Cardinality

From Caelwiki

[13:23] Serendipity: All I'm saying is that a race with no number for 1 would NOT have evolved intellectually along any Terrian race's lines.

[13:24] Lkeas: that is because you are so mundane as to think that the Terran way of understanding mathematics is the only right way :P

[13:24] Genoclysm: Serindipity, 1 is not the same as 1-9.

[13:24] Lkeas: true

[13:24] * Lkeas frowns

[13:24] Lkeas: something about Cardinals...

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[13:25] Lkeas: {2,3,4} and {4,5,6} are both equal to 3

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[13:25] Azerkail: Lkeas.

[13:25] Falerin: The Terran Way?

[13:25] * Lkeas shrugs

[13:25] Falerin: Terra has multitudes of ways

[13:25] Serendipity: Thanks Lkeas- I was pointing something out, not inferring that it was the only way :P And one is not the SAME as 1-9, but it is part of the equation

[13:25] Falerin: Mathematics in their basic understanding change ever 100 years

[13:25] Azerkail: What's the smallest Cardinal Number?

[13:25] Serendipity: Sure, but as far as I know, every culture on Earth started with One.

[13:25] Sonic12040: Aleph Naught

[13:25] Lkeas: yes i saw that Az

[13:26] Falerin: Aleph Null

[13:26] Falerin: Or naught

[13:26] Falerin: But Cantor was on druggs

[13:26] Falerin: Of course....

[13:26] Azerkail: Cantor's on drugs? lol.

[13:26] Lkeas: i can't quite wrap my head around the concept of a "small infinity"

[13:26] Falerin: Ok...

[13:26] Falerin: I will give another lecture

[13:27] * Lkeas pulls out her notepad and pen

[13:27] Falerin: I rarely do this so I demand silence why I do so

[13:27] Serendipity: :D

[13:27] Genoclysm: Serendipity, they can have one without having the rune running 1-9. Look at Roman numerals. They don't have 1-9, but they have a symbol for 1.

[13:27] Falerin: I will do this conceptually

[13:27] Lkeas: guys shut up and listen to the teacher for a minute :)

[13:27] *** Maxx|lunch is now known as Maxx.

[13:27] Falerin: How many possible numbers exist between 1 and 2 Lkeas?

[13:28] Lkeas: what type of numbers? if you include fractions, an infinite number

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[13:28] Falerin: I do include fractions but lets say just Rationals

[13:28] Falerin: So not things like square root 2

[13:28] Falerin: Still infinite

[13:28] Lkeas: yes

[13:28] TDgram: *rubs head* ahh geez...more math..NOOOO

[13:29] Falerin: But the greatest magnitude of that infinity is still 2

[13:29] Azerkail: So would Aleph null for 1 to 2 be like 1.00000...1?

[13:29] Falerin: the biggest number in all of that infinity is still less then the smallest number between 2 and 3

[13:29] Falerin: yes?

[13:29] Lkeas: yes

[13:29] Falerin: Both of these things are infinite sets

[13:30] Falerin: They have in fact the same number of possible correspondences... Infinite... and a line can be drawn between each 1 to 1

[13:30] Falerin: Now... how many numbers are between 1 and 3

[13:30] Falerin: Conceptually its still Infinity

[13:30] Lkeas: yes

[13:30] Azerkail: But 2 is a cardinal number, no?

[13:30] Falerin: But this infinite set has members that the other two do not

[13:31] * Azerkail raises his hand.

[13:31] Falerin: Because it has everything thats in both sets

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[13:31] Falerin: While each individual set may have the same ammount of numbers... the actual contents vary

[13:32] Falerin: {1..2} contains 1.5

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[13:32] Falerin: {2..3} contains 2.5

[13:32] Falerin: {1..2} does not contain 2.5

[13:32] Falerin: it has infinite numbers in it but never contains that 2.5

[13:32] Falerin: vice versa as well

[13:33] Falerin: So then conceptually at least....

