Today on April Fool's Day (by chance) I invented the "God Formula". It is a means of compressing data so revolutionary that it can be used as a basis for the world's most powerful AI (if Marcus Hutter is to be believed).
It sounds like I am kidding but I am not. Only time will tell if I am deluded or if I am onto something big. We shall see..
Great idea! I really hope you are still doing this when my product becomes available. I don't want to waste my time on day to day selling. Your service would be very valuable to me and I like the simplicity. I hope others here will appreciate what you provide. Keep at it and good luck. Talk in (about) a few months when I have a finished product to sell!
I define an infinite number of new sequences twice over in the example above. Between "---universe begin---" and "---universe end---" is listed the state of the universe. After inspecting the current state of the universe one can add new sequences with numeric codes as I have attempted to describe. The example above shows two iterations of this process. In each case an infinite number of sequences is added. So really your program will describe an infinite sequence of infinite sequences. And, if you never stop programming, then you are actively directing an infinite sequence of infinite sequences of infinite sequences!
The purpose of the language is to be "general purpose". Any programming problem can be modeled with this language. Because a number represents anything atall, I would not hesitate to use the language anywhere I have need for an algorithm. Input transformed to number is manipulated by algorithm then presented to the user as output. A gui is typically used to control this process.
Impressively, it is possible to write relations with very little knowledge of specific algorithms. For instance, you may know how to square a number but be totally ignorant of methods to find the inverse function (the square root). Not a problem - you already know enough to direct the computer to take square roots. This might seem magical. It certainly is magnificent, but there is no magic involved. It all has to do with the design of the language.
My favorite programming language is Zero (yes it is my invention). If you have not heard of it then I am not surprised because the language is very new (theoretical actually). You develop programs in it by defining new infinite sequences from existing ones.
All begins with the primitive infinite sequences and the language's means of combination (specified with decimal digits)
---universe begin---
sequence 01: 0 1 2 3 4 5 ...
sequence 03: 1 2 3 4 5 6 ...
sequence 05: 2 3 4 5 6 7 ...
sequence 07: 3 4 5 6 7 8 ...
...
---universe end---
0105: count by two
---universe begin---
sequence 01: 0 1 2 3 4 5 ...
sequence 02: 0 2 4 6 8 10...
sequence 03: 1 2 3 4 5 6 ...
sequence 05: 2 3 4 5 6 7 ...
sequence 06: 1 3 5 7 9 11 ...
sequence 07: 3 4 5 6 7 8 ...
...
---universe end---
01030501: set up some cycles
---universe begin---
sequence 01: 0 1 2 3 4 5 ...
sequence 02: 0 2 4 6 8 10...
sequence 03: 1 2 3 4 5 6 ...
sequence 04: 0 1 2 0 1 2 ...
sequence 05: 2 3 4 5 6 7 ...
sequence 06: 1 3 5 7 9 11 ...
sequence 07: 3 4 5 6 7 8 ...
...
---universe end---
Please note that as the universe grows we always leave space for more sequences by skipping every other sequence designation.
What makes this language useful is that with very little effort any two arbitrary infinite sequences can be defined. As a consequence any mathematical relation is easily defined as a mapping from members of one sequence to the corresponding members of the other sequence.
I will briefly describe the language's means of combination. Writing a sequence designation one next to another will form new sequences by pulling out corresponding members. But all sequences are infinite so after the last designated sequence is visited the member value is used to select the possibly new member in the first designated sequence. Digits that are not able to be confused with sequence designations specify the three other primary means of combination. They are "cons", "car", "cdr". With a proper understanding of their use one can build arbitrary sequences. I will just say that "cons", "car", and "cdr" are used to combine entire sequences which is logically equivalent to combining corresponding members of those sequences.
In the code above I write "0105". The "01" is a sequence designation. The "05" is a sequence designation.
Why is it that Cantor can do an infinite procedure of diagonalization but I can't? If he can diagonalize then I can too. Is it possible that Cantor's "algorithm" is not really an algorithm? Knuth says an algorithm must be correct and must also terminate.
