zelah

It's a type of puzzle game - easy to learn but impossible to master - you can play solo or against an opponent.

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!

Thank you for asking.

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 at all, 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.

good link - thanks

Thanks - that was very fun!

Thank you - I have read this and do realize I am out of my depth.

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.

Regardless, I have a lot to ponder.

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.

So there are natural numbers with an infinite number of digits and that is my point. The average can't be larger than the largest number.

Please answer me this one question: What is the average number of bits that are necessary to represent an arbitrary natural number?

If the average number of bits is finite then I will shut up!