axiomatic sets 40

Basically, mathematics is founded by some axioms of sets.

(1)empty set
(2)extensional
(3)pairing
(4)union of sets
(5)power sets
(6)separation
(7)replacement
(8)regularity
(9)infinite
(10)axiom of choice

I do not think every axioms are clear and definite.

Strictly, for example, by axiom of infinite and separation we will get axiom of empty set.
We are able to reduce ten axioms. 

Next we got the various kinds of numbers,
natural numbers$\mathbb{N}$, integers$\mathbb{Z}$, rational numbers$\mathbb{Q}$, real numbers$\mathbb{R}$, complex numbers$\mathbb{C}$ on sets.

Then, we saw the basic rules of calculation and logics.

Modern mathematics is said to be enough by them. 

base n numbers

The expression of numbers is free.
You might know 'XXVIII' by roman numerals is '28'.

We usually use numbers based 10, which are expressed by 0,1,2,3,4,5,6,7,8,9.
The number '32' means $3\times 10+2$ .
That is, the second digit '3' is '30'.
This writing method is continuous, '4' of '432' means '400',
'5' of '5432' means '5000', and so on.
The number of digits needs $10^n$ .

When we also use numbers based 2 which have 0 and 1, this way has to be kept.
The number '111' is $1\times 2^2+1\times 2+1$ , which is '7' by base 10 numbers,
and is not $1\times 10^2+1\times 10+1$ .

These imply numbers are made by bases, symbols and digits.
When we use base 2 numbers whose symbols are 'i' equivalent to '0'
and 'x'equivalent to '1', 'xxi' means '6' by decimal numbers.

You must remember that originally $0=\left\{ \right\}=\phi$ ,
$1=\left\{\phi\right\},2=\left\{\phi,\left\{ \phi\right\} \right\}\cdots$ .

Generally, numbers has three meanings, of which one is quantity numbers,
 two is ordinal numbers, three is identification numbers.

We can use a most convenient way for numbers.

Binary numbers are most suitable for computers and logic algebra.

axiomatic sets 39 (complex numbers)

We shall define complex numbers from real numbers in set theory.

Let $(a,b)$ be a pair of two real numbers $a,b\in\mathbb{R}$.

Then, we decide four properties on $C=\left\{(a,b):a,b\in \mathbb{R} \right\}$ ;

(1)$(a,b)+(c,d)=(a+c,b+d)$ ,
(2)$(a,b)\cdot (c,d)=(ac-bd,ad+bc)$ ,
(3)for any real number $k\in\mathbb{R}$ , $k(a,b)=(ka,kb)$ ,
(4)if $(a,b)\ne (0,0)$ , then $\frac{1}{(a,b)}=\left( \frac{a}{a^2+b^2},\frac{-b}{a^2+b^2} \right)$ ,

We call the pairs having above properties complex numbers $\mathbb{C}$ .

You must understand immediately $(0,1)\cdot (0,1)=(-1,0)$ ,
$(a,b)\cdot \frac{1}{(a,b)}=(1,0)$ , $(1,0)\cdot (a,0)=(a,0)$ ,
$(1,0)\cdot (0,b)=(0,b)$ , $(0,1)\cdot (a,0)=(0,a)$,
$(0,1)\cdot(0,a)=(-a,0)$ ,
and as we usually write $(0,1)=i$ , then $i^2=i\cdot i=(-1,0)$ .

And $(a,0)$ is fully corresponding to a real number $a$ .
Thus $(0,0)=0, (1,0)=1,(-1,0)=-1,\cdots$ and so on ,
and we also write $a+bi=(a,0)+(0,b)=(a,b)$

Complex numbers are the pair of two real numbers having some properties.
The set $\mathbb{C}$ does not include the set $\mathbb{R}$ formally
same as the relation of natural numbers and integers.
But the fact that $a$ is fully corresponding to $(a,0)$
makes us accept $\mathbb{R}\subset\mathbb{C}$ .

classical logics (an addition)

You must know that $A\subset A$ is always true.
Thus, although an proposition $A$ is false,  $A\subset A$ is true.

With similar, if an proposition $A$ is false, then
$A\sim A$ is true.

However, as an proposition $A$ is false,
$A\cdot A$ is false, and
$A\vee A$ is false.

Please check the truth of the proposition "$A\subset (B\vee C)$ ",
where $A$ is false, $B$ is false, and $C$ is false.

If $A$ is "a day has 25 hours ",
$B$ is "a week has 175(=25×7) hours " and
$C$ is "a month has 750(=25×30) hours" , then
"$A\subset (B\vee C)$ " is true .

