OOTS #1277 - The Discussion Thread

[Devlerbat]

The thing I like best about this is that MitD actually did help. If they never figured out that he falsely marked doors, they would have never found the final dungeon and by making them doubt rooms they have already went through they have to spend time rechecking every room (or almost every room if they are lucky) they had to spend more time they otherwise wouldn't have.

[Tzardok]

You mean integers...the set of whole numbers is just the natural numbers with 0 added, no negative numbers.

According to Wikipedia, it can mean both.

[JT]

Yeah, like I said, it isn't intuitively logical.

It really doesn't matter that a bounded infinity includes other bounded infinities. Infinity x2 is exactly the same size as infinity cubed. They may include different components, but all infinities are equally infinite.

This is not the same as apples and oranges. If one has all the apples, and someone else has all the oranges, combining the two creates a greater whole. Combining two finite things does that.

Infinities are imaginary. They do not have limits such as all the apples or all the grains of sand on Earth, or even all the atoms currently existing in the universe. So no matter how infinite one's imagination, one can always imagine doubling the quantity or halving the granularity, or +1 more imaginary unit.

The quantity of any one infinity is the same as the quantity of any two infinities, or of every infinity combined. Infinity = infinity.

Used to be, there were "countably infinite" sets -- all the same level of infinite -- such as "all integers" ... "all positive integers" ... "all even integers" ... "all rational numbers" ... (an infinite number of countably infinite sets, each of which has the same infinite "amount" of elements... aleph-null). Then, there were non-countable infinity sets -- all irrational numbers ... all imaginary numbers ... (also an infinite number of sets, of different countable levels -- aleph-1, aleph-2, ...).


From the limited amount of attention I've paid to esoteric math, there is now dispute over whether there is an infinite number of non-countably infinite "levels" (aleph-1 through aleph-omega), or just one (i.e. all non-countably infinite sets are the same level of non-countable)... OR ... that infinity doesn't actually exist, except as an imaginary and possibly useful mathematics toy.

Possibly because I haven't paid a lot of attention to it, I'm of the opinion that -- as I was originally taught -- there is an infinite number of infinity levels, but that, for the most part, we only need to care about aleph-0 and aleph-1.

[Windscion]

You lost me at Piers Anthony. After The Color of Magic, I knew better than to go for a second helping.

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.

[Liquor Box]

Well this would appear to put to bed the arguments over whether Team Evil would eventually solve the dungeon or not. An argument which fed into the question of whether Serini was acting rationally by attacking the Order and the Paladins, and also fed into the discussion about which gate was best defended.

[Kish]

Does it? The thing is, without the PCs' intervention, I am not sure if it would resolve that debate at:

1) They would have; eventually they would surely have figured out something was off and gone through all the tunnels.
or
2) They would not have; they would have thought they were at the end and gone "now what?" because of the intervention of the creature in the darkness.

[bunsen_h]

I think that sounds more dramatic than celebratory. Also I seem to be forever associating it with something from the Worms game. Maybe launching a flying sheep?

Try this.

You lost me at Piers Anthony. After The Color of Magic, I knew better than to go for a second helping

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.

My brain automatically substituted it as the first Xanth book, A Spell for Chameleon, vs. the second, The Source of Magic. I trust that KorvinStarmast was not referring to the 15^th, The Color of Her Panties. (Seriously. That's the book's title. That's its subject. No, I haven't read it.)

[HalfTangible]

Well this would appear to put to bed the arguments over whether Team Evil would eventually solve the dungeon or not. An argument which fed into the question of whether Serini was acting rationally by attacking the Order and the Paladins, and also fed into the discussion about which gate was best defended.

If this is an actual discussion I'd just like to point out that Dorukan's gate was the only one whose defenses worked exactly as intended (Elan being dumb enough to set off the self-destruct rune was a fault in our protagonists, not in the gate defenses)

[Grey_Wolf_c]

There are an infinite amount of numbers between 1 and 2, and there are also an infinite amount of numbers between 1 and 3.

