Number

[Anonymous No.15534003]

Why are ordinal numbers and cardinal numbers both called numbers when they have nothing to do with each other? Doesn't that indicate that we're confused about what a number actually is? I imagine that this has an impact on the infinity debate.

[Anonymous No.15534045]

>>15534003
I've never understood the difference

[Anonymous No.15534157]

>>15534003
Mathematicians usually just say "ordinals" and "cardinals"

[Anonymous No.15534468]

bump

[Anonymous No.15534803]

>>15534003
Ordinal numbers
Cardinal numbers
Small numbers
Big numbers
Real numbers
Imaginary numbers
Complex numbers

Numbers is the name of the category. They are all members of the bigger category.

[Anonymous No.15534978]

>>15534803
What defines the category?

[Anonymous No.15535052]

>>15534003
>have nothing to do with each other
I don't understand what you mean here.

[Anonymous No.15535158]

>>15535052
Why are they both called numbers? One deals with counting, the other deals with ordering.

[Anonymous No.15535162]

>>15534003
autism in a nutshell

no sense of abstraction whatsoever

[Anonymous No.15535169]

>>15535158
How can you "order" without "counting"? It's literally the same thing

[Anonymous No.15535305]

>>15535169
I just put things after each other. I'll do it in a way or another way.

[Anonymous No.15535310]

>>15535305
Then there's no order.

[Anonymous No.15535422]

>>15535310
In any set, if every x can be given a > or < with every y, the set has an order. Every total order is isomorphic to some subset of R.

[Anonymous No.15535429]

>>15535422
What does that have to do with ordinal numbers? Do you know what an ordinal number is?

[Anonymous No.15535533]

>>15535429
It's not a number. It's a listing and ranking system.

[Anonymous No.15535541]

>>15535533
>Why are ordinal numbers and cardinal numbers both called numbers when they have nothing to do with each other?
This is the question. (I think you may have misread it?) What's an example of an ordinal number?

[Anonymous No.15536611]

>>15534003
Its just a linguistic distinction, 1st isn't really all that of a separate concept from the number 1

[Anonymous No.15536619]

>>15534003
>Why are ordinal numbers and cardinal numbers both called numbers
Because you can do arithmetic with them.

[Anonymous No.15536757]

>>15536619
So what's more basic than an ordinal or a cardinal?

[Anonymous No.15536953]

>>15536757
Baking soda? What do you mean by basic?

[Anonymous No.15536987]

>>15536953
What kind of number underlies both of them?

[Anonymous No.15537001]

>>15536987
I still don't understand. Underlies?

[Anonymous No.15537194]

>>15537001
What is the primitive class or element that ordinal and cardinal numbers are derived from? What is the basis for grouping them both under the concept "number"? What is the number?

Three more ways of asking the same question.

[Anonymous No.15537244]

>>15537194
Got it.
>What is the primitive class or element that ordinal and cardinal numbers are derived from?
Why would there be something more primitive than a number?
>What is the basis for grouping them both under the concept "number"?
Because it's the same information. You can't have an nth number without counting to n and you can't count to n without having an nth number.

[Anonymous No.15537808]

>>15537244
>Why would there be something more primitive than a number?
I don't know, but that's not what I'm asking for. I'm asking for number, not number primitive, not ordinal or cardinal number.
>Because it's the same information.
Then why are they treated differently?
>You can't have an nth number without counting to n and you can't count to n without having an nth number
I think you have a good point here. Seems like a yin and yang thing going on here.

[Anonymous No.15537832]

>>15537808
>I think you have a good point here.
:)
>Then why are they treated differently?
It's like how blue can be a noun but also an adjective, even though it's the same color.
>I'm asking for number, not number primitive, not ordinal or cardinal number.
You mean how to define the word "number" plain and simple?

[Anonymous No.15537837]

>>15537832
>It's like how blue can be a noun but also an adjective
Getting warmer.
>You mean how to define the word "number" plain and simple?
Yeah, preferably without referring to the members of it.

[Anonymous No.15537854]

>>15537837
Maybe what I don't understand is how it makes sense to define numbers without referring to numbers?

[Cult of Passion No.15537860]

>>15537854
>The definition of Number is A Number...it just is, ok?!

