I’ve emailed so many damn people, it’s not possible to steal this from me anymore…
If the others have received prior emails… They already have thus information.
My technique for sequencing the rationals… I call it the mirroring technique, because i realized that if you mirror all of the natural numbers you have all of the decimals. I discovered this technique, and use it for my sequencing of the reals!
The way it works is that the first ten numbers are counted just as themselves and their negatives:
0,1, -1,2, -2,3, -3,4, -4,5, -5,6, -6,7, -7,8, -8,9, -9
Then after that you count 10 and then the next number is the mirror of 10, which is 01 and then you move the decimal point in once to get the 12th number being 0.1, then the thirteenth number (not counting the negatives which are numbered every other) is 0.(1 repeating). These steps continue until you reach three digit numbers and higher. Once you count 100, you then count the next number as 0.01, then 0.0(1 repeating) then 0.(01 repeating), then you count 101 and it’s mirror. If you keep marching in the decimal point when the number that’s about to be mirrored ends in zero it causes infinite overlap. The number 100 ends in a zero, so after you mirror it you only march the decimal in for one place to the right, if you march it two places to the right, you end up with 00.1, which is the same mirror that you get when the number 10 is mirrored, and will occur an infinite number of times as the zeros expand and you keep marching in the decimal point (which will give you infinite overlap as the sequence expands).
However, if the number doesn’t end in a zero, you keep marching in the decimal point, say the number 102. The next number is mirroring it, so it’s 2.01, then you do the repeating decimals by next counting 2.0(1 repeating) and then 2.(01 repeating), then you march the decimal point in once more to get 20.1, and then you do the repeating decimals by having 20.(1 repeating). (If you march the decimal in one more time, you get 201, and have infinite overlap as well.) Then you count the number 103 and then mirror it and do this over and over again.
I fixed the sequence to be correct.
Formatting this on a phone didn’t work well, but it’s all there…
4 steps of the binary tree sequential algorithm …
Step 1:
1.) 0
2.) 1
Step 2:
1.) 00
2.) 10
3.) 01
4.) 11
Step 3:
1.) 000
2.) 100
3.) 010
4.) 110
5.) 001
6.) 101
7.) 011
8.) 111
Step 4:
1.) 0000
2.) 1000
3.) 0100
4.) 1100
5.) 0010
6.) 1010
7.) 0110
8.) 1110
9.) 0001
10.) 1001
11.) 0101
12.) 1101
13.) 0011
14.) 1011
15.) 0111
16.) 1111
Now technically you need to insert the rationals as well.
Step 1:
1.) 0 (rational)
2.) -0 (rational)
3.) 1 (rational)
4.) -1 (rational)
3.) 0 (binary tree sequence)
4.) -0 (binary tree sequence)
5.) 1 (binary tree sequence)
6.) -1 (binary tree sequence)
Step 2:
1.) 0 (rational)
2.) -0 (rational)
3.) 1 (rational)
4.) -1 (rational)
3.) 00 (binary tree sequence)
4.) -00 (binary tree sequence)
5.) 10 ( binary tree sequence)
6.) -10 (binary tree sequence)
7.) 01 (binary tree sequence)
8.) - 01 (binary tree sequence)
9.) 11 (binary tree sequence)
10.) -11 (binary tree sequence)
11.) 2 (rational sequence)
12.) -2 (rational sequence)
Then you do step 3 of the binary tree… Slowly adding the rational numbers…And continue the list forever…
To actually sequence all of the reals, I use an additional technique, which is to slowly create a new list (second dimension) to march the decimal point to the right for EACH expansion if the binary tree…
So in the more complete list I gave above, it starts at 3.)
You then count a new list called
3.1) 0.000
3.2) 00.00
3.3) 000.0
This will list every single real number in one to one correspondence with finite (countable) enumeration of numbers.
I’m reasonably sure it can all be done on a single list so the multiple dimensions don’t have to be used, but I’m still working it out. Either way, I used a +1 inferential algorithm to prove all the reals are there (and counted), technically it’s in three dimensions even though it looks like 2 because of the new lists on the horizon plane.
I discovered a technique to sequence the rational numbers that I call the mirroring technique. It works wonderfully for the binary tree expansion because it also has a reset point, which is every new whole number.
Please contact me about this, considering that I realize rescuing this problem resolved many incompleteness “theorems / proofs” I think I can solve completeness now, and already have notes to this regard.
I have issues with major depression and psychosis at times, and would really love a research grant or some type of living trust to continue my work… Who doesn’t want this?? But I just sequences all the real numbers!!! Something every mathematician has been trying to do since numbers were first understood. It’s a huge achievement, especially considering paradoxes run on this incompleteness that I just completed!!!
If you have any questions about technique or proof, please contact me.