That second infinity encompasses the first and also has extra - they are not the same size.

Yes they are. Each value v1 between 0 and 1 can be mapped 1-1 to each number v2 between 0 and 2: v2 = 2v1 (add one to both sides to obtain 1 and 2 vs 1 and 3). Therefore they are equally large infinities.

Every infinite subset of any given type of number is as equally infinite as any other infinite subset, and indeed as large as the whole.

The quantity of any one infinity is the same as the quantity of any two infinities, or of every infinity combined. Infinity = infinity.

False. There are infinites that are larger than others - for example, you can use Cantor's diagonal argument to show that you cannot map 1-1 every real number to a natural number. Therefore, the infinity of real numbers is larger than the infinity of natural numbers.

Grey Wolf

[dancrilis]

The quantity of any one infinity is the same as the quantity of any two infinities, or of every infinity combined. Infinity = infinity.

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).

[Grey_Wolf_c]

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.

You cannot compare two infinite sets by subtracting one from the other (in fact, subtraction of infinite sets is not well defined). You can only compare two infinite sets by matching its members. If they can be matched 1-1, the sets are equally large. All four of those sets can be matched 1-1 to each other. Therefore, they are all equally large infinities. First, I'm going to point out than an infinite set + 2 is not bigger than an infinite set plus any other non-infinite number, so s2 = s1.

For the rest, I'll define set 0 as every real number between 0 and 1. (I assume that by "all the numbers" you mean all the reals or all the irrational; proof works equally well with either, but you can't mix and match them, since the latter is bigger than the former)

s0+1 = s1 = s2 = s4
(s0 +1) x2 = s3

They are all identically large infinites.

Grey Wolf

[Liquor Box]

Does it? The thing is, without the PCs' intervention, I am not sure if it would resolve that debate at:

1) They would have; eventually they would surely have figured out something was off and gone through all the tunnels.
or
2) They would not have; they would have thought they were at the end and gone "now what?" because of the intervention of the creature in the darkness.

But your option 2 of going "now what", also shows something is off, which leads back to option one (knowing something was off).

We see in 1263 what happens when Redcloak realises something is off. He notes they'd made zero progress (also an unescapable conclusion if they mark off all the exits without finding anything), speculates it may be due to enemy intervention or their own incompetence (so the line of thought doesn't rely on only the OotS presence) - then comes up with the solution that are now at.

If this is an actual discussion I'd just like to point out that Dorukan's gate was the only one whose defenses worked exactly as intended (Elan being dumb enough to set off the self-destruct rune was a fault in our protagonists, not in the gate defenses)

It was indeed an actual discussion:
https://forums.giantitp.com/showthre...he-gates/page5

The thread is past the 90 day cutoff now, so I guess you'd need to start a new one if you want to discuss though.

From memory I agreed with you that Dorukon's defences were strong, and though ti was between Dorukon's and Soon's gates for best defended.

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).

I agree with Grey Wolf (and others who've said similar). If there are an infinite number of numbers between 1 and 2, and all those numbers plus a whole lot more between 1 and 3, there are still not more numbers between 1 and 3 than 1 and 2. In both cases the number is infinite. In either case there is no end to the quantity of numbers.

Although, it does strike me that it boils down to quite semantic discussion and depends on exactly what words like 'more' or 'equal' mean in this mathematical context.

[dancrilis]

They are all identically large infinites.

I hold that you are wrong - I have no idea how be clearer in showing how you are wrong then I was above (thought about trying to do so with am examples of real life applications but think it would make things less rather then more clear).

If you hold that I am wrong (which you seem to) - then to get me to understand how I am wrong you would need be be clearer in showing this (not sure this is worth your time frankly but have at it if you like).

