my face when I finally get how some infinities are bigger than others

# Mathematical mindfucks

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I wish you would elaborate. Veeky Forums is comprised of users that know more, know about the same, and know much less about the topic you're bringing up and I think you should bring what you know about the topic to the table on your OP.

tl;dr what the fuck are you even talking about

stl;sdr --- dafuq

I heard it well explained how the infinity that is the set of all the whole numbers is a smaller infinity than the number between 0 and 1.

The rule of cardinality is composed of a concept which separates infinite sets by the possibility of infinite continuation. For instance, the set of all real numbers is larger than the set of all whole numbers. This is because the pattern from which you can continue the series grows more difficult which the more possible answers.

1+1=2

+1=3

+1=4

Is easier to surmise than the set of all reals.

the power set of the real numbers has a cardinality bigger than the cardinality of the real numbers

Call me a brainlet or whatever, and it makes sense that theres infinitely many non-integer numbers between two integers for example, but theres still infinitely many integers. And infinity +- anything is still infinity. So I don't understand how one is larger

Isn't this as simple as dividing two numbers by an arbitrarily small value?

e.g. 1/0.000...1 is half as big as 2/0.000...1, yet both are infinite values.

Anyone have that edited manga where the troll presses his face against the window and explains the infinity sizes?

For example, with the positive integers, you can start at 0 and count ("one, two, three...") and eventually reach any positive number, no matter how high. With the positive reals, you can count from zero forever and never get to one. (to see what I mean, try counting all the real numbers between 0 and 1). To describe this we say the positive reals are "countably infinite" and the positive reals are "uncountable infinite." Uncountable infinite things are larger than countably infinite things. Look it up!

First, we have two reasonable definitions for what it means to say that cardinal α is greater than cardinal β. Say set A has cardinality α and B has cardinality β. if there is a function from A onto B, then we say α≥β. If you think about this enough, you'll realize that adding a finite number of elements to the integers, for example, doesnt increase the size of the set. And, there is an onto function from the reals to the integers (floor function, for example) so clearly reals ≥ integers. And there is no onto function from the integers onto the reals so integers < reals.

Without the technicality, the smallest kind of infinity is called countable infinity. Things like all of the positive integers. It's called countable infinity because you can count them one at a time, in order, and eventually, you will get to every number.

This doesn't mean that you are ever done; that just means that no matter which number you pick, you will get there by counting in a finite amount of time. What's the biggest positive integer you can think of? If you count by 1s, you will get there in a finite amount of time. Probably longer than the lifespan of all of the stars in the univers put together, but it will still be finite.

Now there are infinities bigger than this, like all of the real numbers between 0 and 1.

This is because you can't count them all in a row 1 by 1. You can start at 0, but what comes right after 0? 0.00000000000000000001? Nope, that's infinitely larger than the smallest number larger than 0. It would take an infinite amount of steps counting to any of the real numbers between 0 and 1. so t's uncountably infinite, and therefore much larger.

Rational numbers are countable and yet you might not reach 1 by counting in a finite set amount of time. In your explanation it sounds like you assume

|Q|>|N|, which is not true.

Look up Cantor's Diagonal Argument

Sets are considered "equal" (don't know the english word for "równoliczne")

Imagine both infinities are "equal".

That means you can match every, say, natural number to a number between 0 and 1. So we assume you can list every rational and irrational number:

We match 1 with 0,183640..., 2 with 0,1947311....

etc., so that each real number is assigned to a natural number. That means the sets have the same anmount of numbers.

But now we can construct a number like this:

We take number one and change the digit at its first decimal point. So if it was 0,18... we take 0,2

Then we take the second decimal point from the second number and alter it

So if it was 0,194...

We take our original 0,2 and make it 0,28 (or 0,21, whatever. Just not 0,29)

We repeat the process forever.

The number we just got differs from every number on the list by definition:

If a number is nth on the list, our number has its nth digit different. Therefore, we just showed that the number we constructed is not on the list. We will always be able to create one like this. So, there are more real numbers between 0 and 1 than there are in the set of natural numbers.

my face when I finally get how some infinities are bigger than others

There are literally two infinities and no more than two infinities.

