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Topic: [Hyponumber] Building a complete system out of axioms
Replies: 2   Last Post: Jun 26, 2013 4:39 AM

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William Elliot

Posts: 2,637
Registered: 1/8/12
Re: [Hyponumber] Building a complete system out of axioms
Posted: Jun 25, 2013 9:34 PM
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On Tue, 25 Jun 2013, Béranger Seguin wrote:

> Those are the axioms :
> 0. We're talking of hyponumbers. We can name hyponumbers and sets with
> letters.
> 1. A set of hyponumbers is written {x,y,...} and contains hyponumbers, whose
> order matters, and who can occur several times.

That's not a set, it's a sequence.

> 2a. Any hyponumber is made of three hyponumbers.

Circular definition, end of discussion.

> 2b. The set of hyponumbers which "x" is made of can be written "[x]"
> 3a. With three not-all-equal hyponumbers, we can build an hyponumber.
> 3b. The hyponumber made from the hyponumbers contained in the set "x" of three not-all-equal hyponumbers can be written "]x[".
> 3c. ]{x,y,z}[ is called fusion of three hyponumbers.
> 3d. We admitt ]{x,x,x}[=x, but without reciprocity (except in some cases).
> 4a. There are three fundamental hyponumbers : A,B and C.
> 4b. Fundamental hyponumbers are only made of themselves. (for example : [A]={A,A,A})
> 4c. They do not obey to rule 3d because there is reciprocity : [A]={A,A,A} ; ]{A,A,A}[=A (the same applies for B and C).
> 4d. All hyponumbers can be made (with hyponumbers made of hyponumbers made of ... and so on) only with the fundamental hyponumbers.
> ( I have written two axioms 5 and 6 who are a little useless, so those places are currently empty ;) )
> 7. We can mention the n-th element
> Theorems :
> a. Let x be a set of three not-all-equal hyponumbers : []x[]=x (axioms 2&3)
> b. Let x be a hyponumber : ][x][=x (axioms 2&3)

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