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.
2017/01/27
2017/01/12
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" .
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, .........)