Lists of binary sequences and uncountability

Creation of binary lists, discussion about the power set of ℕ, the diagonal argument, Cantor’s first proof and uncountability. Binary system is kind of magic because it can express natural numbers, real numbers and subsets of natural numbers. Below, we will create lists of binary sequences to study the uncountability of the power set of ℕ and real numbers.

1.Infinite list of binary sequences
2.About the Power set of ℕ
3.Frame of Natural Infinity
4.List of numbers smaller than 1
a.Creation of the numbers
b.Denseness of R..
c.Completeness of R..
d.Real numbers in [0,1[
5.About Cantor’s first proof
6.About the diagonal argument
7.Conclusion

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PDF Lists of binary sequences and uncountability
http://pengkuanonmaths.blogspot.com/2016/11/lists-of-binary-sequences-and.html
or Word https://www.academia.edu/30072323/Lists_of_binary_sequences_and_uncountability

Continuity and uncountability

Discussion about continuity of line, how continuity is related to uncountability and the continuum hypothesis.
The real line is made of real numbers which are points. Points are discrete objects, but lines are continuous objects. How does continuity arise out of discreteness when points make line? The idea of uncountability solves this problem. Rational numbers are countable, the line they make contains holes. Real numbers are uncountable, the line they make is continuous. So, continuity must be created by the uncountability of the points of a continuous line. One can imagine that uncountable points are so numerous on the real line that real numbers are squeezed together.

Georg Cantor called the set of real numbers continuum, so he probably thought of creating continuity with discreteness when inventing uncountability. But, what does continuity really mean?

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PDF Continuity and uncountability
http://pengkuanonmaths.blogspot.com/2016/09/continuity-and-uncountability.html
or Word https://www.academia.edu/28750869/Continuity_and_uncountability

 

Cardinality of the set of decimal numbers

Cardinalities of the set of decimal numbers and ℝ are discussed using denominator lines and rational plane. On the rational plane, a vertical line is referred by its abscissa M. Because the points of a vertical line represent the quotients i/M which have the same denominator M, the vertical line at abscissa M is called denominator line of M.

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or Word https://www.academia.edu/23155464/Cardinality_of_the_set_of_decimal_numbers

Prime numbers and irrational numbers  

The relation between prime numbers and irrational numbers are discussed using prime line and pre-irrationality. A rational number is the quotient of 2 whole numbers i and j, coordinates of a points (j, i) in the plane of 2 dimensional natural numbers shown in Figure 1. Each points (j, i) represents a rational number whose value is i/j that equals the slope of the straight line connecting the point (j,i) to the origin (0,0).

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PDF Prime numbers and irrational numbers

http://pengkuanonmaths.blogspot.com/2016/02/prime-numbers-and-irrational-numbers.html

or Word https://www.academia.edu/22457358/Prime_numbers_and_irrational_numbers

On Cantor’s first proof of uncountability

Discussion about Cantor’s first proof using the next-interval-function, potential and actual infinity. Cantor’s first proof of the uncountability of real numbers is the first rigorous demonstration of the notion of uncountability. Countable sets can be put into a list indexed with natural numbers. If a set cannot be listed, then, it has more members than the set of natural numbers and is uncountable. Cantor’s first proof is a proof by contradiction. First, he supposes that all real numbers are listed in any order by the list X=(x1, x2, x3 …). Then, a real number out of this list is found by using a series of intervals, contradicting that X lists all real numbers.

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PDF On Cantor’s first proof of uncountability
http://pengkuanonmaths.blogspot.com/2016/02/on-cantors-first-proof-of-uncountability.html
or Word https://www.academia.edu/22104462/On_Cantors_first_proof_of_uncountability

On the uncountability of the power set of ℕ

This article discusses the uncountability of the power set of ℕ proven by using the out-indexes subset contradiction. Cantor’s theorem proves that the power set of ℕ is uncountable. This is a proof by contradiction. Suppose that the power set of ℕ is countable. This allows us to put all subsets of ℕ in a list. The contradiction will come from the indexes.

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PDF On the uncountability of the power set of ℕ
http://pengkuanonmaths.blogspot.com/2016/02/on-uncountability-of-power-set-of.html
or Word https://www.academia.edu/21601620/On_the_uncountability_of_the_power_set_of_N

Hidden assumption of the diagonal argument

This article uncovers a hidden assumption that the diagonal argument needs, then, explains its implications in matter of infinity. The use of the diagonal digits imposes a condition unnoticed until now. If this assumption were found false, the conclusion of the diagonal argument should be rewritten.

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PDF Hidden assumption of the diagonal argument http://pengkuanonmaths.blogspot.com/2016/01/hidden-assumption-of-diagonal-argument.html
or Word https://www.academia.edu/20805963/Hidden_assumption_of_the_diagonal_argument

Which infinity for irrational numbers?

The value of a decimal number depends on the number of its digits. For irrational numbers that have infinity of digits, their values seem to be definitive. However, the meaning of infinity is ambiguous because there exist several kinds of infinities. If the infinity used to define the number of digits is not clear, the values of irrational numbers will not be well defined. This is why we have to answer the question of the title.

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PDF Which infinity for irrational numbers? http://pengkuanonmaths.blogspot.com/2016/01/which-infinity-for-irrational-numbers.html
or Word https://www.academia.edu/20147272/Which_infinity_for_irrational_numbers

Continuous set and continuum hypothesis

This article explains why the cardinality of a set must be either Aleph0 or |ℝ|.

1. Rational numbers are discrete
2. Real numbers are continuous
3. Collectively exhaustive and mutually exclusive events
4. Continuum hypothesis
5. Cardinality of discontinuous subsets of real numbers

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PDF Continuous set and continuum hypothesis
http://pengkuanonmaths.blogspot.com/2015/12/continuous-set-and-continuum-hypothesis.html
or
Word https://www.academia.edu/19589645/Continuous_set_and_continuum_hypothesis

Cardinality of the set of binary-expressed real numbers

This article gives the cardinal number of the set of all binary numbers by counting its elements, analyses the consequences of the found value and discusses Cantor’s diagonal argument, power set and the continuum hypothesis.
1. Counting the fractional binary numbers
2. Fractional binary numbers on the real line
3. Countability of BF
4. Set of all binary numbers, B
5. On Cantor’s diagonal argument
6. On Cantor’s theorem
7. On infinite digital expansion of irrational number
8. On the continuum hypothesis

Please read the article at
Cardinality of the set of binary-expressed real numbers
PDF http://pengkuanonmaths.blogspot.com/2015/12/cardinality-of-set-of-binary-expressed.html
or
Word https://www.academia.edu/19403597/Cardinality_of_the_set_of_binary-expressed_real_numbers