I am not adverse to being wrong (I have been wrong before and learned things from it, and no doubt I will be wrong again and hopefully learn things from it) - but I could see this conversation going down the route of both of us thinking the other should 'go read a book' (or 'take a course' or some other such related thought pattern) - frankly I am not entirely convinced that we are even using the term infinite in the same manner.

[Peelee]

It seems a bit awkward that I went on to lay out why they didn't receive that specific kind of help, and did receive the type of help that was available and useful at the time, meaning all of this has already been addressed in detail.

Should I assume your quote of me clipped off there because you stopped reading at that point? It's not a convention I'm familiar with, and even the quote specified sending help, not helping period, but at this point we're clearly not on the same page for some reason.

So here's my position: we know there is not going to be a group of people coming in to help the Order. Roy said they rang the alarm bell and nobody cared in Girard's pyramid, we saw the pirates ask why they should keep helping and saw the Order couldn't explain it even without any limitations on communication, and we saw all the High Priests who know exactly what is happening and what it means refuse to offer any help. All indications are that the Order is on their own.

Now, with this being the case, any claims of "but they could call for help" seem, to me, spurious at best. If you want to discuss a hypothetical scenario in which none of these indications happened and what would happen if they just Sent to a bunch of clerics they don't know and described the scenario perfectly and asked for help all within a 25 word limit and said clerics were both inclined and able to come, then i misunderstood and of course feel free to speculate away, i wont take part and i apologize for my misunderstanding and for butting in.

But if the position is "they can still call for help with current canon as-is" then im going to point out where it fails (because it will fail) and not bother with the rest.

Used to be, there were "countably infinite" sets -- all the same level of infinite -- such as "all integers" ... "all positive integers" ... "all even integers" ... "all rational numbers" ... (an infinite number of countably infinite sets, each of which has the same infinite "amount" of elements... aleph-null).

Spoiler: Which culminates in one of my favorite Futurama freeze-frame jokes!

Show

Also, bonus points for it only showing six movies.

Also, my go-to shorthand in explaining cardinality and infinity sizes to friends behind me in math was akways the natural set and the set between 1 and 2. In the natural set, its countable, you can count to the next number easily. From 1 yiu know 2 is next, and 3, and so on. At every number, you know the next number, you can count to it.

Conversely, for the set of numbers between 1 and 2, its uncountable. From 1, there's no next number; 1.1 is there, but then 1.01 is between that and 1, and 1.001 is between those, and.... Its an infinite drop off. And each number in that set has an infinite drop off between it an the next number. Its clearly a greater infinity than the natural number set.

That tends to work well enough for the concept. I also love that set for explaining how infinite does not mean all (eg theres an infinite amount of numbers between 1 and 2, but none of those will be 5).

The thread is past the 90 day cutoff now

45 days, not 90.

[glissle]

If you hold that I am wrong (which you seem to) - then to get me to understand how I am wrong you would need be be clearer in showing this (not sure this is worth your time frankly but have at it if you like).

Suppose S1 is an infinite set of hotel rooms numbered with the integers starting at 14, and S2 is an infinite set of people with social security numbers consisting of the integers starting with 1. If I'm understanding you correctly, you are asserting that S2 has 13 more people than S1 has rooms, and therefore it would be impossible to accommodate every person in S2 with their own room in S1.

However, according to Wikipedia, David Hilbert provided the following proof that every person in S2 can be accommodated in S1:

Finitely many new guests
Suppose a new guest arrives and wishes to be accommodated in the hotel. We can (simultaneously) move the guest currently in room 1 to room 2, the guest currently in room 2 to room 3, and so on, moving every guest from their current room n to room n+1. After this, room 1 is empty and the new guest can be moved into that room. By repeating this procedure, it is possible to make room for any finite number of new guests. In general, assume that k guests seek a room. We can apply the same procedure and move every guest from room n to room n + k. In a similar manner, if k guests wished to leave the hotel, every guest moves from room n to room n − k.