[Anonymous No.15537873]

>>15537860
You're a different anon but I don't understand why you'd paraphrase my post in those terms. The summary seems completely unrelated to what it pretended to summarize.

[Cult of Passion No.15537921]

>>15537873
>define numbers without referring to numbers
Using the term in the term is counter definitional as it creates circular logic.
"It is because it is."
>seems completely unrelated
To the uninitiated and blind.
>pretended
I never pretend (LYING).

Try again, sausage-link.

[Cult of Passion No.15537923]

>>15537921
>term in the definition*

Great...now we both look stupid...

[Anonymous No.15537924]

>>15537921
I'm happy that you're happy.

[Anonymous No.15537926]

>>15537923
I'm sorry that typos make you sorry.

[Cult of Passion No.15537927]

>>15537924
>you're happy
Bold claim.

[Anonymous No.15537932]

>>15537927
How could you not be happy while looking at such a friendly, inquisitive cat?

[Cult of Passion No.15537950]

>>15537932
>inquisitive
The target has been identified, the time for inquisiting has passed.

[Anonymous No.15537956]

>>15537950
Did the cat enjoy that morsel of otoro?

[Cult of Passion No.15537971]

>>15537956
>enjoy
Revenge is always bitter for one must first be wonged in order to enact it.

[Anonymous No.15537982]

>>15537971
>one must first be wonged in order to enact
That's what she said.

[Anonymous No.15537987]

>>15537854
How can you refer to numbers without knowing what a number is? Otherwise numbers could be anything.

[Cult of Passion No.15537989]

>>15537982
>she
Found the problem.
Feeling wronged and being wronged are not the same thing.

My Judgement is final.
[clack]

[Anonymous No.15537991]

>>15537987
Are you this anon? >>15537837
I mean how do you define a word that refers to something without referring to the thing it refers to?

[Anonymous No.15537993]

>>15537989
Reread the quote :)

[Cult of Passion No.15537998]

>>15537993
Tell that to my divorce attorney.

[Anonymous No.15538000]

A number is any element of a partially ordered set that transforms like a number.

[Anonymous No.15538005]

>>15537998
He'll just agree you should have wonged her better.

[Anonymous No.15538097]

>>15537991
Yes.

By defining what you're looking for that would bind the things it would refer to. The thing is, concrete objects exist regardless of the name you give them, so they beg for names and names they shall receive without much controversy. Hey, what is that? Oh, that's an eagle. Eagle you say? Okay that's easy.

Numbers are a different story. You can show me four eagles. You can show me a 4th eagle. You can't show me four. To even ascribe "four" on to a group of things called "eagles" assumes they're all the same species (or similar enough in some way) and not falcons or something.

So what is four? What is a number in general?

[Anonymous No.15538933]

bump

[Anonymous No.15539033]

>>15538097
>bind the things it would refer to
But in the case of cardinal numbers and ordinal numbers there aren't really two things to be bound. They both refer to the same thing. Keeping track of repetition, perhaps, if you want to avoid the words counting, ordering, and number.

[Anonymous No.15539037]

>>15534978
If you know what an ordinal is you took enough set theory to answer than question yourself dumbfuck

[Anonymous No.15539047]

>>15535422
R^2 with lexicographical ordering is total order not isomorphic to any subset of R.

[Anonymous No.15539087]

>>15539037
I don't know what a number is desu. That's why I made this thread.
>But in the case of cardinal numbers and ordinal numbers there aren't really two things to be bound. They both refer to the same thing. Keeping track of repetition, perhaps, if you want to avoid the words counting, ordering, and number.
But they're not the same thing.

[Anonymous No.15539170]

A "number" is a word in the english language commonly used to describe elements of specific sets; including (but not exclusively) fields, groups or rings.

[Anonymous No.15539186]

>>15539087
>But they're not the same thing.
You quoted me under someone else's post number. But they are, they refer to the exact same information. What difference do you see?

[Anonymous No.15539415]

>>15539186
>But they are, they refer to the exact same information. What difference do you see
Cardinals and ordinals have different properties. They may be similar but they are not the same.