[13:33] Azerkail: So... the integers serve as the boundaries between each infinite set?

[13:33] Falerin: set {1..3} has MORE infomration

[13:33] Falerin: then either of the two other sets

[13:33] Falerin: Of course if one draws correspondences one can derive via a proof that that the sets are not actually larger they are of identical size

[13:34] Falerin: None of these appropach Cardnality

[13:34] Falerin: These are all still ordinals

[13:34] Lkeas: what is the difference?

[13:34] Falerin: Well now lets look at {1..postive infinity}

[13:34] Falerin: It not only has an infinite number

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[13:34] Falerin: but it has an infinite number between each ordinal

[13:34] Falerin: as well

[13:34] TDgram: 1 + 1 = 11 = 1 + 1

[13:35] Falerin: Lets consider this

[13:35] Falerin: {1,2,7} {8,20,17}

[13:35] Falerin: These are two sets

[13:35] Falerin: They have the same cardinality

[13:35] Lkeas: yes, both have cardinal numbers of "3" right?

[13:35] Falerin: its 3

[13:36] Falerin: You can make a 1 to 1 correspondence to determine the number of memebers of the set

[13:36] *** Serendipity has signed off IRC (Vanished: Web IRC Disconnected).

[13:37] Falerin: Cantor demonstrated via complex proofs that no matter what the numbers in the set... that if a set is unbounded i.e. infinite

[13:37] Falerin: Then any subset of that set

[13:38] Falerin: will have identical cardinality to the set itself

[13:38] Falerin: I.e. the cardinality of 1..2 is identical to 1..200

[13:38] Falerin: its called Aleph Naught

[13:38] Falerin: Its an infinity

[13:39] Falerin: However the real number set can be proved equally complexly to defy 1 to 1 correspondence

[13:39] Falerin: it has a different Cardnality then the set of Natural Numbers

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[13:39] Falerin: Its in fact GREATER in cardinality then the set of Natural Numbers

[13:40] Falerin: If you want the proof there are too.. one that uses nested intervals

[13:41] Falerin: and one that uses the diagonal argument

[13:41] Falerin: The latter is perhaps easiest to understand

[13:41] Falerin: But basically it states that within bounds there are degrees of infinity

[13:41] Falerin: and they behave in specific ways

[13:41] Lkeas: behave?

[13:41] Falerin: So while indeed 1..2 has infinite possibilities

[13:41] Falerin: NO number in that infinity > 2

[13:41] Falerin: Not one

[13:42] Falerin: The function describing that set has a limit at 2

[13:42] Falerin: and it will never pass beyond that

[13:42] Falerin: When the function is drawn graphically therefore it will never pass 2 at all either

[13:42] Lkeas: hmm.

[13:43] Falerin: But its no less infinite then a number that CAN pass 2

[13:43] Falerin: its just less cardinal

[13:43] Falerin: Which is messy but...

[13:43] Falerin: And its a very lay understanding as well

[13:43] * Azerkail raises his hand to ask a question.

[13:43] Falerin: Yes?

[13:44] Azerkail: The same can be said that no number in the set of 1..2 is less than one?

[13:44] Falerin: Thats correct

[13:44] Rimblade: Are we describing straight lines here?

[13:44] Genoclysm: I understand all of that which you said well, but one thing. Could you define Cardinal Number for me?

[13:44] Falerin: It works both for positve and negative infinity

[13:45] Falerin: Cardinal Numbers are measurements of "Magnitude"

[13:45] Falerin: How much is in the set

[13:45] Genoclysm: Alright.

[13:45] Lkeas: i'm still confused about the subsets of unbounded sets thing

[13:45] Falerin: {1,2,4,5} {1,7,4,7}

[13:46] Falerin: Both of them are the same cardinal number

[13:46] Falerin: 4

[13:46] Genoclysm: Got it.

[13:46] Azerkail: Cantor proved that any unbounded subset of N has the same cardinality as N, even if this might be against intuition at first.

