Continuity of the Real Numbers

[Anonymous No.15183235]

If you take the set of real numbers from 0 to 1, and multiply them each by 2, you'd get the set of real numbers from 0 to 2. For each real number from 0 to 2, there would be a unique real number from 0 to 1 that it would be mapped from, and from each real number from 0 to 1, there would be a unique real number from 0 to 2 that it would be mapped to. There's a one to one correspondance.
If there's a one to one correspondance, then the number of real numbers from 0 to 1 would be the same as the number of real numbers from 0 to 2, they would be the same size. and if you imagined them each as lines, a line of length 1 and a line of length 2, they would have the same number of points, the same length, they would be the same size.
This is of course not true, so one of things mentioned before was wrong. It is probably one of these following two things:
>one to one correspondance means being the same size
>there is a one to one correspondance from the segment 0 to 1 to the segment 0 to 2

[Anonymous No.15183236]

>>15183235
part two:
In mathematics, it is said that the first one is the one that is false. But is that truly correct? Think about any two lines, one bigger than the other, it would make sense to say the bigger one has more points, it's bigger. One to one correspondance, is the way people can say if two things are the same size or not, saying they can be mapped to each other one by one, is the same thing as saying they have the same amount of things in them. It's the same as matching. Match up each one in one group with each one in another group.
So now, look at the second option. That there isn't a one to one correspondance between 0 to 1. to me, this seems like it makes more sense. it would mean that there would be some number from 0 to 2, that is not a multiple of two of a number from 0 to 1. And that that number can't be halved. You might say then, all the real numbers can't be halved. But then that's the problem. If you are choosing the second option you are saying the real number aren't perfectly continuous. You might say that it doesn't make sense that a number can't be halved. But look at the natural numbers. The number 3 in the natural numbers isn't the product of 2 and another natural number. So it wouldn't be farfetched to say there are real numbers that aren't the product of 2 and another real number. I am still undecided on this matter, and am so making this thread to ask ye on /sci/. Which option do you think is false, the first greentext option, or the second? Or maybe its some other assumption about the real numbers.

[Anonymous No.15183240]

>>15183235 (OP)
part two:
In mathematics, it is said that the first one is the one that is false. But is that truly correct? Think about any two lines, one bigger than the other, it would make sense to say the bigger one has more points, it's bigger. One to one correspondance, is the way people can say if two things are the same size or not, saying they can be mapped to each other one by one, is the same thing as saying they have the same amount of things in them. It's the same as matching. Match up each one in one group with each one in another group.
So now, look at the second option. That there isn't a one to one correspondance between 0 to 1. to me, this seems like it makes more sense. it would mean that there would be some number from 0 to 2, that is not a multiple of two of a number from 0 to 1. And that that number can't be halved. You might say then, all the real numbers can be halved. But then that's the problem. If you are choosing the second option you are saying the real number aren't perfectly continuous. You might say that it doesn't make sense that a number can't be halved. But look at the natural numbers. The number 3 in the natural numbers isn't the product of 2 and another natural number. So it wouldn't be farfetched to say there are real numbers that aren't the product of 2 and another real number. I am still undecided on this matter, and am so making this thread to ask ye on /sci/. Which option do you think is false, the first greentext option, or the second? Or maybe its some other assumption about the real numbers.

[Anonymous No.15183245]

>>15183235
part two:
In mathematics, it is said that the first one is the one that is false. But is that truly correct? Think about any two lines, one bigger than the other, it would make sense to say the bigger one has more points, it's bigger. One to one correspondance, is the way people can say if two things are the same size or not, saying they can be mapped to each other one by one, is the same thing as saying they have the same amount of things in them. It's the same as matching. Match up each one in one group with each one in another group.
So now, look at the second option. That there isn't a one to one correspondance between 0 to 1. to me, this seems like it makes more sense. it would mean that there would be some number from 0 to 2, that is not a multiple of two of a number from 0 to 1. And that that number can't be halved. You might say then, all the real numbers can be halved. But then that's the problem. If you are choosing the second option you are saying the real number aren't perfectly continuous. You might say that it doesn't make sense that a number can't be halved. But look at the natural numbers. The number 3 in the natural numbers isn't the product of 2 and another natural number. So it wouldn't be farfetched to say there are real numbers that aren't the product of 2 and another real number. I am still undecided on this matter, and am so making this thread to ask ye on /sci/. Which option do you think is false, the first greentext option, or the second? Or maybe its some other assumption about the real numbers.

