Why the real numbers don't exist

but some infinities are greater then others and its simply depends on how small of a piece of break it into
sp less to break into pieces

> it is clear that [0, 2] has more numbers than [0, 1].
prove it.

>>9153325
cardinality ≠ number of the things when infinity is concerned.
They have the same cardinality because you can make a bijection between them, in this case;
f: [0.1] -> [0.2]; f: x -> 2x
As you can see our function is injective (for each value in the co-domain there is at most one value in the domain which maps there) and surjective (each value in the co-domain is covered by at least one value in the domain)
∴ It is a bijection
As f is a bijection from [0.1] to [0,2] they must have the same cardinality even though the concept of count is meaningless for infinite sets

>>9153335
but "numbers" that are a fraction can be infinitely small and since they are no expanding like the universe they dont have a rate of going to infinity

but it seems that which one is bigger is a meaningless question how is one to have more number if they are both infinite

>>9153325
>Because we added more numbers, it is clear that [0, 2] has more numbers than [0, 1].
That's not addition, so your argument fails.

>>9153341
so OPs entire shitpost is based on the (false) claim that [0,2] has more numbers then [0,1] while any brainlet undergrad on his first semester analysis course will tell you its not.

>>9153325
"""more numbers""" is not well defined, making your argument ignorable

>>9153340
To prove that two sets A and B have the same size you must both give an injection from A to B and an injection from B to A.
You only gave one from [0,1] to [0,2] which is not enough. You must also find one from [0,2] to [0,1].

>>9153355
Inverse of f

>>9153355
come on, son
g: [0,2] -> [0,1]; g: x -> 0.5x

>>9153355
No, A proving the existence of a bijection f: A -> B necessarily implies the existence of an inverse bijection f^-1: B->A such that f f^-1 = Id_A and f^-1 f = Id_B.

>>9153325
>the number of numbers in that subset
Like which people claim?

>>9153385
The Jews.

>>9153350
His argument is that one can create the set [0,2] by "adding" [0,1] and [1,2], therefore [0,2] should have more numbers than [0,1], since adding a positive number to a positive number creates a larger number. This argument fails because addition is not defined on infinite sets.

there are actual arguments as to why the reals are ill-defined, but this isn't one.

>>9153325
The reals are not discrete. It is a continuum

>>9153983
Zeno's paradox is only resolved by a discrete universe. Real numbers are unnecessary mental masturbation.

>>9153987
of course they're a construct. No one is pretending they aren't.

>>9153325
let's call the set of [0,1] a
and [0,2] b

For all a, there exists the number 2a that will always fall into set b
For all b, there exists a number b/2 that will always fall into set a

I'm a rusty brainlet dropout but I'm sure a 15yo could follow this

>>9153325

[0, 0...0] is not a number, so ofcourse there are less numbers in [0, 1] then there are in [1, 2].

[1, 1...1] is a number.

So there are more numbers in [1, 2] then there are in [0, 1]

>>9153325
> we can conclude that the real numbers don't exist
No, you can't.
But you might be able to allege a problem with the cardinality methods used with infinite sets.

See Hilbert's Hotel Paradox for how infinite sets can be added to other infinite sets while retaining the same cardinality.

Babby's first quiz - can you rigorously prove that the powerset of S has a strictly greater cardinality than S without looking online? Please be include of infinite sets.

[math] \psi: (U: \mathbf{Vect}_\mathbb{R} \to \mathbf{Set}) \Longrightarrow^{\cong} \mathbf{Vect}_\mathbb{R}(\mathbb{R}, --) [/math]

>>9154057
eh, close enough

>>9153325
beautiful example of circular reasoning

>>9153325
>Now suppose, like some people claim, that the number of numbers in that subset is the same as the number of real numbers in total
That's not what they claim. They claim that both sets are un-countably infinite. They have the same cardinality, but infinity is not a number, so you cannot say they have the same number of elements.

The cardinality of both [0,1] and [0,2] is in fact equivalent because there exists a function that maps from one set to the other, and an inverse function that maps the second set back to the first.
>>9153340
>>9153362

Running with this, observe the interval [-1, 1]. We can map this from [0, 2] with the function f(x)=x-1 (inverse is x+1). We can then map this to an interval [-2, 2] with f(x)=2x (with .5x as the inverse). You can iterate f(x)=2x on the interval [-2, 2] again and get [-4, 4] with equivalent cardinality. You can iterate on that interval, and on each subsequent interval. The cardinality remains the same no matter how many times you iterate it, meaning that across any interval [-n, n] in the real numbers, the cardinality is the same as [0, 1].

If we continue to iterate this, the lim of n as the iterations approach infinity is: n -> infinity. This further implies that the interval [-n, n] approaches [-infinity, infinity] with each iteration, which is the set of all reals, implying that the set of all reals would have the same cardinality as [0, 1].