[Anonymous No.15539421]

>>15539415
But what difference do you see, in terms of what they refer to? As a starting point, I guess, what don't you like about "keeping track of repetition"?

[Anonymous No.15539442]

>>15539421
Nothing. That's what an ordinal is, but not necessarily a cardinal. Repetition loses sight of what quantity is.

[Anonymous No.15539459]

>>15539442
>Repetition loses sight of what quantity is.
I don't understand what you mean here. How does keeping tracking of repetition lose sight of what it's keeping track of?

[Anonymous No.15539463]

>>15539459
*track

[Anonymous No.15539935]

>>15539459
I have five cows. Just imagine five cows in a field. Which one is the third cow? Which one is the fifth cow?

[Anonymous No.15539974]

>>15539935
You can't know until you know. Maybe I don't understand your point. The information is the same..?

[Anonymous No.15539994]

>>15539974
*post

[Anonymous No.15541337]

>>15539974
There's no order here.

[Anonymous No.15541538]

>>15541337
You can't know there's five cows until you count them and counting them creates an order.

[Anonymous No.15541554]

>>15541538
I can point with two fingers to count two cows at once.
I can use my whole hand and count all five cows at once.

[Anonymous No.15541567]

>>15541554
But how do you know you have five fingers unless you've already counted them? That orders the cows by the order of the fingers.

[Anonymous No.15541605]

>>15541567
Through subitizing them. Believe it or not, but your mind can subitize small quantities to grasp them all at once.

[Cult of Passion No.15541611]

>>15541567
Bro, he isnt using any order, he is just referencing his mental catalog of approximations.

[Anonymous No.15541616]

>>15541605
Subitizing isn't counting.
>>15541611
Neither is approximating.

[Cult of Passion No.15541617]

>>15541616
>Neither is approximating.
"perceive the number"
I wasnt asking.

t.Mathematical Linguist

"I wasnt counting, I was generalizing, I wasnt generalizing, I was percieving, I wasnt percievng, I was intuiting, I wasnt intuiting, I was dreaming..."

[Anonymous No.15541625]

>>15541617
That's a lovely free-verse poem.

[Cult of Passion No.15541631]

>>15541625
The thought-loop of someone meaning what words dont contain, with each misuse they choose another word, to which will exclude some aspect of the desired definition so another is chosen.

A higher level of Vocabulary skill points can solve this but it cannot solvea formless expression of the user...only "ūnt ärt" can.

Gotta grind dem skills to level up and get that lvl 99 cape to gloat to the noobs.

[Anonymous No.15541637]

>>15541616
>Subitizing isn't counting.
Then what it is? It is the direct intuition of quantity. We have to account for that.

[Anonymous No.15541639]

>>15541631
No cats today? :(

[Anonymous No.15541643]

>>15541637
It's a guess. You don't know the number for sure until you count.

[Cult of Passion No.15541646]

>>15541637
>Then what it is?
Its generalizing a shape and referencing it to known numerical values. After a small number it becomes too much to intuitively number so it is recorded on an emotional level (since none of this is using the analytics of the brian) and it becomes "hUgE/Waaa pikachu" and has the tendency to exagerate since the emotional conveyance to another is the one used internally.

[Cult of Passion No.15541648]

>>15541639
Today is Jersay Animal day.

[Anonymous No.15541722]

>>15541643
And then you don't know the count for sure until you count again. And so on. I guess we can't know number!

[Anonymous No.15541751]

>>15541722
Why would you need to count again?

[Anonymous No.15541760]

>>15541751
Because maybe you got the count wrong. Measure twice and cut once they say!

[Anonymous No.15541764]

>>15541760
How do you count wrong?

[Cult of Passion No.15541768]

>>15541760
>maybe you got the count wrong
I dont understand, I didnt count wrong.

[Anonymous No.15541783]

>>15541764
>>15541768
Mistakes happen.

[Anonymous No.15541788]

>>15541783
How, though?

[Cult of Passion No.15541824]

>>15541783
I dont understand, I didnt make a mistake.