Infinitely many new guests
It is also possible to accommodate a countably infinite number of new guests: just move the person occupying room 1 to room 2, the guest occupying room 2 to room 4, and, in general, the guest occupying room n to room 2n (2 times n), and all the odd-numbered rooms (which are countably infinite) will be free for the new guests.

[Sir_Norbert]

I hold that you are wrong - I have no idea how be clearer in showing how you are wrong then I was above (thought about trying to do so with am examples of real life applications but think it would make things less rather then more clear).

If you hold that I am wrong (which you seem to) - then to get me to understand how I am wrong you would need be be clearer in showing this (not sure this is worth your time frankly but have at it if you like).

I am not adverse to being wrong (I have been wrong before and learned things from it, and no doubt I will be wrong again and hopefully learn things from it) - but I could see this conversation going down the route of both of us thinking the other should 'go read a book' (or 'take a course' or some other such related thought pattern) - frankly I am not entirely convinced that we are even using the term infinite in the same manner.

The term requiring precise definition is not "infinite" but "size", or "same size".

Mathematically speaking, two sets are said to have the same size if they can be put in one-to-one correspondence: for each element of set A, you can pair it with a unique element of set B -- no two elements are paired with the same element, but every element has a pair.

Having a precise definition, we can now show that two different infinite sets are the same size, for example the set of positive integers and the set of positive integers greater than 7. We can't write out in full an infinite mapping, but we can show a scheme for one:

1 -- 8
2 -- 9
3 -- 10
4 -- 11
and so on

and since every element of the first set is paired with exactly one unique element of the second set, they are the same size.

[Sir_Norbert]

Also, my go-to shorthand in explaining cardinality and infinity sizes to friends behind me in math was akways the natural set and the set between 1 and 2. In the natural set, its countable, you can count to the next number easily. From 1 yiu know 2 is next, and 3, and so on. At every number, you know the next number, you can count to it.

Conversely, for the set of numbers between 1 and 2, its uncountable. From 1, there's no next number; 1.1 is there, but then 1.01 is between that and 1, and 1.001 is between those, and.... Its an infinite drop off. And each number in that set has an infinite drop off between it an the next number. Its clearly a greater infinity than the natural number set.

It's worth noting that this is an example of arriving at a correct conclusion from faulty reasoning. What you have described, the property of the real numbers that between any two distinct reals is another real (from which it follows that between any two reals there are infinitely many reals) is termed being dense.

Now, the infinite set of reals is greater than the set of natural numbers, but this does not follow from its being dense. Proof: the set of rational numbers is both dense and countable.

[Peelee]

It's worth noting that this is an example of arriving at a correct conclusion from faulty reasoning. What you have described, the property of the real numbers that between any two distinct reals is another real (from which it follows that between any two reals there are infinitely many reals) is termed being dense.

Now, the infinite set of reals is greater than the set of natural numbers, but this does not follow from its being dense. Proof: the set of rational numbers is both dense and countable.

Probably why I have two of my friends have doctorates in math and I don't.

[dancrilis]

If I'm understanding you correctly

You are not.

A hotel with an infinite number of rooms with an infinite number of guests is fine whether you call something room 1 or room 14 doesn't change anything, that is different then having a known set which contains an infinite amount of numbers and another known set of containing an infinite amount of numbers which has the entirety of the numbers from the first set and also some additional numbers and knowing that the second set effectively contains the first set but the first set does not contain the second.

You can then remove the common values from the second set and arrive at the additional values which were only contained in the second set - which may be finite or infinite.

[Diachronos]

This is my question. I can't think of how this works at all.

And come on Kraagor, really? A gauntlet where you don't actually have to fight anything? He should be forcing you to pile up the corpses on a scale or something!

Regarding the implementation, my guess is that there's some sort of trigger that activates when you enter the dungeon (probably something connected to the teleportation trap), and you have to hit every single trigger in order to make the path to the gate appear. Most likely within a certain amount of time; can you really call it a proper gauntlet if you can just waltz through it at whatever pace is most convenient for you?