[Anonymous No.15183249]

>>15183235
>multiply them each by 2, you'd get the set of real numbers from 0 to 2
Wouldn't you get the set of "even" real numbers from 0 to 2?

[Anonymous No.15183264]

>>15183235
>0 to 1, and multiply them each by 2,
You get 2 0's tho

[Anonymous No.15183274]

>>15183249
can you name a real number that is not "even"?

[Anonymous No.15183298]

>>15183245
>So it wouldn't be farfetched to say there are real numbers that aren't the product of 2 and another real number.
Naturals don't have multiplicative inverses. The (nonzero) reals do. If x is not equal to 2k for any k, then x/2 must not be a real, and yet the reals are closed under multiplication

[Anonymous No.15183299]

>>15183235
Intervals of real numbers have no physical size. Show me one. You can't.

>And that that number can't be halved.
Multiplication defines a real number from any 2 chosen real numbers. The product of X and 1/2 is guaranteed to be a real number because multiplication is defined to be a function. If 0 < X < 2 then the number is provably greater than 0 and less than 1.

> it would make sense to say the bigger one has more points, it's bigger
You can say whatever you want but it's not math.
>Or maybe its some other assumption about the real numbers.
It's your assumption that the way you spatially organize a set is some kind of invariant that is dependent on the set. It's not. It's just factually true that you can continuously map a real interval into a real line with the endpoints arbitrarily far apart.

[Anonymous No.15183300]

>>15183235
>If you take the set of real numbers from 0 to 1, and multiply them each by 2, you'd get the set of real numbers from 0 to 2.
False and gay.
Homosexual bait thread do not respond any further.

[Anonymous No.15183312]

>>15183274
What I mean is, if you run through the real numbers between 0 and 1 and multiply them by 2, as a process, in the result there will be a "gap" between each resulting number as if you skipped an "odd" number that's normally between 0 and 2. Of course, the gap eventually gets filled in because there are infinite real numbers between 0 and 1. It just seems to me that because of the multiplication process, there are numbers in between 0 and 2 that aren't contained in 0 to 1 times 2.

I understand that infinity is weird and infinity x 2 = infinity. It just doesn't sit right with me to say that the set of real numbers between 0 and 2 is completely mappable to the set of real numbers between 0 and 1.

[Anonymous No.15183316]

You shouldn’t be allowed to talk about numbers if you can’t include sets.

[Sage No.15183319]

>>15183312
>>15183316
Stop taking the bait. This is bait. Stop posting. This thread is bait.

[Anonymous No.15183328]

>>15183235
But the same holds with the rational numbers Q in place of R.

[Anonymous No.15183341]

>>15183235
this is fuckin topology shit

[Anonymous No.15183346]

>>15183299
>Show me one
any interval in spacetime

[Anonymous No.15183347]

>>15183319
it isn't bait. i made the thread seriously. how is it bait.

[Anonymous No.15183398]

>>15183235

[Anonymous No.15183457]

>>15183312
There is no gap, as you can always divide a real number by 2. If x is in [0, 2] then x/2 is always in [0, 1].

[Anonymous No.15183476]

>>15183235
>they would have the same number of points, the same length
How does one imply the other? Intuitively I'd argue that one can make lengths of arbitrarily size using infinitely many points.

[Anonymous No.15183483]

>>15183235
>This is of course not true, so one of things mentioned before was wrong. It is probably one of these following two things
>leaves out the actually wrong one

[Anonymous No.15183499]

>>15183312
>through real numbers between 0 and 1 and multiply them by 2, as a process, in the result there will be a "gap" between each resulting number
this is not true. this is not how the number line works. the number of real numbers between each two real numbers is infinity.

pose your problem the opposite direction. Taken the reals between 0 and 2, divide them "all" by 2. If the resulting set is somehow "twice as dense" as your
>set of real numbers from 0 to 1
then that first set is necessarily incomplete.