I will just diagonalize forever before I give the set to Cantor. If he claims that he can diagonalize the set to generate a new number not in the set then I will say "Oh, yeah - I missed one. I wasn't really done after all. Give it back to me and I will add some more to it before you get your hands on it!". As you can see, Cantor will never get a list because I will never really be done generating it.
I could just give him the list and say that whatever he adds is precisely what I would have added had I continued. This is like an infinite lazy list in programming.
>But the set N does not have transfinite natural numbers, so q does not map F to N, it maps F to something else.
How many natural numbers are there? How many bits does it take to represent the average natural number? If you believe the natural numbers do not include transfinite numbers then how do you pick a successor when counting? There are infinite picks to be made so some of the picks must be transfinite. What I am calling a transfinite natural number must exist in N because N is an infinite set.
Assume that N has only finite numbers in it but is itself an infinite set. Would you care to tell me which number (or numbers) are listed twice? But then it is not really a set!
>I's much easier to consider the infinite strings of digits like "0.765653625367523765..." or "0.5265362556..." or "0.000073468763478..." and also the one with repetitions like "0.0006767000000..." or "0.0072257822222222...". This is essentially a copy of the real number, but in this copy "0.2999999999999..." is different from "0.300000000000000..."
This trick makes much easier to prove that the diagonal ´+1 in each one digit is not in the list. Then it's possible to fix the details and use the real numbers instead of the infinite strings of digits.
What am I missing?
"0.765653625367523765..." could be assigned the transfinite natural number beginning "1765653625367523765..."
"0.000073468763478..." could be assigned the transfinite natural number beginning "1000073468763478..."
Transfinite natural numbers must exist otherwise you do not have an infinite set.
It is possible. The inverse of m is called q. The function q takes an infinite sequence of coin flips and one by one changes every heads to 1 and every tails to 0. The infinite string of zeros and ones is then prepended with a 1 and interpreted as a transfinite natural number in binary notation. This will not take forever because each change will only take half as long as the previous one. A transfinite natural number can exist since there are an infinite number of natural numbers. If we stop at 1-bit numbers then the natural numbers are not an infinite set. Likewise, if we stop at 2-bit numbers then the natural numbers are not an infinite set. Our only option is to concede that transfinite natural numbers do actually exist and that they can be put into correspondence with all sequences of coin flips.
Hundreds of thousands of mathematicians are wrong.
The claim is that there are more real numbers in the range from zero to one than there are natural numbers. To see that this is false simply realize that you don't actually have to write a decimal point to specify the real numbers in this range. Without the decimal point these real numbers just become natural numbers. Can a rational person believe that there are infinite sequences of digits in the form of real numbers but not infinite sequences of digits in the form of natural numbers? The natural numbers are just an infinite sequence of finite numbers. If you believe n is a natural number then you must also believe that n*10 is a natural number. One more digit! There is always one more digit (that is what infinity implies). If there really are an infinite number of natural numbers then some of them must be of a transfinite number of digits or else you would be including numbers in the list more than once.
The problem with Cantor's argument comes down to the fact that the procedure he uses to find a number not in the set is essentially the same as the procedure he uses for creating the infinite set in the first place. The only difference is our understanding of randomness. His procedure for finding a number not in the set may not seem very random but it might be as random as any other. A truly random coin could theoretically come up heads every time. The important part of his argument is that the infinite list of real numbers has no repeats. The diagonalization procedure similarly ensures that there are no repeats. On the one hand he claims the infinite set of real numbers exists. On the other hand he argues that the diagonalization that yields a number not in the set has not already been done. He takes away infinity and then gives it back!
There is only one infinity. It means "repeat". It is simply the interplay of finite state with process. You can think of it as an "infinite loop" in programming. To say that one infinity is smaller than another is to deny that the smaller is infinite. Infinite means without bound.
It is not true that all computers are digital. A computer made out of Tinkertoys or DNA is no less of a computer although the rules of operation may be different.
You argue that an "organism" is not a computer (as it does not compute). But why should I believe that a computer is not necessarily an "organism" in your sense of the word? Or would you have us believe that organisms and computers are not made from the same universal stuff?
It sounds like I am kidding but I am not. Only time will tell if I am deluded or if I am onto something big. We shall see..