[13:47] Falerin: Lkeas is back to the infinity thing..... the cardinality of of the set of ALL rationals between 1..10 is the same exact thing as the cardinality of all rationals between 2..3....

[13:47] Lkeas: why?

[13:47] Azerkail: Oh.

[13:47] Genoclysm: Infinite Lkea.

[13:47] Falerin: No matter the fact that 2..3 contains different numbers

[13:47] Genoclysm: Lkeas*

[13:47] Falerin: You can pair them up exactly the same way

[13:48] Falerin: And there is no place where pairing via diagonal or interleaves is impossible

[13:48] Lkeas: but i thought we had established that {1..3} contained more information than {1..2} or {2..3}

[13:48] Falerin: Well not actually

[13:48] Lkeas: how is the cardinality the same?

[13:48] Falerin: it has no greater magnitude of information

[13:48] Falerin: it has simply DIFFERENT informationm

[13:49] Lkeas: hmm.

[13:49] Azerkail: I think I get it.

[13:49] Azerkail: Infinity exists between 2..3 and 1..10, but the "flavor" of infinity is different?

[13:49] Falerin: Yes

[13:49] Falerin: Ahem

[13:49] * Lkeas wonders what infinity tastes like

[13:49] Falerin: The flavor is different

[13:49] Falerin: but its still infinite

[13:49] Falerin: HOWEVER

[13:49] * Azerkail likes Strawberry Infinity.

[13:49] Falerin: when you get much larger then this

[13:50] Falerin: the Actual SIZE of infinity gets different

[13:50] Lkeas: larger than what?

[13:50] Falerin: And for then you cannot be comparing between reals

[13:50] Azerkail: Thank God Infinity's a concept and not a number.

[13:50] Falerin: You have to be comparing between TYPES

[13:51] Falerin: the higher numbers are uncountably infinite

[13:51] Falerin: and...

[13:51] Falerin: http://en.wikipedia.org/wiki/Uncountable_set

[13:51] Falerin: Just... there

[13:51] Rimblade: What about negative infinity sets?

[13:51] Falerin: Behaves identically

[13:51] Falerin: there is no conceptual distinction in sets in either direction

[13:52] Falerin: Negative Cardinality is a nonsense idea

[13:52] Falerin: because Cardinality measures magnitude....

[13:52] TDgram: Cardinals?

[13:52] Lkeas: a measurement..

[13:52] Genoclysm: What is that idea anyways?

[13:52] TDgram: Oh i remember this

[13:52] Falerin: so {-1,3,-4} still is Cardinality 3

[13:52] Azerkail: Magnitude is the value of the number, not its polarity.

[13:53] Lkeas: right

[13:53] Falerin: It measures the quanity of numbers that exist in the set

[13:53] Falerin: It is like taking the absolute value

[13:53] Lkeas: so where does aleph-null fit in to this? i think that part flew over my head

[13:53] Falerin: shrug

[13:53] TDgram: (Link:

[13:53] TDgram: :D

[13:53] Rimblade: So, how does #i relate to all this? Sorry, I've got a dynamic (dynamic= dumb) head for mathematics

[13:53] Falerin: Aleph Null is the name given to the smallest Cardnial infinity

[13:53] Falerin: the infinity that exists in the natural numbers

[13:54] Falerin: Thats all Lkeas

[13:54] * Azerkail raises hand to ask another question.

[13:54] Lkeas: oh

[13:54] Falerin: Yes?

[13:54] Azerkail: Aleph Null is the same as Aleph Naught, correct/

[13:54] Falerin: Yes

[13:54] Falerin: Null and Naught are synonymous

[13:54] Azerkail: Kay, cool.

[13:54] Genoclysm: I thought he already said that.

[13:54] ForeverEndeavor: So Aleph Null would be {1...1.000(infinite 0's)001}?

