one set can then be matched against the other.

Now, here's the tricky part, get ready.

I match 1 to 1, 2 to -1, 3 to 2, 4 to -2, 5 to 3, 6 to -3, 7 to 4, 8 to -4, 9 to 5, 10 to -5, 11 to 6, 12 to -6, and so on. I can do this forever. Every even number matches one of your negative numbers infinitely.

The sets are the same size. One contains items not found in the other, but like the set of oranges and bananas, it is not larger than the set with just oranges.

[carrion pigeons]

Kindof all I have been saying (however poorly I might have been saying it).

I just stumbled on this discussion, and wanted to point out that the boundary condition he's talking about puts us squarely back into non-infinite sets. If you say all positive even numbers smaller than 100 vs all natural numbers smaller than 100, then yeah, obviously there are 50 more of the latter. But that argument only works because there's an endpoint for you to reach. When counting through a set never gets to an end, how can you possibly make the same argument? It'll take you exactly as long to count 1,2,3,4,5,... forever as it will to count 2,3,4,5,6,... forever, so the fact that the set described by the former includes a number that isn't in the set described by the latter is fully irrelevant to the question of whether one is bigger than the other.

I'm not going to get into a conversation about this like the Wolf guy did because I don't care if you want to be wrong, and I don't check this site nearly often enough to keep up with it anyway, but any discussion about infinity size that involves *getting to the end* of one or both sets in order to quantify anything about either one of them is never going to hold water, because that's the exact thing that characterizes an infinite set: you can't do that. Making claims like "there are exactly as many numbers from 1 to 2 as there are from 1 to 3" is provably true, but it remains a silly thing to say because the comparison you're making is the same as saying "non-number X has the same numeric value as non-number Y, because the numeric value of both X and Y do not exist".

[KorvinStarmast]

TCOM was Pratchet. No idea which PA you were thinking of, and it doesn't actually matter, since they are all written in very much the same style.

Sorry, I was remembering incorrectly. It's been over 40 years.
The Source of Magic

[HalfTangible]

Okay, let's try this:

You are saying that all whole numbers is a smaller set than all integers. To prove this you are matching 1 to 1, 2 to 2, 3 to 3 and so on. At the end of your sequence you declare that all these unmatched negative numbers demonstrate that the set of all integers is larger.

This appears true because in grade 1 you learned about sets and how a set with three oranges is larger than a set with two. Cancel the same number of oranges in both sets and what is left over demonstrates that set to be larger. But which is larger, a set with three oranges and a banana, or a set with four oranges?

Whole numbers and integers are different things, so they don't directly cancel. One must find a common denominator to allow cancellation. Let's call these things numbers, and numbers in one set can then be matched against the other.

Now, here's the tricky part, get ready.

I match 1 to 1, 2 to -1, 3 to 2, 4 to -2, 5 to 3, 6 to -3, 7 to 4, 8 to -4, 9 to 5, 10 to -5, 11 to 6, 12 to -6, and so on. I can do this forever. Every even number matches one of your negative numbers infinitely.

The sets are the same size. One contains items not found in the other, but like the set of oranges and bananas, it is not larger than the set with just oranges.

One of these sets has twice as many numbers to draw upon as the other, therefore they are not the same size. Your own set shows this.

[lio45]

One of these sets has twice as many numbers to draw upon as the other, therefore they are not the same size. Your own set shows this.

For the [N→∞]th time, this is wrong.

I'll reiterate what I posted above: stop thinking about the common sense definition of "size".

But hey, you're in decent company when you're finding this counter-intuitive:

https://www.scientificamerican.com/a...r-than-others/

At first sight, this definition of size seems to lead to contradictions, which were elaborated by the Bohemian mathematician Bernard Bolzano in Paradoxes of the Infinite, published posthumously in 1851. For example, Euclid’s “The whole is greater than the part” appears self-evident. That means if a set A is a proper subset of B (that is, every element of A is in B, but B contains additional elements), then A must be smaller than B. This assertion is not true for infinite sets, however! This curious property is one reason some scholars rejected the concept of infinite sets more than 100 years ago.

For example, the set of even numbers E = {0, 2, 4, 6, …} is a proper subset of the natural numbers ℕ = {0, 1, 2, …}. Intuitively, you might think that the set E is half the size of ℕ. But in fact, based on our definition, the sets have the same size because each number n in E can be assigned to exactly one number in ℕ (0 →0, 2 →1, 4 →2, …, n →n/2, …).

Consequently, the concept of “size” for sets could be dismissed as nonsensical. Alternatively, it could be termed something else: cardinality, for example. For the sake of simplicity, we will stick to the conventional terminology, even though it has unexpected consequences at infinity.