[Anonymous No.15541867]

>>15541788
How? Being distracted, having unclear sense-perception, having your short-term memory overwhelmed, being confused over what you're counting, finding out new information after you're finished counting, etc.
Why? Beats me. The best answer why mistakes happen, despite evolutionary pressures against it, is because the Lord Almighty finds it amusing and that sometimes mistakes turn into adaptive bonuses.
>>15541824
No, what if you did make a mistake?

[Cult of Passion No.15541874]

>>15541867
>what if
Lady, it aint tho...

[Cult of Passion No.15541876]

>>15541874
DANMIT!
NOW I LOOK A LIKE A FOOL. THANKS ASSHOLE...

[Anonymous No.15541887]

>>15541867
But that just means you failed to keep track. It seems like "keeping track of repetition" is a pretty solid working definition so far.

[Anonymous No.15541904]

>>15534003
As long as you only consider finite numbers (i.e. elements of [math]\mathbb{N}[/math]), ordinals and cardinals are essentially the same thing. The difference occurs once you consider infinite "numbers".

What do I mean by "number" here. In our context, a "number" is by definition an ordinal.

What is an ordinal? Let's assume we work in ZFC. Then one can show that every set can be well-ordered, i.e. for every set X there exists a partial order [math]\leq[/math] such that
i) every two elements can be compared via [math]\leq[/math], and
ii) every non-empty subset of X contains a minimal element (which is necessecarily unique).

Now let's consider pairs (X,[math]\leq[/math]) where X is a set and < is a well-order on X. Call two such pairs (X,[math]\leq[/math]) and (X',[math]\leq[/math]') isomorphic if there exists a bijection between X and X' that identifies [math]\leq[/math] and [math]\leq[/math]'. Then one way to define an ordinal is as an isomorphism class of well-ordered sets.

[Anonymous No.15541910]

>>15541904
Now convince yourself that if you restrict yourself to finite well-ordered sets, you just get the natural numbers as ordinals. In other words, every two finite well-ordered sets with the same number of elements are isomorphic.

Now the crucial observation is that when you pass to infinite sets, this doesn't hold anymore. For example, consider the set [math]\mathbb{N}[/math] of natural numbers with the usual order. Let X be the set obtained by adding one element to [math]\mathbb{N}[/math] which is larger than all the other elements. Then X is a well-ordered set which is not isomorphic to [math]\mathbb{N}[/math]. On the other hand, there exists a bijection between X and [math]\mathbb{N}[/math]. So X and [math]\mathbb{N}[/math] have the same cardinality, but don't define the same ordinal (when equipped with the above well-orders).

Recall that a cardinal is just an equivalence class of sets, where two sets are equivalent iff the exists a bijection between them. We saw that different ordinals can define the same cardinal, but not vice versa.

To summarize, for an ordinal you need a specific order for your set (intuitively an order that allows one to "count" the elements), while a cardinal only cares about the underlying set (so one might think about cardinality as the "number of elements" of your set).

[Anonymous No.15541926]

>>15541887
How does one keep track? What is that process like?

[Anonymous No.15541930]

>>15541904
>Let's assume we work in ZFC.
That's gonna be a no from me dawg. Check their early life section.

[Anonymous No.15541939]

>>15541926
Let's start by calling it a faithful record. What would be your criticism?

[Anonymous No.15541971]

>>15541910
>To summarize, for an ordinal you need a specific order for your set (intuitively an order that allows one to "count" the elements), while a cardinal only cares about the underlying set (so one might think about cardinality as the "number of elements" of your set).
How can you have an order that needs to be counted without the count you needed for the order?

[Anonymous No.15541996]

>>15541939
How does one keep a faithful record?

[Anonymous No.15542005]

>>15541996
I don't know. But if you do have a faithful record of repetition, then it seems to me like you have a number. (I'm obviously open to criticism.)

[Cult of Passion No.15542014]

>>15541996
>How does one keep a faithful record?
>I don't know.
This conversation bounces back and forth between bewildering and funny.

[Anonymous No.15542019]

>>15542014
The more cats you post the more I like you. Don't make me hate you

[Anonymous No.15542032]

>>15534003
I assume it has been pointed out, but they do have to do with each other:
Cardinals are a set theoretical concept.
Ordinals can be modeled in set theory.
Ordinals form a line (order) and you can associate any cardinal with some ordinal (not constructively) (it even makes sense to define caridnal number in terms of the set theoretical model of the ordinals), and in a way that ordinals essentially mark intervals along that line. In that canonical sense there's more ordinals than cardinals, and the ordinal order determines the cardinal order.

