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

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andAare sets.B

andqare members.p

Option 1:andqare members ofp, but thenAis not equal toq.p

Option 2:is a member ofq,Ais a member ofp.B

= Discreteness =DXORq= a localized element = {.}p

= Continuum =Ctoqcorrespondence = a non-localized element = {.___.}p

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 nos inD, its base value = 0, but because it exists (unlike the emptiness), its cardinality = 0^0 = 1.C

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 aelement. Any line aDelement.C

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

XOR ratio between LINES to POINTS

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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:

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= Discreteness =DXORq= a localized element = {.}p

= Continuum =Ctoqcorrespondence = a non-localized element = {._.}p

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 definingthe correspondence itselfas 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 consept 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

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

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