The Concept of Infinity in the Continuum Hypothesis
This is not as commonly known, infinite value. But countless in different contexts.
Assumed a series of integers {1, 2 , 3, n} to any extent (infinite).
Then, the real numbers {1.5, 2.6 3.9563, r} to any extent (infinite)
However, the sequence of numbers of any type is always infinite, in the sense it's unreachable.
But George Cantor said differently.
He said that there were countless numbers that were more than another countless set of numbers.
There are two infinite series of numbers, that's what it means.
Example:
- ...all the stars in the sky are countless, and as countless as the number of hairs on everyone’s head
- ...or, all the planets that exist are innumerable, and as countless as the number of particles
- ... more axiomatically, all the incalculable possibilities, are the same as all the changes that exist.
Surprisingly George Cantor claims there is an infinite-2 which is greater than infinity-1.
That's the concept of George Cantor.
Two Kinds Infinite
Isn’t infinity enough innumerable? Then how can there be two kinds of infinite, where one "infinity" is greater than the second "infinity".
That’s the concept of George Cantor.
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
____
In simple language ... there are the highest that are still more than previously the highest. There are two at highest at the same thing?
IF THERE ARE TWO THINGS THAT WERE Equally DIFFICULT, THEN HOW TO MEASURE ONE THING MORE DIFFICULT THAN THE OTHER.
Back to previous example ..
- ...all the stars in the sky are countless, and as countless as the number of hairs on everyone’s head
- …or, all the planets that exist are innumerable, and as countless as the number of particles
- ... more axiomatically, all the incalculable possibilities, are the same as all the changes that exist.
If simplified...
...counting the stars in the sky and counting the number of hairs on all heads
IF THERE ARE TWO THINGS THAT WERE Equally DIFFICULT, THEN HOW TO MEASURE ONE MORE DIFFICULT THAN THE OTHER?
Deeper and axiomatic...
If there are two things that are impossible, then how to measure one which is more impossible than the other impossibility?
How to measure something more 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 cheated, but they deceived themselves, then made this a trick for the next generation
ILLUSIONS 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?
If you give labels to distinguish one to another , then you may label these as infinite-1 & infinite-2.
Aleph-0 (infinite-1) & Aleph-1 (infinite-2)
DECEIVED BY METHODS - Without Knowing
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.
Similar to Godel’s method which states that mathematics is fragile (it caused a commotion in the world of mathematical intellectuals). Yes, it’s not a problem, sometimes there’s an uproar, so that it seems like an intellectual challenge (even though what happens is a sophisticated intellectual trick)
THE TRUTH IS THAT MATHEMATICS IS RIGOROUS, only that they saw the flexibility of mathematics in such a way. and said that math wasn’t rigorous.
This is similar to someone saying that iron is malleable and water is strong, only because heated iron can be bent & high pressure water can split iron.
Whereas between iron & water have different properties. Thus everything seems to be blurry, even though everything remains solid as it should follow the situation (solid iron as iron, solid water as water).
Method Full of Deceit...
ILLUSTRATION
What was the method for measuring that between the two things that are Equally DIFFICULT to attain, but THERE IS THE MOST DIFFICULT in between both?
What was the method of measuring that something unmatched could be broken by another unmatched?
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?"
=> But hot water & cold water are the same water, so nothing is more watery than the other one (between hot water & ice water)
But there were those who insist that hot water was more watery than cold water. That’s weird.
Then we asked, "how come hot water was more watery than cold water?" The answer? wew
He said, "hot water is hotter, it’s good to make hot coffee. It brings more intimacy. It’s good to take it to make coffee at a coffee shop while discussing". Wow it’s funny answer. OUT OF CONTEXT . That’s what happened in the case of the continuum hypothesis.
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?
Continuum Hypothesis Method
The method is simple:
- Take a set of numbers which thoroughly countless a lot.
Examples of integer numbers (not
decimal), {1. 2, 3, 4, ...} | Realize
first that the series of these numbers
has obviously countless {1. 2, 3, 4, 5, ...,
1000000, ..., 12542863, ...}
2. Take a set of numbers with different type which the total amount is also uncountable a lot, the same previous set of numbers which also countless.
Examples of real numbers {1, 1.5, 2,
2.5, 3, 3.5, ...} | Realize first
that this series of numbers has
obviously countless {1, 1.5, 2, ..., 12543,42288, ... , 1000000, ...,
12542863.000001, ...}.
Point 1. represents innumerable numbers of non-decimal type (aleph-0)
Point 2. represents innumerable numbers of real type involving decimal digits (aleph-1)
Point 1. & point 2. show the existence of a set of numbers of different types, each of which has an innumerable number.
Well, here’s the weird thing. How to measure that one of the two is more than the other, when both are included as many as innumerable?
Bijective
Simply by making a relationship (connecting in) between each member of the two sets.
So I was curious, why was it chosen that way? Where’s the accuracy?