[Anonymous No.15183532]

>>15183483
what is the actually wrong one

[Anonymous No.15183547]

it conflates "the number of real numbers between a and b" with "the number line distance between a and b"

[Anonymous No.15183553]

>>15183235
It's because the size of the real numbers, that being continuity, doesn't work like traditional addition.
[math]2^{\omega}+2^{\omega}=2^{\omega}[/math]
That is, the cardinality of the continuum plus itself is still the cardinality of the continuum. In fact, for non-finite cardinalities, adding two of them is the same as taking the maximum of the two.

>if you imagined them each as lines, a line of length 1 and a line of length 2, they would have the same number of points, the same length, they would be the same size.
Length is completely different from cardinality. Length is a metric on Euclidean space, while cardinality is set theoretic.

[Jealous No.15183557]

The Time Travel Interpretation of the Bible
>https://vixra.org/abs/2104.0068
We describe the Biblical work of ages as a time travel program for saving humanity from extinction. God's existence is proven as a consequence of the existence of time travel, which is supposed. We present the case that Abraham's grandson Jacob, also called Israel, is Satan. We make the case that the Israelites are described as God's chosen people in the Bible despite their identity as the children of Satan because God's Messiah is descended from Abraham through Satan. They are chosen as the ancestors of the Messiah rather than as Satan's children. We propose an interpretation in which God commanded Abraham to kill his son Isaac to prevent Isaac from becoming the father of Satan. We suggest that God stayed Abraham's hand above Isaac because preventing the existence of Satan would also prevent the existence of Satan's descendant the Messiah. The history of the Israelites is summarized through Jesus and Paul. This paper is written so that the number of believers in the world will increase.

[Anonymous No.15183839]

But real numbers behave this, accept it goy

[Anonymous No.15183854]

>>15183235
>If there's a one to one correspondance, then the number of real numbers from 0 to 1 would be the same as the number of real numbers from 0 to 2, they would be the same size.
Yes, if by "size" you mean cardinality.
>and if you imagined them each as lines, a line of length 1 and a line of length 2, they would have the same number of points, the same length, they would be the same size.
No. You're conflating cardinality with the concept of measure. Don't post again until you complete real analysis up through measure theory.

[Anonymous No.15184186]

>>15183235
>one to one correspondance means being the same size
no it doesn't. Also size without predefined measure doesn't mean anything.
Pls inform yourself about basic shit before posting about stuff like this

[Anonymous No.15184326]

>>15183854
>>15183553
These are thorough explanations including all the terms you need to Google to inform yourself

>>15183547
>>15183499
These are excellent explanations for laymen which essentially say the same thing, less precisely, without those terms.

[Anonymous No.15184351]

>>15183235
>This is of course not true, so one of things mentioned before was wrong. It is probably one of these following two things
Infinity, that's what's wrong.

Infinity does not exist
https://philpapers.org/archive/SEWTCA.pdf
http://theorangeduck.com/page/infinity-doesnt-exist

[Anonymous No.15184402]

>>15183235
It entirely depends on what you're defining as size. This is what we call a measure. Many people would say the 'size' in this case is the largest number in the set minus the smallest number.

[Anonymous No.15184558]

>>15183299
>Show me one. You can't.
Literally look at your own fucking hand. The interval between any two points on it.
Any interval ever in the real world fits, and you out of sheer supreme retardation dared say none exist.

[Anonymous No.15184590]

>>15184558
mathematicians deny this reality and say that number and measure are distinct. look at how many replies in this thread are repeating this fact. they live in a fantasy world.

[Anonymous No.15184631]

>>15184558
>Literally look at your own fucking hand. The interval between any two points on it.
your hand is not an interval of real numbers. it exists in euclidean space. By measuring it with a real-graded ruler (really, rationally-graded), you are necessarily applying a metric. This could be true regardless of what you measured it with in the "real world" because the "real world" comes with a euclidean metric.