Delve into the intriguing world of set theory as we explore the Continuum Hypothesis. Unravel the mysteries of this mathematical conjecture, its implications on the infinite universe of real numbers, and its profound impact on the foundations of mathematics.
Mathematics is Rigorous
Godel and Einstein saw flexibility in mathematics and said that math wasn’t rigorous.
- The truth is that math is rigorous, it’s just that they saw the flexibility of mathematics in such a way that they didn’t have to be rigorous
If there are no 50^80 or there is no “krohntirtoir” then there can be no “inn@&@?!”
Seremonia
- Yes, but what if there are not?
- J Kusin
- No, but thanks for the appreciation
Delete.
You’re holding on infinites, not on aleph-null.
- Seremonia
- Cantors diagonal numbers don’t overlap with the ones on the list either.
- Delete.
- You have to relate math on reality to make sense of it.
- The fictionalist and pure formalists do not claim how it relates, just that it is effective.
This is similar to saying that iron is malleable and water is strong, only because heated iron can be bent & high pressure water can split iron.
The difference between iron & water have different properties
- Everything seems blurry, even though everything remains solid as it should follow the situation
How can you formally determine that there is an infinite series of numbers that is more infinite than another infinite?
Continuum Hypothesis Method
- Take a set of numbers which are countless a lot.
- Realize first that the series of these numbers has obviously countless.
- Take another example of numbers with different type which are also uncountable a lot, and the total amount is the same for both sets of numbers
- How do you measure that one set is more than the other?
- Make a relationship (connecting in) between each member of the two sets
The Concept of Infinity in the Continuum Hypothesis
George Cantor said that there were countless numbers that were more than another countless set of numbers.
- 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”?
- There are two at the highest at the same thing.
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 exist are innumerable,… all the incalculable possibilities, are the same as all the changes that exist.
ILLUSIONS in Math
A series of numbers is always uncountable, so how can there be a number that is greater than the one previously counted?
- If you give labels to distinguish one to another, then you may label these as infinite-1 & infinite-2.
- Decide by methods – without knowing.
1 <---> 1.5.
Similar to buttoning a shirt, only the difference is that the length is the same between the left and right sides, but on the left side the buttonhole is a less than as it should be compared to the right side of the shirt with greater amount of buttons.
The Continuum Hypothesis is FALSE
When you point to any number to the left of the set, you are actually pointing to an existence.
- Even from the left side there are connecting points as decimal numbers to the right side, so this confirms axiomatically that there is an equality of numbers, that the two sets are sets with the same possible number of UNCOUNTABLE numbers.
Cantor formulated a pattern of comparing distances as above and applied it to two sets:
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, 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 one, so the value of the infinite is greater than the other infinity
- There is an infinity that is shorter than another infinity
Method Full of Deceit
What was the method for measuring the difference between the two things that are equally DIFFICULT to attain, but there is the most difficulty in between them?
Continuum hypothesis is failed
If we can do multiple calculations on different areas of infinity and that looks like we are doing things differently…
These axioms tackle this issue:
If the two things overlap perfectly, then they are really just one thing
- Two infinites overlap each other perfectly then it’s just one.
- We are playing at the same area, there is no bigger infinity, it is just one infinity
- How do the integers and even integers perfectly overlap?
If there are two things that are impossible, then how to measure one more difficult than the other?
The root of this problem lies in the fact that both impossible things cannot be equal, and the only way to distinguish between them is to measure something more difficult when both are equal
