Continuum Hypothesis - Deepstash

# Continuum Hypothesis

seremonia.medium.com

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22 IDEAS

6

## New Concept of Infinity

The Concept of Infinity in the Continuum Hypothesis

This is not as commonly known, infinite value. But countless in different contexts.

However, the sequence of numbers of any type is always infinite, in the sense it's unreachable.

There are two infinite series of numbers, that's what it means.

16

## Two Infinites

Surprisingly George Cantor claims there is an infinite-2 which is greater than infinity-1.

Isn’t infinity enough innumerable? Then how can there be two kinds of infinite, where one "infinity" is greater than the second "infinity".

Yet intuitively, the infinite is pretty much weird. Then how many kinds of infinite are more than the previous infinity?

This is where the understanding of the continuum hypothesis begins
____

16

## Illusion In Math

A series of numbers is always uncountable, so how can there be a series of numbers that is more uncountable than the one that was previously uncountable?

How to measure one thing than the other when both are equal? Thus the method was found which became the root of this problem. Some have objected to this.

It’s not that they’ve been cheated, but they deceived themselves, then made this a trick for the next generation.

16

## The Method - Parable

What is the method for measuring that something that is innumerable can challenge another that is also innumerable?

I will give an example before going into the formal method, so that we can see the weird of the continuum hypothesis in this case.

It’s like between hot water & ice water, then asked,

• ... "which is more characteristic of water, hot water or ice water?"

15

## The Formal Method

How can, according to George Cantor’s version, formally determine that there is an infinite series of numbers that is more infinite than another infinite?

Simply by making a relationship (connecting in) between each member of the two sets.

So, if the members of the two sets can be connected one to one (bijective), then it means that they are the same amount of number, both have the same amount of infinite.

Put simply: there is an infinity that is shorter than another infinity. You see, this reasoning is much more absurd, but forcing it, as this is a mathematical structure.

15

## An Illustration

It’s like when we button a shirt, so if all buttons are buttoned, it is assumed that the left and right sides are the same length.

But when the buttons don’t fit properly, so you can see that there is a longer side of the shirt. But actually THE LEFT SIDE & RIGHT SIDE IS TOTALLY THE SAME LENGTH!!! That’s it

15

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Similar to the illustration of buttoning a shirt, only the difference is that the length is the same between the left and right sides, but on the left side of the shirt, the buttonhole is a less than as it should be compared to the right side of the shirt with greater amount of buttons. So there are buttons that don’t fit in the holes (because there are fewer buttonholes) …

15

## First Analysis

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They thought there were points that couldn’t be connected (marked by a red cross) one to one (bijective) from left to right (from the member of the set of integer numbers to the set of real numbers.

But they forgot that what was (red) crossed was actually a location, an existence, not nothingness. So this confirms axiomatically that from the left side and from the right side, they can always be connected one by one.

15

## Second Analysis

When you point to the number one, you are actually pointing to something that exists.

The question is "how wide is the “number 1" you’re pointing at?"

• ... Of course the number 1 is the area of ​​0 to 0.999999999999999 ...
• ...Similarly the number 2 is
area from 0 to 1.999999999999999 ...

It can be seen here that even from the left side there are connecting points as decimal numbers to the set of real numbers on the right. So?

15

## Problem Solved

So that between the members of the set of integers and the members of the set of real numbers (involving decimals), it can always be paired (connected) one to one (bijective) for all members of the two set of different types of numbers.

This also has confirmed that from the left side to the right side there is an equality of numbers, that the two sets are sets with the same possible number of UNCOUNTABLE numbers.

Yes, that "Continuum Hypothesis" is FALSE.

That there is no "countless" greater than another “countless”.

15

## What About George Cantor's Diagonal?

Actually, whether we are trying to do cantor’s diagonal, or multiplying power set of aleph-null, but it’s actually we are doing on the same numbers as whole numbers, as one infinity.

Although you can create multiple infinities, still we are doing on the same range.

• It’s just that our point of view seeing all of those as different areas, illusion of separation, since we are dealing with multiple cardinality. but in essence?

12

## The Same Area

Those all sets are just one set of infinite number, and we did arranging those numbers into different set of point of view, so we thought we were dealing with something different increasingly at different stage of infniity.

It’s just a set. Only they did interchange in between numbers, so it looks like they did two or more moment, but actually just one moment interchangeably.