This was actually the example I was about to (completely independently!) give today: the set of even numbers being the same size (under that definition of size that doesn't preclude one from being totally contained within (so "smaller than", in layman's terms) the other) than the set of natural numbers.

Why are they the same size? Because that's how size is defined: if it's guaranteed you can match/pair the elements from the sets in a one vs one relationship, as Grey Wolf said like 10+ times now, then they are the same size.

So, name any natural number, make it as big as you want, I'll still be able to name you that number's unique 1-to-1 even number pairing (i.e. 2x your number).

[Fyraltari]

One of these sets has twice as many numbers to draw upon as the other, therefore they are not the same size. Your own set shows this.

Infinity times two is still infinity. The very same kind of infinity.

Because infinity isn't a number the notion that a particular infinite amount may be twice, thrice or any other -ice as big as another particular amount is nonsensical.

[Jacky720]

Here's a viewpoint that's wrong:

Consider the set of naturals, and the set of positive evens.
Consider the function, N(x) = x represents the number of natural numbers between 0 and x, and E(x) = x/2 represents the same for evens.
We can say that the relative sizes of a finite portion of these sets is given by N(x)/E(x), which is a 2:1 ratio for any x.
I might say that "the set of evens covers half as many numbers on the way to 47 as the set of naturals does."
And even if you take the limit of N(x)/E(x) as x approaches infinity... it remains 2.
So why can't I take that limit, and say N is twice the size of E?

Unfortunately, I lack the understanding to get why that's wrong. I like limits, and they usually don't lead me astray. Maybe it's just the idea of dividing infinity by a finite value that's considered a no-go. But I feel like the rate at which these two functions approach infinity ought to be a valid concern, beyond "they are both infinite and both countable and both have the same cardinality of being countable, whatever a cardinality is."

[Fyraltari]

Here's a viewpoint that's wrong:

Consider the set of naturals, and the set of positive evens.
Consider the function, N(x) = x represents the number of natural numbers between 0 and x, and E(x) = x/2 represents the same for evens.
We can say that the relative sizes of a finite portion of these sets is given by N(x)/E(x), which is a 2:1 ratio for any x.
I might say that "the set of evens covers half as many numbers on the way to 47 as the set of naturals does."
And even if you take the limit of N(x)/E(x) as x approaches infinity... it remains 2.
So why can't I take that limit, and say N is twice the size of E?

Unfortunately, I lack the understanding to get why that's wrong. I like limits, and they usually don't lead me astray. Maybe it's just the idea of dividing infinity by a finite value that's considered a no-go. But I feel like the rate at which these two functions approach infinity ought to be a valid concern, beyond "they are both infinite and both countable and both have the same cardinality of being countable, whatever a cardinality is."

This only works because of your specific example.

Take the Naturals (N) and the Rationals (Q). All Naturals are Rationals so by your definition, Q has a greater size than N. How much bigger is it? You cannot answer that because there is an infinite number of Rationals betwen each natural.

Yet, Rationals and Naturals are both countable infinities, which is why we can say they have the same size.

[HalfTangible]

Infinity times two is still infinity. The very same kind of infinity.

Because infinity isn't a number the notion that a particular infinite amount may be twice, thrice or any other -ice as big as another particular amount is nonsensical.

You can have infinities of different sizes.

[Jacky720]

This only works because of your specific example.

Take the Naturals (N) and the Rationals (Q). All Naturals are Rationals so by your definition, Q has a greater size than N. How much bigger is it?

I would say "bigger at a ratio of the countable infinity of rationals between zero and one, to one", but you're right that I couldn't calculate that with my stated method.

I did a little more Googling. Would the concept of "natural density" be relevant to this conversation? It seems to map more closely onto that hard-to-shake intuitive idea of size, but it only works with subsets of N.

[lio45]

One of these sets has twice as many numbers to draw upon as the other, therefore they are not the same size. Your own set shows this.

You can have infinities of different sizes.

Your own link is proving you incorrect. The naturals, the evens, the rationals are all infinities of the same size.

[HalfTangible]

Your own link is proving you incorrect. The naturals, the evens, the rationals are all infinities of the same size.

Incorrect .

In other words, p is a real number without a natural number partner—an apple without an orange. Thus, the one-to-one correspondence between the reals and the naturals fails, as there are simply too many reals—they are "uncountably" numerous—making real infinity somehow larger than natural infinity.

[lio45]

Incorrect .

Lol.

You were wrong when disagreeing with Brian when he (correctly) said that ℕ and ℤ are infinities of the same "size", and you're doubling down on not knowing what you're talking about by insisting that since ℝ is a "greater" infinity than ℕ (which isn't anywhere near news to anyone who knows what they're talking about), it "proves your earlier point".