The number of infinities is unbounded (the power set of any set is guaranteed to have a higher cardinality including infinite sets), the usual representation of these is the sequence of aleph numbers

There exist countable and uncountable infinity. All countable infinities have the same number of elements. The same holds for uncountable infinities.

Not true. The power set of the real numbers (the set of all subsets of real numbers) has greater cardinality than the set of real numbers.

Taking the power set always 'increases' cardinality. There are countabpe infinitely many types of uncountable infinites.

There are countable infinitely many types of uncountable infinites.

Can you expand on that ? How would you prove it ?

Are both of you guys baiting right now?

There are countably many countable infinities: 0,1,2,3,..., and #N.

How many uncountable infinities are there? It'd have to be more than uncountably many. Actually it's a number too big for set theory to handle. The set of uncountable infinities, is not a set, and it doesn't have a cardinality. It's actually a class

No infinity is truly bigger than others. We use "bigger" to illustrate that the supposedly bigger infinite sets have subsets which are too infinite, but to use the word "x infinity is bigger than y infinity" implies that y has a finite value, which is wrong

So what about an "infinite set"? Well, to begin with you should say precisely what the term means.

Okay if you don't, at least someone should. Putting an adjective in front of a noun does not in itself make a mathematical concept.

Cantor declared that an "infinite set" is a set which is not finite. Surely that is unsatisfactory, as Cantor no doubt suspected himself. It's like declaring that an "all-seeing Leprechaun" is a Leprechaun which can see everything. Or an unstoppable mouse is a mouse which cannot be stopped. These grammatical constructions do not create concepts, except perhaps in literary or poetic sense. It is not clear that there are any sets that are not finite, just as it's not clear that there are any Leprechauns which can see everything, or that there are mice which cannot be stopped.

Certainly in science there is no reason to suppose that "infinite sets" exist. Are there and infinite number of quarks or electrons in the universe? If physicists had to hazard a guess, I am confident that the majority would say: No. But even if there were an infinite number of electrons it's unreasonable to suppose that you can get an infinite number of them all together as a single data object.

Relevant to this. It blew my mind when I learned that the cardinality of irrational numbers is bigger than that of rationals.

In fact, I still find it hard to believe. Irrationals was a mistake.

C'mon, guys.

This thing achieved already the popular math status quite a while ago. I mean, even VSauce made a couple videos about it back in the days it was still good.**youtube.com****youtube.com**

I get that same look every time I show them that the head is BIGGER than their fist.

That one econometrics paper where that dude proved that for certain time series processes, a larger sample size makes your estimates less accurate

don't worry user, Veeky Forums is a safe space, we're devoted to diversity and strive to include people from all areas of the intelligence spectrum

It depends on how you count. If you form a table and count the diagonals you will definitely count every single rational number eventually.

1/1

2/1, 1/2

3/1, 2/3, 1/3,

etc.

and that goes for any countable infinity. That's why they're countable.

my face when I finally get how some infinities are bigger than others

Now for something even deeper.

There are an infinite number of real numbers between any two rational numbers

There are an infinite number of rational numbers between any two real numbers.

There are infinitely more real numbers than rational numbers.

This is not true (at least under ZFC with GCH which is the most reasonable set of axioms to discuss this). Uncountable infinity refers to aleph-0 and aleph-0 only, the cardinality of the natural numbers (and integers and rationals etc.). Any larger cardinality (of which there are infinitely many) is known as an uncountable infinity. To prove this we can use the fact that, in general we can always go to the next aleph number using an operation known as the power set. This is taking the set of all subsets of your given set. For example the power set of {1, 2, 3} = {{}, {1}, {2}, {3}, {1, 2}, {1, 3}, {2, 3}, {1, 2, 3}}. It can be shown that the power set has a strictly greater cardinality than the original set even for infinite sets (see **en.wikipedia.org**

so each number has a "higher resolution" in a way? Very good explanation user.