[13:55] Azerkail: I read back and Falerin said Aleph Naught instead of Null, and I thought there was a specific reason in the change of diction.

[13:55] ForeverEndeavor: There are still infinite #'s in between, but it is the smallest and most exclusive infinity, correct?

[13:55] Falerin: The cardinality of the Real numbers is actually INFINITELY more Cardinal so its Called Beth Null

[13:56] Lkeas: the infinity that exists in the natural numbers... as in the cardinality of {natural numbers}?

[13:56] Falerin: Numbers betwenn them are called aleph one

[13:56] Falerin: aleph two

[13:56] Falerin: etc

[13:56] Lkeas: and beth null would be the cardinality of {real numbers}?

[13:56] Falerin: Aye

[13:56] Azerkail: I knew a girl named Beth Null. T_T

[13:56] * Lkeas 's pencil breaks

[13:56] * Azerkail hands her a pen.

[13:57] Lkeas: you use pen for math?

[13:57] Azerkail: I do.

[13:57] Falerin: Anyway

[13:57] Lkeas: sorry

[13:57] Falerin: this is just a way of saying that beyond vast and uncountable

[13:57] Falerin: there is EVEN MORE VAST and uncountable

[13:58] Genoclysm: brb

[13:58] Falerin: How then can any finite being... conceive or percieve... that entire magnityde

-End of Lecture-

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[13:58] Rimblade: We can't- that's why we don't think about all those decimals :P

[13:58] Lkeas: heh.

[13:59] * Azerkail smites the decimal system.

[13:59] Lkeas: hmm. so i guess the kicker here is, is all of this somehow related to Maxwell's "Let's go Cardinals!" comment?

[14:00] Lkeas: that is why i began looking them up in the first place...

[14:00] Azerkail: I was hoping he was going to make a LOST reference. =P

[14:00] Rimblade: I wonder if a Faceless could make Azerkail stop talking about lost? >.> <.< >.> ^.^

[14:01] Maxx: Is it possible the Cor Dem count in terms of relative magnitudes?

[14:01] Lkeas: and if so... does 755 have something to do with cardinality?

[14:02] TDgram: ..

[14:02] Rimblade: Err... that's what arabic numerals mostly do, isn't it?

[14:03] Rimblade: I mean, 3 is only different from 1 in that it represents a seperate magnitude, right?

[14:03] Maxx: Sorry, it didn't quite come out as I thought it.

[14:03] TDgram: our best guess was 755 had to do with the equality of nature.. 5's being birth and death 7 being life

[14:03] Rimblade: And magnitude is simply 'how far from 0', right?

[14:03] Maxx: I'm trying to match the runes to constellations (post on the forums).

[14:03] TDgram: ..

[14:03] TDgram: Good luck

[14:03] Lkeas: yeah i saw that, it was pretty interesting :)

[14:04] * Azerkail is about to make a LOST reference. To the bunkers.

[14:04] *** Sacred has signed off IRC (Distracted: ).

[14:04] TDgram: Runes matching constilations? = Stargate movie ripoff

[14:04] Maxx: So, a particular rune would identify the magnitude of life at the particular time the constellation is visible.

[14:04] Maxx: *corresponding time

[14:04] Azerkail: In LOST, the focal numbers (4 8 15 16 23 42) add up to 108, another focal number. In AQ, would the runes, if they were numbers, add up to 755, since its a "holy" number to the Cor Dem?

[14:05] Lkeas: well... we already established a set of numbers that add up to 755

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[14:05] Azerkail: Trinity, no?

[14:06] Azerkail: What numbers were they?

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[14:06] Maxx: Why would the faceless use Arabic numbers and the Cor-Dem use runes?

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[14:06] Lkeas: 84 114 105 110 105 116 121

[14:06] TDgram: The Faceless doesnt use numbers...all it knows is emotion

[14:07] Azerkail: All seperate numbers... I smell a LOST reference. What if the runes line up to those numbers?