As far as the gauntlet itself... honestly, I'm not sure that Kraagor was even alive when construction on it was started; he'd already been killed by the Snarl when the Order of the Scribble split up and started building their defenses around the gates. At most he might've suggested it while they were in the process of constructing the gates themselves.
Even if he was involved in designing the place, it's highly unlikely that he would've considered the possibility of someone being able to blitz through the dungeons without having to kill anything. Even if someone did get the idea to just trap the monsters instead of fighting them (which a barbarian isn't that likely to consider), a party still has to deal with the fact that they're working off of limited resources and can't pull that stunt indefinitely. Sure, Redcloak summoned something that can do it, but the odds that someone would spend the resources on summoning a quinton instead of, say, something that can more directly contribute to the efforts of fighting through the place, are very low.

[danielxcutter]

Also, as I’ve mentioned earlier, this is at least as much a tribute to his memory as much as it is actual defense, plus it’s only the first line. And it’s mostly automatic aside from monster replenishing anyways. I think it’s a bit early to write it off just yet.

Incidentally I’m kinda getting frustrated again at Serini. If it wasn’t so ridiculously implausible I’d have thought she was working with (for a loose definition of such; being duped or blackmailed also counts) the IFCC or something.

[glissle]

You can then remove the common values from the second set and arrive at the additional values which were only contained in the second set - which may be finite or infinite.

Did you skip this:

Infinitely many new guests
It is also possible to accommodate a countably infinite number of new guests: just move the person occupying room 1 to room 2, the guest occupying room 2 to room 4, and, in general, the guest occupying room n to room 2n (2 times n), and all the odd-numbered rooms (which are countably infinite) will be free for the new guests.

Even if the second set has a million members for each member of the first set, the same thing can be done by moving the person in room 1 to room 1 million+1, the person in room 2 to room 2 million+2, etc.

[Grey_Wolf_c]

I hold that you are wrong - I have no idea how be clearer in showing how you are wrong then I was above (thought about trying to do so with am examples of real life applications but think it would make things less rather then more clear).

It should be trivial to prove me wrong - you just need to give me a value from the bigger set that does not have a matching value in the smaller. Be aware, before you give me "2.3", that the matching value I will give you back is x = (y-1)/2 +1, where y is the number you give me (assuming we are sticking with s1 and s3*). So, for 2.3, I'll give you 1.65. So, knowing my "secret" of how I'll pair every number from what you claim is a bigger set, if you are right, there should still be infinitely many numbers I can't match to one in the smaller set. So, give me one.

If you can't, it's because they are the exact same size.

(BTW, this same approach is the one I like best to prove that 0.999... = 1; if they weren't the same number, you could give me any of the infinite numbers between them. But you can't, therefore they must be the same number. Infinities are weird that way)

Grey Wolf

*If you want, we could switch to s'1, the set of real numbers between 0 and 1, and s'3, the set of real numbers between 0 and 2. Then, the formula is the much simpler x' = y'/2

[brian 333]

Infinity is inherently uncountable, so logic fails to analyze it. For logic to work the compared items must be countable.

Example: a set of whole numbers is infinite. A set of integers is infinite. If, by cancellation, we can remove all the whole numbers from both sets, it appears that we have 0 whole numbers remaining, but exactly as many integers left as we removed from the whole numbers set. Proof that the set of integers is twice as large as the set of whole numbers.

But:

We have still not counted all the whole numbers. No matter how many are cancelled, we'll make more. The fact that they are uncountable makes the number very large. A quintillion is an uncountable number. The fact that they can never, by any means, be demonstrated to have all been counted, makes them an infinity.

So what if for every whole number we can assume two integers? Both are exactly the same size: neither can be counted completely, and both sets continue without end. Integers = whole numbers x 2, excluding -0. But the set of whole numbers can be multiplied by 2, and now it equals the set of integers. But doing so also implies integers are doubled, and we're back where we started, except again we x2 while numbers, and again...

