# 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.

PDF On the uncountability of the power set of ℕ
http://pengkuanonmaths.blogspot.com/2016/02/on-uncountability-of-power-set-of.html

# 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.

PDF Hidden assumption of the diagonal argument http://pengkuanonmaths.blogspot.com/2016/01/hidden-assumption-of-diagonal-argument.html

# 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.

PDF Which infinity for irrational numbers? http://pengkuanonmaths.blogspot.com/2016/01/which-infinity-for-irrational-numbers.html

# 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

PDF Continuous set and continuum hypothesis
http://pengkuanonmaths.blogspot.com/2015/12/continuous-set-and-continuum-hypothesis.html
or

# 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

Cardinality of the set of binary-expressed real numbers
PDF http://pengkuanonmaths.blogspot.com/2015/12/cardinality-of-set-of-binary-expressed.html
or

# A 1.95 m long solenoid exerting Aharonov–Bohm force on a coil

This experiment shows the magnetic force on a coil exerted by the magnetic field of a long solenoid (1.95 m, 6.4 foot) that should be zero.

A 1.95 m long solenoid exerting Aharonov–Bohm force on a coil
http://pengkuanem.blogspot.com/2015/10/a-195-m-long-solenoid-exerting.html
or

# Disc magnet parallel action experiment

The magnetic field of the earth can rotate a flat coil in its plane. This is explained in Earth’s magnetic field and parallel action. Will a disc magnet rotate a flat coil the same way? The magnetic field of a disc magnet is central symmetric and the field lines are contained in median planes (See Figure 1). If a flat coil is coplanar with a median plane, the Lorentz forces on the currents will be perpendicular to the plane of the coil. So, Lorentz force could not rotate the flat coil in its plane. But in my experiment the coil rotates.

Disc magnet parallel action experiment
http://pengkuanem.blogspot.com/2015/06/disc-magnet-parallel-action-experiment.html
or Disc magnet parallel action experiment (video included)

# Earth’s magnetic field and parallel action

Earth’s magnetic field and parallel action

The magnetic field of the earth is uniform on its surface. The resultant Lorentz force a uniform magnetic field exerts on a coil of any shape is zero. The torque perpendicular to a flat coil is also zero. So, a current carrying coil in the magnetic field of the earth should stay immobile. However, my experiment shows that the test coil rotates in its plane. See the video of this experiment: http://youtu.be/JKMG8jY1RRg

Earth’s magnetic field and parallel action
http://pengkuanem.blogspot.com/2015/06/earths-magnetic-field-and-parallel.html
or Earth’s magnetic field and parallel action (video included)

# Solenoid parallel action experiment

In “Q: Parallel action with a solenoid” I have asked a question: the magnetic field of a long solenoid being zero, will it make a coil rotate in its plan? This experiment will investigate the parallel action of a long solenoid. I have done this experiment and here is my result.

Solenoid parallel action experiment
http://pengkuanem.blogspot.com/2015/06/solenoid-parallel-action-experiment.html
or Solenoid parallel action experiment with video included

# Q: Parallel action with a solenoid

According to my corrected magnetic force law, parallel currents attract each other and anti-parallel currents repel each other, see Theory about parallel action experiment. So, for the rectangular coil near the solenoid in Figure 1, the upper current is parallel to that in the front side of the solenoid and the lower current is anti-parallel. So, there would be a torque created on the coil and the coil should rotate.

However, the magnetic field outside a solenoid is zero and cannot act any force on a current. Moreover, there cannot be force parallel to current for classical theory.

So, If I do this experiment, will the coil rotate or not?