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Saturday, February 19, 2011

Foundation of the natural numbers

At last I found a more in depth explanation of the natural numbers. In this idea numbers are labels assigned to sets and adding 1 to a number represents a map between two sets. If we would analyze the mathematics of a hypothetical alien civilization we would see that they have natural numbers too, they just used different labels in their definition.

Mathematically it roughly works like the following:
Let A be the collection of all sets.

Let f be a map
 a: |-> a U {a}

Assign label '0' to Emptyset
If a has label 'x' then assign label 'x+1' to f[a]

Label a f[a]  a U {a} 
0 Empty 

1  f[Empty] Empty U {Empty}
    = {Empty}
    = {0}

2 {0} f[{0}]  {0} U {{0}}
    = {0, {0}}
    = {0, 1}

3 {0,1} f[{0,1}] {0,1} U {{0,1}}

More later. ( Confirmed as an article in the NT Wiki. )


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(Raumpatrouille – Die phantastischen Abenteuer des Raumschiffes Orion, colloquially aka Raumpatrouille Orion was the first German science fiction television series. Its seven episodes were broadcast by ARD beginning September 17, 1966. The series has since acquired cult status in Germany. Broadcast six years before Star Trek first aired in West Germany (in 1972), it became a huge success.)