How do you feel it?

classical logics (the truth table)

We shall call a basic proposition or formula an atom.
For example, an atom is "human beings are animals",
or "sin(x+y)=sin(x)cos(y)+cos(x)sin(y)" and so on.

We can give a true or false value to an atom.
These above are both true.

However, "$y=x+2\quad (x\in [0,1],y\in [0,1])$" is false.

Next we basically introduce 4 connectors and 1 operator for atoms.

(1)A→B : if A,then B.
(2)A・B : A and B.
(3)AＶB : A or B.
(4)A～B : A→B and B→A.
(5)$\neg$A : not A.

By using connectors and operator,
we are able to make a more complicated proposition than an atom.
"A" is "$x$ is a real number."
"B" is "$x$ is a rational number or a irrational number."
We get a new proposition "C" which is "A→B" .
(we have already known "A～B".)

Then, how we get the true or false value of "C" ?

The truth table is gotten.

A B  →
t  t   t
t  f   f
f  t   t
f  f   t

A B  ・
t  t    t
t  f    f
f  t    f
f  f    f

A B   Ｖ　
t  t     t  
t  f     t  
f  t     t  
f  f     f  

A B   ～　
t  t     t  
t  f     f  
f  t     f
f  f     t  

A  $\neg$A　
t    f
f    t  

coffee break 17 (darksouls3)

I am playing darkouls3 the fire fades edition using about 200 hours.
Now, I have defeated many enemies in the main game, dlc1, and dlc2 except the last boss .

The last boss is Soul of Cinder in Kiln of the First Flame.
If I have beaten the final boss, I will see the ending of the game.

I want to see the one of multi endings, however I also do not want to finish the game.

axiomatic sets 38 (axiom of choice 2)

(1)No matter how easy or no matter how difficult operation it is,
despite the operation never ending,
are we able to look infinite process obvious ?

(2)Although we can not always show the function,
we have to always accept the existing of the function. is it obvious?

Axiom of choice asserts  these are obvious and true.

axiomatic sets 37 (axiom of choice)

For any non-empty set $\mathcal{A}$ , there is a function $f$
which can pick up an element $a$ from every each set $A$ in $\mathcal{A}$.
The collection of all $a$ is a set, and $f$ is called a choice function.


Do you think it is obvious or a self-evident truth?

coffee break 16 (Playback)

Recently I am reading Philip Marlowe series of Raymond T.Chandler again.

The Long Goodbye, The Big sleep, Farewell, My lovely, The Little Sister,,,

In Japan as the translations by Haruki Murakami became famous,
I am also reading his works.

I love hard boiled detectives by Robert B.Parker, Ross Macdonald, S.Dashiell Hammett,,,

Haruki Murakami is not my favourite author.
However, his translation is very amenable to originals.

In Playback, the statement of
"If I wasn't hard, I wouldn't be alive.
 If I couldn't ever be gentle, I wouldn't deserve to be alive. "
is most famous.

His translation is not bad.

axiomatic sets 36 (a totally ordered relation)

We have already gotten the equivalent relation.
For a set $\Omega$ and elements $a,b,c\in \Omega$ ,

(1)if $a\sim b$ ,then $b\sim a$ .
(2)for all $a$ , $a\sim a$ .
(3)if $a\sim b$ and $b\sim c$ , then $a\sim c$ .

We get the ordering relation by changing the condition.
The difference is only (1).

(1)if $a\preceq b$ and $b\preceq a$ , then $a\sim b$
(2)for all $a$ , $a\preceq a$ .
(3)if $a\preceq b$ and $b\preceq c$ , then $a\preceq c$ .

Namely, $a\preceq b$ does not always mean $b\preceq a$ .

A totally ordered relation of the set is for any $a,b\in\Omega$ , $a\preceq b$ or $b\preceq a$ is true.

If it is not true, the set is incomparable and called the partially ordered set.

In axiomatic set theory, we often use a pair $(\Omega,\preceq)$ of a set $\Omega$ and a relation $\preceq$ .
(you must note that a relation is also a set. )

axiomatic sets 35 (ordinal numbers)

You may remember the structure of natural number $\mathbb{N}$ ,
\[ 0=\left\{ \right\},1=\left\{0\right\},2=\left\{0,1\right\},3=\left\{0,1,2\right\},\cdots , \]
based on axiom of infinity ($n+1=n\cup\left\{n\right\}$) .

We will also define
\[ \mathbb{w}=\left\{0,1,2,\cdots \right\}    \]
as all natural numbers $n$ .

