June 20, 2019
Debunking my own Stephen Hawking variety problem solution:
So Stephen Hawking wanted a way for the Universe to start where matter wasn’t uniform so it would form into a variety of shapes.
The Real numbers don’t do it.
My solution, or so I thought, was the defined class of numbers. Take ‘0’,’1′,’2′,…,’9′,’.’,’+’,’-‘,’*’,’/’,’^’,’root’, ‘ln’, ‘cos’, ‘sin’, sum, sequence, product,'(‘,’)’, and maybe a delta function. Some variety is possible in the definition but the point is the same. By going through the digits like a number base, you get all the defined numbers. We want to divide all the numbers between concrete, defined numbers and random numbers that spread themselves everywhere and cannot even be defined. We don’t need to know if numbers are infinite or if we are repeating numbers and should be able to tell if the expressions are legitimate.
Again, by going through the digits like a number base, you get all the defined numbers. Do they form a variety? You might think so. Consider just these three points.
0 ln (2) 1
………………………..
It seems that irrational numbers like logarithms will be in different arrangements around different rational numbers, providing a variety.
Finite decimals exist. Repeating decimals exist, and for things like logarithms there are infinite endings for which the endings stick even if a repeating decimal part modifies them systematically.
It seems that these infinite endings will provide the variety trick.
Let’s say we had defined just 0, ln(2) and 1. We can include the extension of ln (2) to 0 and 1 to twice as far. We get -ln(2) and 2-ln(2).
Each time we introduce a logarithm, in the process of counting, we add and subtract its value to all the other non-logarithm points. We extend all line segments involving the logarithm to twice their length and add the endpoint. Then we combine each pair of two logarithms, each set of three, until we have used all the logarithms so far.
It takes almost forever to count but not forever and the point is unmistakeable. Every rational point has all the other rational points and the same logarithm parts on either side of it and extending as long as wanted. Every logarithm point also has all the the other rational points and the same logarithm parts on either side of it and extending as long as wanted. Perfectly balanced Universe.
Logarithms should be the only thing that provides trouble but we can do this with other things.
So we conclude two things:
(1) The defined class of numbers does not provide the variety Stephen Hawking wanted in his initial Universe and
(2) I should give you a discount on my books. However at least I suggested different probabilities based on the number of symbols to a number at the end of my book.