13

## Venn Diagram

Continuum hypothesis is failed, because of these reasons:

If we can do multiple calc on different areas of infinity and that looks like we are doing things differently, or we're dealing with different infinity, but actually we're doing at the same area, the same numbes were used interchangeably.

These axioms tackle this issue ...

• 🧩 If the two things overlap each other perfectly, then they are really just one thing (venn diagram)
• ​​​​​​🧩 Two infinites overlap each other perfectly, then it's just one thing

We're playing at the same area & there is no bigger infinity, it's just one infnity

11

## Discussion @StackExchange

What are the useful outcomes of denying the Continuum Hypothesis?

This opens new perspective of how we think on math.

_____

How do the integers and even integers perfectly overlap?

– J Kusin

yesterday

When infinite, that means the farthest length of integers just the same the farthest lenght of any kind of numbers. do you think the farthest of the first infinity lower than another kind of inifinity. try not making comparison the other way around, just focus on the farthest of infinity and make comparison on those field

– Seremonia

22 hours ago  Delete

8

Don't make quick conclusion on this case. you can prolong your question in this issue, so that we can synchronise understanding better on this issue –

Seremonia

22 hours ago   Delete

Consider you are holding aleph-null with full of infinities from any fields. the question is does aleph-null bigger than all infinites? no, since aleph-null is just a label. you're holding on infinites, not on aleph-null –

Seremonia

22 hours ago   Delete

8

I’m focusing on “overlap”. Cantors diagonal numbers don’t overlap with the ones on the list either. Even and don’t overlap odd either. Since we can order {0,1,3..} and {0,1,2,3,4…} by the time we get to 2 in the second list, we know we will never find it in the first. Both are infinite yet different. Can you agree to that? –

J Kusin

22 hours ago

Let's do thought experiment. consider there is huge room. do you thing only one of both kind of number can fill the entire space? no. both (integers & real numbers) can fill the entire space in a single room. –

Seremonia

22 hours ago   Delete

9

Okay i will try following you –

Seremonia

22 hours ago   Delete

Any cantor's diagonal trial, actually can be connected bijective, simply by understanding that both numbers can be divided. are we on the same page on this?. make a detail question. we try to slice this sharply to zoom the issue. but we have to relate this with reality, so we can make sense of this case –

Seremonia

22 hours ago   Delete

Remember that any time we talk about numbers, then we talked about things (not just numbers) –

Seremonia

22 hours ago   Delete

8

When someone said about "infinity" don't be tricked by the cardinality, but try seeing on infinity itself as it's expanding to the entire possible space –

Seremonia

22 hours ago   Delete

We don’t know how math relates to the physical. I can write down 50^80 yet what is the physical meaning. We won’t come to terms here. I can see we differ at this step. –

J Kusin

22 hours ago

8

You have to relate math on reality. if you reject this, try with small step. consider a number must be related to a thing, otherwise we're dealing with nonsense. although someday someone accept math explaining other dimention, or relativity, still it has to do with things. there is nothing simple than this –

Seremonia

22 hours ago    Delete

8

A single finite number may be related to a banana. two finite numbers may be related to two bananas. more & more numbers may be related to more and more bananas, rocks, books, atom, and so forth. more & more infinites number then, must be related to all of possible things that fill the entire possible space. this is the consequence of following the logic behind math –

Seremonia

22 hours ago   Delete

8

There is nothing weird about this kind of thinking. it's all make sense, and that's the way we must analysis math –

Seremonia

22 hours ago    Delete

Math is part of reality I’m not arguing that. I’m saying we don’t know how. The fictionalist and pure formalist do not claim how it relates just that it is effective. You are a physicalist about math. Not everyone is. –

J Kusin

22 hours ago

50^80 may be related to huge things as much as 50^80. but thanks anyway for your appreciation to my perspective rather than just downvoting. thumbs up –

Seremonia

22 hours ago   Delete

9

Yes but the argument is what if there are not 50^80 things. Anyway I’m not saying you couldn’t be right. No camp seems completely satisfactory –

J Kusin

22 hours ago

If there are no 50^80 or there is no "krohntirtoir" but the 50^80 is much more make sense to be related to "possibility", while "inn@&@?!" much more harder to think of the possibility –

Seremonia

21 hours ago   Delete

Although i agree with you that there could be math field can't be translated into physical understanding, but at least we can try on this issue, one step at a time –

Seremonia

21 hours ago   Delete

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Beware of Illusion in Math | Denial of CH Determines that We are not Living On Discrete World

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