- Boots <---> Men
- High Heels <---> Women
- Flip Flops <---> Little Boy
There seems to be a one-to-one relationship between the left <---> right. This is a sign that the two have the same amount.
Another example:
1 <---> 2
2 <---> 4
3 <---> 6
4 <---> 8
5 <---> 10
6 <---> 12
7 <---> 14
8 <---> 16
9 <---> 18
The set of number on the left {1, 2, 3, 4, 5, 6, 7, 8, 9} has the same equal entity as on the right side {2, 4, 6, 8, 10, 12, 14, 16 , 18}. Although at first glance it appears that the total set of numbers on the right is more than the total number of sets of numbers on the left. Even more, at a glance when viewed from the largest number from the left & right, also seen, the last number on the right side (18) is larger than the last number on the left side.
But the calculation to determine which one is more than the other one, is not from common way.
This (according to George Cantor) is not calculating a total of finite numbers, but calculating its POSSIBILITY AS A WHOLE. It measures how many total of uncountable can be.
Yes, the answer is actually easy. The total of probability can be as many as the set of numbers on the left as compared to the set of numbers on the right. But George Cantor insisted that the amount of infinite number of two sets could be different. There is an amount of infinite which is more than another infinity.
What is the basis of his reasoning?
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.
This is similar to reasoning comparing a 100 meter rope vs a 200 meter rope, and of course the longer rope is 200 meters.
So Cantor formulated a pattern of comparing distances as above and applied it to two sets. Therefore?
- If the members of two sets can be connected one-to-one, it means that the distance between them is the same length and regardless of how far, the two sets are considered to have the same probability of infinity.
- However ...
However, if the members between two sets cannot be connected at all, in the sense that some are not connected, then? There is greater sst of numbers than the other one, so the value of the infinite is greater than the other infinity.
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.
Even though the structure of reasoning like this is no longer in the mathematical category, but is in the "absurd" category
I give an example...
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
Continuum Hypothesis
(Measurement)
Similarly, Cantor measures which of the two members has greater amount of infinity, by drawing a connecting line between the two points of the two set of numbers
That is by making a simulation between integers vs real numbers involving decimal numbers. Like this:
1 <---> 1
❌<---> 1.5
2 <---> 2
❌<---> 2.5
3 <---> 3
❌ <---> 3.5
4 <---> 4
❌ <---> 4.5
5 <---> 5
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) …
So it is concluded that there are more buttons on the right side of the shirt (even though the distance between the two sides is still the same.
This is where the continuum hypothesis is weird.
This is just an example with an illustration to make it easier to understand, so that you can see where the confusion of reasoning is.
Although using illustrations, but still the illustrations are able to show clearly where the chaos of reasoning.
Later, in the end I will explain axiomatically, so that the understanding through this illustration becomes clearer convincingly, that the reasoning on the hypothesis continuum is WRONG! ❌ :
- that the reasoning on the continuum hypothesis is absurd
- that the reasoning on the continuum of the hypothesis does not characterize mathematical reasoning and does not abrogate the certainty of math
Axiomatic Analysis "Continuum Hypothesis"
Check out the following formats:
1 <---> 1
❌<---> 1.5
2 <---> 2
❌<---> 2.5
3 <---> 3
❌ <---> 3.5
4 <---> 4
❌ <---> 4.5
5 <---> 5
First Analysis
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. For nothingness cannot exist among existences. This means that there are actually no cracks, no voids on the left side of the set, so this confirms axiomatically that from the left side and from the right side, they can always be connected one by one, so that both sides viewed from any angle of calculation show the equality of incalculable possibilities, an equal infinity.
Second Analysis
When you point to any number to the left of the set, you are actually pointing to an existence.
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?
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.
Problem Solved. So?
Yes, that "Continuum Hypothesis" is FALSE.
That there is no "countless" greater than another “countless”
What about 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?
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.
Continuum hypothesis is failed, because of these reasons:
If we can do multiple calculations on different areas of infinity and that looks like we are doing things differently, and again it looks like we’re dealing with different (many) infinity, but actually we’re doing all of those interchangeably at the same area, the same numbers 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 are playing at the same area and there is no bigger infinity, it’s just one infinity.
THIS WAS MY CONVERSATION ABOUT THIS ISSUE.
Hope, you can see a different perspective on this issue
How do the integers and even integers perfectly overlap?
– J Kusin
5 hours ago
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
3 hours ago Delete
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
3 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
3 hours ago Delete
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
2 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
2 hours ago Delete
okay i will try following you
– Seremonia
2 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
2 hours ago Delete
remember that any time we talk about numbers, then we talked about things (not just numbers)
– Seremonia
2 hours ago Delete
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
2 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
2 hours ago
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
2 hours ago Delete
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
2 hours ago Delete
there is nothing weird about this kind of thinking. it's all make sense, and that's the way we must analysis math
– Seremonia
2 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
2 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
2 hours ago Delete
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
1 hour 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
1 hour 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
1 hour ago Delete