If we go to the next number from $\mathbb{w}$, then
\[ \mathbb{w}+1=\left\{0,1,2,\cdots,\mathbb{w}  \right\},  \]
Furthermore,
\[ \mathbb{w}+2=\left\{0,1,2,\cdots,\mathbb{w},\mathbb{w}+1  \right\} , \]
and $\cdots\cdots\cdots$

Then,
\[ \mathbb{w}+\mathbb{w}=\left\{0,1,2,\cdots,\mathbb{w},\mathbb{w}+1,\mathbb{w}+2,\cdots  \right\}  \]
and we naturally put
\[ 2\mathbb{w}=\mathbb{w}+\mathbb{w} .   \]

Further and furthermore,
\[ 2\mathbb{w},3\mathbb{w},\cdots, \mathbb{w}\mathbb{w} \]
\[\mathbb{w}^2=\mathbb{w}\mathbb{w}=\mathbb{w}+\mathbb{w}+\cdots  \]
\[ \mathbb{w}^3=\mathbb{w}^2+\mathbb{w}^2+\cdots  \]

(you must also remember that $3^2=3\times 3=3+3+3,4^2=4\times 4=4+4+4+4$ and $3^3=3^2\times 3$)

Although we can not write anymore,
we are formally able to get $\mathbb{w}^{\mathbb{w}},\mathbb{w}^{\mathbb{w}^{\mathbb{w}}}\cdots$ .

These are called an Ordinal number.
It is the expansion of the property of Natural number.

When you find out eternal infinite repetitions of an infinite number,
could you feel the depth of real number $\mathbb{R}$ ?

axiomatic sets 34 (regularity 3)

We shall give the easy proof in which Axiom of regularity
\[ \forall x[x\ne \phi\rightarrow \exists y[y\in x\wedge y\cap x=\phi]]  \]
means that there is not a infinite sequence $a_1\ni a_2\ni a_3\ni\cdots$ .

Suppose that there is a infinite sequence $a_1\ni a_2\ni a_3\ni\cdots$ .
Then, we are able to make the new set $A=\left\{ a_1,a_2,a_3,\cdots  \right\}$ .

If we accept axiom of regularity,
then there must be a $b$ such that $A\cap b=\phi$  in $a_1,a_2,a_3,\cdots $ .

Let $b$ be $a_i$ . However, as $\cdots\ni a_{i-1}\ni a_i\ni a_{i+1}\ni \cdots$ ,
$a_{i+1}\in b(=a_i)$ and $a_{i+1}\in A$ are true.

It is a contradiction, because $a_{i+1}\in(A\cap b)$ .
Therefore, there is not a infinite sequence $a_1\ni a_2\ni a_3\ni\cdots$ .

Conversely, we accept axiom of regularity.
For two arbitrary sets $A,a$ , even if $A\cap a=a_1$ ,$A\cap a_1=a_2$ ,$\cdots$ continue,
by axiom of regularity, the sequence will stop somewhere.

If $A\cap a_i=\phi$ , then by $a_i=b$ , axiom of regularity will be satisfied.

axiomatic sets 33 (regularity 2)

Axiom of regularity is also called axiom of foundation.
\[ \forall x[x\ne \phi\rightarrow \exists y[y\in x\wedge y\cap x=\phi]]  \]

This axiom requires that even infinite sets are bounded
and all collections of things is not necessarily a set.

Example(1); $A=\left\{ x: x\notin x \right\}$  is not a set.

$A$  is a collection of a set which does not contain itself.
(you are able to consider that $A$  is a very huge collection. )

If you accept that $A$  is a set, then a contradiction occurs.

We will give you a question.
Which do you prefer, $A\in A$  or $A\notin A$ ?

This is Russell's paradox.

However, using axiom of regularity we have to recognize $A$  is not a set.
If $A\in A$ is accepted, as there exists the sequence $A\ni A\ni A\ni\cdots$ ,
$A$ violates the axiom.

If $A\notin A$ is accepted, $\left\{ x: x\notin x \right\}$ is not satisfied.


Example(2); The collection $A$ of all sets is not a set.

$A$ is the largest collection in all sets.

However, as $A\in A$ , there is a sequence $A\ni A\ni A\ni\cdots$ .
Therefore, by axiom of regularity $A$ is not a set.

The collections which are not set are called classes. 

axiomatic sets 32 (regularity)

This axiom of regularity was offered by J.Neumann.

\[ \forall x[x\ne \phi\rightarrow \exists y[y\in x\wedge y\cap x=\phi]]  \]

All sets must satisfy this axiom.
A collection of things which can not satisfy this axiom is not a set.

At a glance, you might see it difficult to understand.
This axiom requires that infinite sets are not unbounded.

Suppose a infinite sequence of the sets $a_1\ni a_2\ni a_3\ni\cdots$ .
If $a_i=\left\{ a_{i+1}  \right\}$ ,then
$a_1=\left\{ a_2 \right\}=\left\{ \left\{ a_3 \right\} \right\}=\cdots $ .