[14:07] Lkeas: the problem with that set is that there are two 105's

[14:07] Maxx: Yes, but they know 755 -- a set of Arabic numbers.

[14:07] Rimblade: Can emotion be used in leiu of a numerical system?

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[14:07] Lkeas: they who, maxx?

[14:07] Azerkail: Maxx: Kinda like how the Germans used the Enigma but the Japanese used the Purple machine in WW2. They were allies but even they had secrets from each other.

[14:08] Azerkail: Just a thought.

[14:08] ForeverEndeavor: Why is 2 105's a problem?

[14:08] Genoclysm: I'm back now.

[14:08] Maxx: The faceless, Lkeas. Maxwell said they know "755", which is a set of Arabic numbers.

[14:08] TDgram: ..

[14:08] Azerkail: The AQ Numbers would be 84 114 105 110 116 121 755.

[14:09] Rimblade: They know 755... but they might know it as a meaning, not as the actual '755'

[14:09] Lkeas: the faceless may see 755 as a symbol of trinity, that does not mean they equate 7 with "seven"

[14:09] Lkeas: it could be a symbol for lightning or Avatar for all we know

[14:09] TDgram: *patiently waits for them to hit a dead end*

[14:09] ForeverEndeavor: That could very well be a jumble of figures for them

[14:09] Rimblade: If you recall, Truth often sent us messages encoded with numbers- and yet, we are told that they don't understand them. Therefore, that can't have been his intent.

[14:09] Lkeas: Truth was being run by the Cor Dem though

[14:10] ForeverEndeavor: They could relate it the exact same way that many people relate a cross to Christianity and David's star to Judaism etc.

[14:10] Azerkail: Theory: 755 is a understandable to both us and the Cor Dem, thus our only shared word, for now.

[14:10] Azerkail: Somewhat of a calling card, if you will.

[14:10] Rimblade: So, do the Cor Dem understand Truth?

[14:10] ForeverEndeavor: It's also known to the rest of the Network

[14:10] Azerkail: NOVA is in the Network then.

[14:11] ForeverEndeavor: Not necessarily

[14:11] Azerkail: OMGz revalation.

[14:11] ForeverEndeavor: NOVA knows 755 but that doesnt mean that they are in the Network

[14:11] Genoclysm: The Cor Dem know numbers, and the Faceless use them to use numbers for them.

[14:11] Azerkail: The Network's calling card, their beacon, the symbol that red flags messages to them is 755?!

[14:11] ForeverEndeavor: We know it and we aren't in the network

[14:11] Rimblade: How do the Cor Dem controll the Faceless then?

[14:11] Azerkail: NOVA is investigating it their own way, given that Maxwell has been warning us lately to keep it from NOVA.

[14:12] Genoclysm: They don't Rimblade.

[14:12] TDgram: To my understanding the Faceless control Cor dem

[14:12] Rimblade: Well, that's my point- they can't.

[14:12] Genoclysm: The Cor Dem provide the numbers for the messages.

[14:12] Lkeas: the Cor Dem don't control the faceless Truth... they control a program called "Truth" that sends messages to the forums

[14:13] *** Rimblade is now known as Rimeating.

[14:13] Genoclysm: The Faceless use the Cor Dem to send their message numerically.

[14:13] TDgram: ^ yes

[14:13] Azerkail: Doesn't one have the numbers and the other has the brains?

[14:13] TDgram: ..

[14:13] Genoclysm: no

[14:13] TDgram: One has the brains and numbers the other has emotion

[14:13] Genoclysm: One has numbers, the other has emotion.

[14:13] Genoclysm: ah, he beat me.

[14:14] Azerkail: Right.

[14:14] Lkeas: well, that's what i'm wondering... are the messages originally from the faceless and the Cor Dem are just messengers? or are the Cor Dem sending messages against the will of the faceless in order to try to get us to help them escape slavery?

[14:14] TDgram: :p

[14:14] Genoclysm: A few might have been vying for freedom.