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>>9154082
Nah that doesn't rigorously prove |R|=|(0,1)|
You only ever prove for a finitely long segment of |R| of arbitrary length, but that segment, no matter which iteration you are on, is infinitely shorter than the entire Real Line(in terms of the standard measure, not cardinality).

Just use:
f(x)=tan(pi*(x+0.5))

but blasted BLUMPF babys on suicide watch cause their shitty continuum can't KEEP UP

WILDBERGER 2020 YYYEEEEEAAAAAAAAAAAAAAAAHHHHHHHHHHHHHHHHHHOOOOOOOOOOOOOOOOOOOOO

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Also, if you want to include the end caps in your function - it's trivial but slightly tedious to do.

Map [0,1]->(0,1) like so:

g(x) = {1/2 x=0
1/(n+2) x is of form 1/n, n=int
x x!=0 and x!=1/n

This is bijective

Then use f(x) of above to map (0,1) to R

>>9154674
God this was supposed to be a nicely formatted piecewise function but 4chig killed my spacing

>>9154661
kek

>>9153325
I have literally no idea about higher maths, I just saw this post featured on the 4chan-main page.
I read the OPs hypothesis and tried to follow it logically. After doing that, I decided that this somehow sounds like shitposting.
Is OP shitposting or does it only look like it?

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Ok, so maybe OP phrased what he is saying a bit awkwardly, but he is actually onto something highly controversial. There is healthy skepticism as to the nature of set theory, or even if we should have set theory, within the mathematical community. (See attached pic.). There is, right now, a debate as to the resolution of the continuum hypothesis. Basically, is there an infinity between the infinity of the real numbers and the natural numbers? This debate can literally be resolved either way, as a mathematician named Cohen proved. The underlying troubles with set theory, on the other way, (e.g. Russell's paradox), relating to whether infinity is actual (like a thing that you can manipulate) or potential (an abstraction that you can extend however far you want, as Aristotle would say). OP, if you're out there, don't get discouraged, you're onto something sublime. Look at Poincaré's skepticism of Cantor's work.

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>>9153325
Check this out, then go read some Aryanstotle.

>>9154082
>>9154656

Simple partial/intuitive proof Cardinlity(0,1)=Cardinality(R)

You can use the diagonal argument to show any supposed list of all real numbers between 0 and 1 is incomplete.
So Card(0,1) = uncountable.
There is no cardinality defined that is between the Card(naturals) and the Card(reals).
So we can assume Card(0,1) >= Card(R).
But, why would Card(0,1) be greater than Card(R) since it is a mere interval of R ?
So we can assume Card(0,1) = Card(R).

>>9155220
Its shit posting, he tries to say the "number" of things, when it's clear that the size of these sets can't be expressed by a number. Once you start dealing with infinite sets, you can't assign an integer to the size and they don't work intuitively.

>>9155227
The continuum hypothesis has nothing to do with this post. Someone startes making babby issues with the idea of infinite sets, and you conflate that with a high level mathematical issue.

>>9155426
This gets a little dicey because you are assuming the CH is true.

You can just use the Conway 13 function to show that any nonempty open interval can be projected onto the entirety of R

https://en.m.wikipedia.org/wiki/Conway_base_13_function

This proves |(a,b)|>=R, a<b

Now you are done since R already contains that interval, so the cardinality is equal.

>>9153987
Zeno's paradox can be resolved with either. Since in reals you have the definition of a limit and a measure, notions of velocity and continuous distance are well defined. For a discreet system, it takes some exterior information to convey what should happen in the next instant. Reals don't have instants, so it doesn't have the problem of a still photo of a car not moving.

>>9155443
Here's a way that doesn't use silly functions and provides some intuition as to "how" a set of finite (Lebesgue) measure can have the same cardinality as a set with infinite measure:

You can bijectively map (a,b) to the 1-sphere minus a point by using a simple parametrized equation. Then you can map the 1-sphere minus a point bijectively to R by using the stereographic projection.

Since the composition of these two functions is bijective, this proves |(a,b)|=R, a<b

>>9155479
Pretty much this
>>9154656
>f(x)=tan(pi*(x+0.5))
But stretched accordingly for any interval

>>9153325
>Now suppose, like some people claim, that the number of numbers in that subset is the same as the number of real numbers in total
dropped

proof by counterexample

Rationals exist
Irrationals exist

there are an infinite amount of rationals between two finite intervals
there are an infinite amount of irrationals between two finite intervals

QED

>>9155227
>Basically, is there an infinity between the infinity of the real numbers and the natural numbers?
hold the fuck on
Isnt N countable why R isnt? And in that way the infinities are definetly distinct.
I mean it could just be that the concept of countability is beig quesioned too but I tought countability is pretty straightforward and rigorous

>>9153325
>it is clear that
brainlet

>>9153400
>"adding"
valid in "math"

>>9153400
Of cource it is defined. Look into cardinal arithmetics.