The biggest problem I have is the fact that all of you in this field are treating infinity as a finite number. It's very irritating to me because you're waltzing into this field as if your premise has even the slightest ground.

Nothing is more insane than this field. I'm sorry but when Newton said "as X approaches 0", I don't think you are to go any further than the mere fact that you can assume that the Endless tends toward a value but will never reach it. You can call it the result of infinite measures but to say that is still flawed because it implies infinity is finite. There shouldn't even be a word for infinity because you're encompassing the concept with boundaries which the concept stretches beyond. If any of you take this field seriously, you need to have your heads examined.

Just wait until you realize that some infinitesimals are smaller than others.

Conventions shmentions

It's not a convention. You've just tried to express a nonsense thought. Invisible pink unicorn.

mfw solving real intergrals using huge ass half circles in the complex plane

Naturals have infinite subsets as well, like all naturals bigger than 3 or all even numbers

The sizes of infinite sets are defined by where you can find a one to one mapping from the set to say the natural numbers or the real numbers.

For example we can map even numbers to the natural numbers this way by sending 2x to x, and x to 2x in reverse. Hence we have the same number of even numbers as there are natural numbers.

Meanwhile you cannot map the real numbers to the natural numbers bijectively so the size of the real numbers, whilst still infinite is larger. You can look up a proof of this by reading about Cantor's diagonal argument.

You can formalise it all if you study complex analysis in depth. It's a useful tool in applied mathematics without needing to know why it works though.

Learning the Fourier Transform blew my mind. Basis vectors in linear algebra made a lot of sense and were intuitive for me, but I never imagined that sinusoids/complex exponentials could form an infinite basis set for basically any function (except for some really weird ones). It's also crazy how important this little piece of math ended up being.

On that note, thinking of sinusoids as complex exponentials is also pretty cool, but doesn't seem as beautiful to me.

Relative size only applies to finite things. OP is confused because he understands this, but the mainstream narrative with regards to infinite sets in relation to each other makes use of the words "smaller" and "bigger". His confusion is semantic

You are convoluting systems built upon different sets of axioms and insisting that perfectly rigorously defined notions of infinity in one does not hold in the other, which is true but completely misses the point. In set theory the idea that two infinity cardinals can be different simply means you cannot draw a 1 to 1 relationship between the members of their respective sets, a perfectly well defined notion. Perhaps the axioms of set theory do not coincide with the real world as well as the axioms of number theory but this is again completely irrelevant, mathematics can be 'wrong' just because you don't find it useful.

desu, if you have a more "algebraic/combinatorial" mind, I think just showing N<P(N) is must "cleaner".

With the positive reals, you can count from zero forever and never get to one.

This is sort of misleading since this statement is also true for rational numbers (think about the sequence 1-1/n), yet the rational numbers have the same cardinality as the integers.

Biggest mindfuck:

AC [math] \Rightarrow [/math] reals can be well-ordered.

On top of that:

You can't explicitly give a well-ordering (philosophically: there exists something, that doesn't exist)

but that's wrong you fucking retard

you can find every number from 0..infinity in 0.0 to 0.infinity

But not the other way round (and "0.infinity" is grabage notation):

[math]

[0,1] \subset \mathbb{R} \hookrightarrow \mathbb{N}

[/math]

does not exist.

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set of all sets cannot include itself

y'all are stupid cunts dressed as smart pricks

this is actually pretty much dogshit and just bases on made up rules for made up mathematical concepts.

not real math.

They do. Look up the different cases of a dampened pendulum. You can't treat them with a purely real function.

OP has no confusion. He understands what it means for an infinity to be bigger than another infinity. You, on the other hand, don't.

Yeah, you'd think 0 to infinity would have 3333... in there somewhere but guess not

fucked by definitions/markup once again

There are only two sizes of infinity. Unless you know something that I don't.

Shiet that was actually pretty good. I always thought Vsauce was just a meme.