[Cult of Passion No.15542037]

>>15542019
Goats it is.

[Anonymous No.15542054]

>>15542037
Such a perfect animal. The foundation of symbolism qua poetry.

[Anonymous No.15542074]

>>15542032
>there's more ordinals than cardinals
How so? (In terms of why they are both called numbers.)

[Anonymous No.15542135]

>>15542005
That seems fair. But the lack of knowledge is not great.

[Anonymous No.15542155]

>>15542135
I don't think I understand what you mean here.

[Anonymous No.15542449]

>>15542155
You don't know how to tell if you have a faithful record so we're stuck at an impasse.

[Anonymous No.15542455]

>>15542449
If you do, it's a number. If you don't, it's not.

[Anonymous No.15542529]

>>15542455
How to tell?

[Anonymous No.15542531]

>>15542529
I don't see how it matters?

[Anonymous No.15542552]

>>15542531
You don't it matters if you can't tell what a number is or how to point to one?

[Anonymous No.15542556]

>>15542552
We've solved number up to "keeping track of repetition." If you've kept track of repetition, you might have a number. If you didn't, you don't.

[Anonymous No.15542557]

>>15542556
*probably don't

[Anonymous No.15542571]

>>15542556
Yeah but you can't tell if you've kept track so what's the point?

[Anonymous No.15542579]

>>15542571
Why is there a point to defining a word?

[Anonymous No.15542596]

>>15542579
Because we have to work with what we're trying to describe.

[Anonymous No.15542603]

>>15542596
But what's wrong with "keeping track of repetition"?

[Anonymous No.15542606]

>>15534045
"Number" is a view on the data. It's actually just the reification of your effort to enumerate the elements.

[Anonymous No.15542646]

>>15535169
Counting is to attain a quantity. Ordering is to arrange known quantities in relation to each other. Very different. Take that Poli sci degree back to Twitter

[Anonymous No.15542658]

>>15542646
>Counting is to attain a quantity.
>Ordering is to arrange known quantities
How do you do 2 without 1?

[Anonymous No.15542666]

>>15542658
All you need for an ordering is the ability to arrange two elements by any consistent criterion, it needn't be according to quantity. If you can do this with any arbitrary pair of elements, you can put all of them in order according to this criterion, and by iterating over the elements in order you are counting. The possibility of ordering creates the ability to count. The counting is an iteration over the persisted results of the finished ordering process.

[Anonymous No.15542674]

>>15542666
No, the difference here:
>attain a quantity
>arrange a quantity
I don't see how you can have one without the other.

[Anonymous No.15542684]

>>15542674
Having 1 is a consequence of doing 2.

[Anonymous No.15542687]

>>15542684
True.

[Anonymous No.15542692]

>>15542687
It's actually funny that it's also true in the sense that, "1" being a counting number, it literally is the result of doing ordering and thus labelling any element "1".

[Anonymous No.15542700]

>>15542692
Totes. I love this thread. Even the namefags here are beautiful people with beautiful minds.

[Anonymous No.15542789]

>>15542646
... which makes ordinal numbers more like analogies between numbers than things in themselves. When you describe things ordinally (1st, 2nd, 3rd, etc.), you're applying an analogy between a collection of objects and your memory of quantities.

[Anonymous No.15542828]

>>15542789
Can't have one without the other.

[Anonymous No.15542869]

>>15542828
True. But now I know why and in what priority they exist. I also don't think relations are things. This clarifies a lot, actually.

[Anonymous No.15543468]

>>15539037
but Black Dynamite, what is an ordinal number?

[Anonymous No.15544244]

it's ogre

[Anonymous No.15544961]

>>15544244
not yet fiend!!!

[Anonymous No.15544987]

>>15542869
What's the priority?

[Anonymous No.15545977]

>>15544987
Cardinal > ordinal. Ordinals are not things in the proper sense. In other words, I have upgraded my arsenal of finitism autism.