(remember $\mathbb{N}$ or axiom on infinity .)

On the other hand, let us consider $A=\left\{ a_1,a_2,a_3,\cdots \right\}$ .
As this $A$ violates the axiom of regularity, $A$  is not a set.

We have to call a class (not a set) $A$ .

Although $\mathbb{N}$ is a set, $\left\{ \infty\right\}$ is not a set.
(please do not misunderstand. )

This axiom makes too big huge collections of things be not a set.

This axiom is equivalent to
" for all $x$ , there does not exist the sequence $x\ni x_1\ni x_2\ni x_3\ni\cdots$ ",
or $x$ such that $x\ni x_1\ni x_2\ni x_3\ni\cdots$  is not a set.

axiomatic sets 31 (infinite sets)

In number system, We have already used the axiom of infinity.

\[ \exists x[0\in x\wedge \forall y[y\in x\rightarrow (y\cup \left\{ y\right\})\in x] ]  \]

 ,where $(y\cup \left\{ y\right\})$ is called the successor set of $y$ .

In axiomatic set theory, an infinite set is the set $x$ such that if an element $y$ is in $x$ , then the successor set of $y$ is also in $x$ .

Namely, if $0\in x$ , then the successor set $1$ of $0$ is in $x$ .
As $1\in x$ , the successor set $2$ of $1$ is in $x$ , $\cdots$ and so on.

This infinite set has no maximum number in the set.

You must have seen by this axiom there is $\mathbb{N}$ as the base of numbers.

This axiom asserts there are sets whose elements are infinite.

axiomatic sets 30 (making a subset 2)

By axiom of separation, we are able to make a new little set whose elements are chosen from a given set.
\[ \forall x\exists y\forall y[z\in y\leftrightarrow z\in x\wedge P(x)]  \]

For example, if $P(x)$  is $z\in w$ ,then
\[ \left\{z\in x:z\in w \right\}=y=x\cap w .  \]

However, we are only able to make a few new sets by using this axiom.
We want to make a bigger set freely.

A.Fraenkel offered an alternative axiom "axiom of replacement".
\[  \forall x\exists y \forall z[z\in y\leftrightarrow \exists w[w\in x \wedge \phi(w,z)]]  \]
 ,where $\phi$ is a formula or a statement of properties or a function.

That is to say, if
\[  \forall x\in a,\quad \exists y ,\quad \phi(x,y) \]
 is satisfied, then
\[  z=\left\{ y:\phi(x,y),x\in a \right\} \]
becomes a set.

More simple, using a function $\phi(x)=y$ ,
it is possible for us to make a new set $(y\in) z$ by a given set $a$ whose elements are $x$.

A function (this is a set, too) for a given set can make a new set. it is very strong extension.

Of course, axiom of separation can be proved by axiom of replacement easily.

axiomatic sets 29 (restart)

For a long time, we discussed the number system or the structure of numbers.
It means that all numbers is sets.

As a function or a relation has already been defined by sets,
all statements in mathematics can be written by sets.

We shall return to the axiom of sets.

We have already addressed 6 axioms of sets.

1.empty set
There is a empty set.

2.extensionality
All elements is same if and only if the two sets are same.

3.pairing
There is a set such that if a element is in the set, the element is equal to one of two elements of the set.

4.union of sets
There is a set such that if a element is in the set, the element is in one of two other sets or in other both two sets.  

5.power sets
There is a set whose elements are all subsets of the set.
(subsets are made by the axiom of extensionality. )

6.separation
We are able to make a set whose elements are selected from a given set.
(for a selection, we need a formula or a statement of properties. )


There still remain several axioms in ZFC.

In next posts, we will discuss the axiom of replacement which is chosen behalf of the axiom of separation.

coffee break 15 (Happy new year)

Although this winter vacation was long, as I was very busy, I could not do anything.

Despite I bought many games, some of the packages are not open.
It is easy to buy games, but it is difficult to complete a game.

Hyper light drifter
final fantasy X/X-2
Dying light
Shu
Assassin's creed 3
Assassin's creed chronicles china
Dust force
Zup2

Now I am playing "salt and sanctuary" .
It is called 2D clone of "Dark souls" .

All enemies are very strong, even tiny enemies in fields kill my character easily.
The salts which my character has are deprived,
and by second death before getting back the salts, my salts (experiences) vanish.

As the sense of loss of the salts is much huge,
I thought I wanted to throw away the game many times.
(However, I am still playing the game. )

Actually it is the feelings same as "Dark souls" .

(Lost, lost, lost, .........)