A formal solution to Hilbert's 1st and 6th problems

  • Thread starter Doron Shadmi
  • Start date
ShadmiIn summary, the conversation discusses the concept of discreteness and continuum in mathematics, and how they are defined by their structural concept rather than their quantity concept. The speaker presents a theory of numbers that focuses on structures built from associations between opposite concepts. They also mention the distinction between sets and members of sets and discuss the meaning of "XOR ratio" in their language. The conversation ends with a link to more detailed information on their theory.
  • #1
Doron Shadmi
Dear researcher,




A formal solution to Hilbert's 1st and 6th problems
---------------------------------------------------

A and B are sets.

q and p are members.

Option 1: q and p are members of A, but then q is not equal to p .

Option 2: q is a member of A , p is a member of B .

D = Discreteness = q XOR p = a localized element = {.}

C = Continuum = q to p correspondence = a non-localized element = {.___.}

In the Common Math 0^0 is not well defined, because each member is D.

Let us say that power 0 is the simplest level of existence of some set's content.

Because there are no Ds in C, its base value = 0, but because it exists (unlike the emptiness), its cardinality = 0^0 = 1.

There are now 3 kinds of cardinality:

|{}| = 0 = the cardinality of the Empty set.

|{._.}| = 0^0 = 1 = the cardinality of C.

|{.}| = 1^0 = 1 = the cardinality of D.


Any point is a D element. Any line a C element.

It means that there is a XOR ratio between LINES to POINTS.


XOR ratio between LINES to POINTS
---------------------------------

0(LINE) 0(POINT) -> 0-(No information) -> no conclusion.

0(LINE) 1(POINT) -> 1-(Clear Particle-like information) -> conclusions on points.

1(LINE) 0(POINT) -> 1-(Clear Wave-like information) -> conclusions on lines.

1(LINE) 1(POINT) -> 0-(No clear information) -> no conclusion.



Some explanation:
-------------------
D = Discreteness = q XOR p = a localized element = {.}

C = Continuum = q to p correspondence = a non-localized element = {._.}

By the above definitions, for the first time in modern mathematics, there is clear and sharp distinction between the Continuum and the Discreteness concepts, not by their Quantity Concept, but by their Structural concept.

By real analysis the Continuum is "infinitely many elements with no gaps between them".

By defining the correspondence itself as a legitimate member, I redefine the original lexicographical meaning, back to the Continuum concept, and change the perception of Continuum and Discreteness concepts in Modern Mathematics.


Then, in the detailed menuscript, I clearly show that the Structure concept has more interesting information than the Quntity concept in Mathematics, in general.


For more detailed information, please see:

http://www.geocities.com/complementarytheory/CATpage.html

I know that it is hard to understand, because I have changed the most abvious paradigm, which says that Math is first of all, to deal with Quantities.

By my new theory of numbers, that follows this opening on Hibert's 1st and 6th problems, I clearly show that Math first of all is, to deal with Structures that are built from associations between oppiste conceptsts.




Sincerely yours,

Doron Shadmi
 
Last edited by a moderator:
Physics news on Phys.org
  • #2
"q = a , but if q = {} then q = |a| ."

What does this mean? You said that a was a member of set A. Are you assuming that A is a set of sets? What is |a|?

You seem to be seriously confused about the distinction between sets and members of sets.
As usual you are terms in non-standard ways without defining them. I am still wondering what an "XOR ratio" is.
 
  • #3
HallsofIvy

Thank you for your reply.

What does this mean? You said that a was a member of set A. Are you assuming that A is a set of sets? What is |a|?
I am not assuming that A OR B are sets of sets. all what I do is not to conclode {} as a member, but its cardinality, which is 0.

Capital A or capital B are sets.

The lowercase letters: a,b,q and p are members.

|a| = the cardinal of a

XOR is an exclusive OR.
 
Last edited by a moderator:
  • #4
If the members of A are not sets then what do you mean by "q = a , but if q = {} then q = |a|".

I KNOW what XOR means. I even know what a ratio is! I do not know what "XOR ratio" means.
 
  • #5
Dear HallsofIvy,

Again, thank you for your reply.

A or B can be any kind of set, which means 3 options:

1) Set of sets.
2) Set of members that are not sets + members that are sets.
3) Set of members that are not sets.

Code:
{0,1,2,3,...}={[B]{ }[/B],[B]{[/B]{ }[B]}[/B],[B]{[/B]{ },{{ }}[B]}[/B],[B]{[/B]{ },{{ }},{{ },{{ }}}[B]}[/B],[B]{[/B]...  
               |0| |-1-| |----2----| |----------3----------| |--4 
                |    ^        ^                 ^               ^    
                |____|        |                 |               |
                  |           |                 |               |
                  |___________|                 |               |
                        |                       |               |
                        |_______________________|               |
                                    |                           |  
                                    |___________________________|

From the above example you can learn that any number (but not 0) has an internal structure that is built by Empty set recursion.

Through this point of view, I can check 1 to 1 correspondance between members that are sets.

More than that: 0.101101... and |{}|.|{{}}| |{}| |{{}}| |{{}}| |{}| |{{}}| ... , are the same.


Please give the same meaning to "XOR" and to "XOR ratio".

In my language, Hebrew, we add the word "ratio"(in Hebrew it is "YACHAS") before the logical condition. So, maybe I wrongly translated it to English. Please tell me if I can omit it.

Another important thing is that I have added some explanation to my 1st message in this thread, and I hope it will help you to understand the meaning of my work.

Yours,

Doron
 
Last edited by a moderator:

What are Hilbert's 1st and 6th problems?

Hilbert's 1st problem asks for a method to determine whether a given polynomial equation with integer coefficients has a solution in integers. Hilbert's 6th problem seeks a proof for the transcendence of certain numbers, including pi and e.

Why are Hilbert's 1st and 6th problems important in mathematics?

Hilbert's 1st and 6th problems are important because they address fundamental questions in number theory and algebra, and their solutions have implications for other areas of mathematics.

What is the significance of a formal solution to Hilbert's 1st and 6th problems?

A formal solution to Hilbert's 1st and 6th problems would provide a complete and rigorous proof for the existence of a method to solve the problems, thus settling two long-standing questions in mathematics.

What is the current status of a formal solution to Hilbert's 1st and 6th problems?

As of now, there is no known formal solution to Hilbert's 1st and 6th problems. Mathematicians continue to work on these problems, and progress has been made in various related areas, but a complete and formal solution has not yet been achieved.

What are some possible implications of a formal solution to Hilbert's 1st and 6th problems?

A formal solution to Hilbert's 1st and 6th problems could have far-reaching implications for number theory, algebra, and other branches of mathematics. It could also potentially lead to new insights and techniques in solving other unsolved problems in mathematics.

Similar threads

Replies
11
Views
258
  • Set Theory, Logic, Probability, Statistics
Replies
4
Views
980
  • Advanced Physics Homework Help
Replies
1
Views
722
  • Other Physics Topics
Replies
4
Views
2K
  • Introductory Physics Homework Help
Replies
1
Views
780
Replies
7
Views
2K
  • Classical Physics
Replies
4
Views
675
  • Special and General Relativity
Replies
12
Views
734
  • Differential Equations
Replies
2
Views
1K
Back
Top