[Anonymous No.15545982]

>>15545977
The fifth stick isn't a stick? But five sticks are all sticks?

[Anonymous No.15546009]

>>15534045
Some would say that having sex and understanding the difference between cardinals and ordinals are mutually exclusive.

[Anonymous No.15546016]

>>15545982
The fifth stick is just a stick. There's no "fifth-ness" to it. That's just a convenient mental fiction, similar to what Aristotle calls an accident.

[Anonymous No.15546028]

>>15546016
But you can't have five sticks without having a fifith stick. So there's no "five-ness" either. Just sticks.

[Anonymous No.15546154]

>>15546028
I would agree to that. Quantities are abstracted from entities in general. But if we had to make it a thing, a cardinal would be an idea. And an ordinal would be comparing ideas.

[Anonymous No.15546216]

>>15546154
>a cardinal would be an idea. And an ordinal would be comparing ideas
What is a cardinal an idea of, though? It's an idea of the comparison. So maybe we could say that an ordinal compares ideas and a cardinal idealizes comparing? I'm not convinced that the one has priority over the other.

[Anonymous No.15546780]

>>15546216
>What is a cardinal an idea of, though? It's an idea of the comparison.
Wait, can you explain?

[Barkun Demi-GodofSCI !hax6YPhEF6 No.15546789]

>>15546780
Good. I support you.

[Anonymous No.15546812]

>>15546780
Explain what? (I'm the same anon you quoted here)

[Anonymous No.15547100]

>>15546812
Why you associate cardinals with comparisons.

[Anonymous No.15547111]

>>15547100
I'm not sure if I do. But I think I'm close to understanding what the other anon means when he writes
>a cardinal would be an idea. And an ordinal would be comparing ideas.

[Anonymous No.15548043]

>>15547111
The cardinal would be an idea in itself. Ordinals would be a comparison between instances of ideas.

[Anonymous No.15548809]

>>15548043
>The cardinal would be an idea in itself
How? Cardinals are also a comparison between instances of ideas. You need the idea of a stick before you can have an idea of five sticks (or an idea of a fifth stick).

[Anonymous No.15548823]

>>15548809
That's an astute observation, and one I've been wanting to tackle. I was wondering when somebody was going to bring that up.

There's also the tension of "concentrating" or "diluting" the definition of an idea to constrict or expand quantity respectively. In some sense, quantity is impossible without making some compromises, at least when it comes to concrete and particular details. Quantity only deals with the abstract and the general.

[Anonymous No.15548826]

>>15548809
>>15548823
To make it simpler, there are only unique things until you start applying criteria and start casting out "irrelevant" details. Before we count, there is only stick_fasfds stick_afafaf, stick_affaf, stick_rafkifa, and stick_lfkzlk (it's cheating to preemptively label each thing as a stick, but I don't want to get too lost in the weeds),

[Anonymous No.15549588]

bump

[Anonymous No.15549930]

bump

[Anonymous No.15550858]

BUMP

[Anonymous No.15550903]

>>15548823
>>15548826
I think I'd agree with all of this. Except maybe the part about concentration and dilution. (It's not immediately clear to me how the idea of a stick could be affected by how many there are.)
However, I'm still quite convinced that you can't know
>stick_fasfds stick_afafaf stick_affaf stick_rafkifa stick_lfkzlk
is five sticks before you know the fifth stick. You could put them all in a bag, shake it up, forget which stick is the fifth, and still know you have five sticks in the bag. But you have to pass through "fifth" before you get to "five." So in terms of priority, I'd offer that while the cardinal may outlast the ordinal, they both happen at the same time.

[Anonymous No.15550952]

>>15550903
>(It's not immediately clear to me how the idea of a stick could be affected by how many there are.)
Other way around! The idea of a stick affects how many there are! Another great example is the bachelor's degree.

[Anonymous No.15550963]

>>15550952
I think I get what you're saying. But if so, we're on to a third layer. A meta-meta-idea. If an idea can affect the repetition of an idea, then we're no longer talking about a number that does exist but a number that can exist.

[Anonymous No.15551819]

>>15550963
You know what? I think you're absolutely right. I'm not sure what more I could add or question. I've clarified all I wanted to know about numbers. Thank you for participating with me.