[HalfTangible]

Lol.

You were wrong when disagreeing with Brian when he (correctly) said that ℕ and ℤ are infinities of the same "size", and you're doubling down on not knowing what you're talking about by insisting that since ℝ is a "greater" infinity than ℕ (which isn't anywhere near news to anyone who knows what they're talking about), it "proves your earlier point".

I say "you can have infinities of different size".

You say that my own link proves me wrong.

I point you to the part of the link that shows exactly what I was saying: that infinities can be of different sizes.

And now you are saying I'm "doubling down on not knowing what I'm talking about"

You also put quotations around something that as I recall I never said in any capacity in this thread.

We're done.

[Breccia]

I admit I'm a little late to the party, but, why are we talking about any kind of infinity when it's all but been confirmed there's a finite number of doors, and that Team Evil will eventually open them all? It was kind of an important plot moment.

I really like the idea of the Gate being behind a gauntlet. I'm even guesing the teleportation traps force you to do the arenas in order no matter which door you open first. But Team Evil seemed to suggest they were re-visiting cleared arenas, so I'm not confident in that.

[Sir_Norbert]

I say "you can have infinities of different size".

You say that my own link proves me wrong.

I point you to the part of the link that shows exactly what I was saying: that infinities can be of different sizes.

And now you are saying I'm "doubling down on not knowing what I'm talking about"

You also put quotations around something that as I recall I never said in any capacity in this thread.

We're done.

It's further up this page. Reply #372. You said: "One of these sets has twice as many numbers to draw upon as the other, therefore they are not the same size. Your own set shows this."

You said that in response to brian333's correct claim that the set of positive integers and the set of all integers are the same size.

[Windscion]

The idea that [1:3] has "twice as many" real numbers as [1:2] is incorrect, because infinite numbers in general do not work that way.

0 <= counting number < countable infinity (Aleph₀) < size of real number set (c or Aleph₁)

All countable infinities are the same size; they are all greater than any finite integer and all are smaller than the size of the set of real numbers. You cannot say that 2*Aleph₀ > Aleph₀. In fact,

Aleph₀ + Aleph₀ = Aleph₀ * Aleph₀ = sqrt(Aleph₀) = Aleph₀.

Aleph₀ - Aleph₀ doesn't exist
Aleph₀ / Aleph₀ doesn't exist.

The reason the rules work this way is logical, and has been explained already, but to summarize: if two sets can be mapped onto each other, they are the same size. Here onto means covering the entire set, nothing left out. A 1-to-1 mapping is NOT required to prove this, it's just simpler to use a mapping that covers both directions.

Using (a:b) to means all values x such that a < x < b and [a:b] to means all x such that a <= x <= b, we get

(0:1) maps onto [0:1] by multiplying x in (0:1) to x' = 1.5*x-0.25, covering (-0.25:1.25) which includes all the points in [0:1]
[0:1] maps onto (0,1) via the identity map.
In both cases "having stuff left over" doesn't matter. They are the same size.

I pray this sheds some light on the matter.

[The MunchKING]

I admit I'm a little late to the party, but, why are we talking about any kind of infinity when it's all but been confirmed there's a finite number of doors, and that Team Evil will eventually open them all? It was kind of an important plot moment.

Because someone else brought up infinite planes in a dream realm.

[Breccia]

Because someone else brought up infinite planes in a dream realm.

Appreciated, saved me the trip.

[lio45]

I say "you can have infinities of different size".

You say that my own link proves me wrong.

Yep. Your own link proved your earlier post wrong. (As Sir Norbert tried to clarify for you.)

I point you to the part of the link that shows exactly what I was saying: that infinities can be of different sizes.

And now you are saying I'm "doubling down on not knowing what I'm talking about"

Exactly. Because if you were at least somewhat familiar with the subject and/or had read your own link, you'd know that you were wrong with your response to brian333's correct claim that the set of positive integers and the set of all integers are the same size.

We're done.

Yeah, I should probably follow the excellent advice of "carrion pigeons" earlier on this page: "I'm not going to get into a conversation about this like the Wolf guy did because I don't care if you want to be wrong (...)"

[Fyraltari]

You can have infinities of different sizes.

Yes, I know. We all here know that some infinities are bigger than others. That's what cardinality measures. R is bigger than N. But do you know how big? It isn't twice as big, it is ten times as big, it's infinitely bigger. Because the notion of an infinity "twice the size" of another infinity is nonsensical which is what I was talking about.

[1:2] isn't "half the size" of [1:3], they are of the same size, simple as that.

[Edreyn]

Because someone else brought up infinite planes in a dream realm.

If I only knew what it will cause.