Logic fails. Logic can usefully apply the concept of infinity, and logic can demonstrate differences in bounded infinite sets, but so long as both sets are infinite, neither can be shown to be larger than the other. Given the set with fewer observable units, one can always multiply those units and catch up.

[Grey_Wolf_c]

Infinity is inherently uncountable, so logic fails to analyze it. For logic to work the compared items must be countable.

This is incorrect. There is both countable infinities and uncountable infinities, and the latter are bigger than the former. There is nothing illogical about either. Unintuitive, plenty. Illogical, nothing.

GW

[Jasdoif]

False. There are infinites that are larger than others - for example, you can use Cantor's diagonal argument to show that you cannot map 1-1 every real number to a natural number. Therefore, the infinity of real numbers is larger than the infinity of natural numbers.

The commentary on the relevant Irregular Webcomic gives a thorough treatment on this (as well as the comic itself stretching a Star Wars reference into it).

[HalfTangible]

Infinity is inherently uncountable, so logic fails to analyze it. For logic to work the compared items must be countable.

Example: a set of whole numbers is infinite. A set of integers is infinite. If, by cancellation, we can remove all the whole numbers from both sets, it appears that we have 0 whole numbers remaining, but exactly as many integers left as we removed from the whole numbers set. Proof that the set of integers is twice as large as the set of whole numbers.

But:

We have still not counted all the whole numbers. No matter how many are cancelled, we'll make more. The fact that they are uncountable makes the number very large. A quintillion is an uncountable number. The fact that they can never, by any means, be demonstrated to have all been counted, makes them an infinity.

So what if for every whole number we can assume two integers? Both are exactly the same size: neither can be counted completely, and both sets continue without end. Integers = whole numbers x 2, excluding -0. But the set of whole numbers can be multiplied by 2, and now it equals the set of integers. But doing so also implies integers are doubled, and we're back where we started, except again we x2 while numbers, and again...

Logic fails. Logic can usefully apply the concept of infinity, and logic can demonstrate differences in bounded infinite sets, but so long as both sets are infinite, neither can be shown to be larger than the other. Given the set with fewer observable units, one can always multiply those units and catch up.

That's not how logic works but also haven't we been showing for the last several pages that you can have infinities of different size?

[The MunchKING]

the 15^th, The Color of Her Panties. (Seriously. That's the book's title. That's its subject. No, I haven't read it.)

To be fair, the titular panties were a relatively minor subplot in the overall story of the book. They were only relevant because it was set up in a previous book as an Impossible Question. Most of the time she runs around naked anyway, because she's a mermaid.

[LuisDantas]

Excellent strip. I always like an intelligent Vaarsuvius moment, and this one delivered that in spades. And his logic is impeccable, too.

In fact, nearly everyone with a speaking part was well-written. Roy, Haley, Serini. Even the Beholder.

[BloodSquirrel]

If this is an actual discussion I'd just like to point out that Dorukan's gate was the only one whose defenses worked exactly as intended (Elan being dumb enough to set off the self-destruct rune was a fault in our protagonists, not in the gate defenses)

I would argue that Soon and Girard's worked as intended as well. Soon's gate was only taken as the result of an overwhelming invasion backed by an epic Lich, and they still would have won if Miko hadn't Miko'd. Girard's gate was doing fine until its defenders were wiped out by a completely unrelated spell that was well beyond what he could have planned for.

Serini continues to demonstrate that her antagonistic attitude is counter-productive to her own stated goals. If she'd just given them a roughly accurate time, V wouldn't have had the information to guess how it actually worked. They'd probably just have assumed that it was one of the last gates. And what was her endgame here, anyway? Either they directly interfere, or Team Evil reaches the end, and the Order follows them into the last dungeon, both things she wanted to avoid.