The power of finite and the weakness of infinite binary point numbers
Take a sequence of the numbers 0 and 1. You can interpret this sequence as a real number between zero and one.
Just write 0.( your sequence).
This can be interpreted as a binary number.
For instance 0.1101 is .
Now, I have read here about the might of irrational numbers, whose sequences go on and on, never ending, containing all the knowledge in the world.
I see two difficulties with that form of thinking.
It is not at all difficult to construct an irrational number, that contains nearly nothing at all. Take 0.101001000100001000001...… as an example. I seriously doubt it contains even one little story, let alone all books in the world.
To contain all books in the world plus all things in the internet plus some imaginary books as well a finite sequence is more than enough. Taking a few trillions of bits or more will do the trick.
Now look at the set of all real numbers between zero and one, that can be written as a finite sequence in this way: These numbers are all rational numbers regardless of the length of the sequence. Take as long a sequence as you need and then a few trillions more. They will remain rational numbers.
To our mind these two sets—the real numbers between zero and one and the rational numbers, that can be written as finite sequences, look suspiciously alike, since you can choose any length you like for the second set.
Yet they are mathematically completely different.
The second set contains not even all rational numbers. For instance is not in this set.
Now, Solomonoffs induction says:
“More precisely, suppose that a particular infinite input string x0 is about to be fed into U.
However, you know nothing about x0 other than that each term of the string is either 0 or 1.”
That is, where Solomonoff induction looses me. The sentence ” take a infinite input string and do something with it” is where my imagination simply stops.
For me it is practically identical to : Go to the axiom of choice and ask for a real number between zero and one with certain characteristics.
All irrational numbers between zero and one are in the set. Now, the set of irrational numbers is uncountable. There are so many that they dwarf the rational numbers, the constructable numbers, the computable numbers to practically nothing.
Therefore, you would get with what my intuition would describe as a “probability of one” an irrational number that is neither rational or computable or has any description at all.
In other words you would get a number that makes no sense.
And then, of course, all bets would be of.
Another way of thinking about that is the following: All information a human, an ant, humanity as a whole, every Turing machine, every neural network has will be always finite.
To use the mathematical powers to think of some sort of “infinite information” is not something I would recommend.
That we can think about the set of real numbers or an infinite universe or other infinite things does not at all goes against this principle. The information we have about all these things is still finite.
This feels a bit loose on the definitions. I agree that finitism seems intuitively reasonable, and that the finite amount of information we can have about the terrain of the world out there around us is a reason that, as far as I can currently see ahead, it ought to never be possible to conclude the greater multiverse is able to contain infinities.
But I also don’t see any way to strongly conclude the multiverse must not contain infinite-sized objects. I know folks who insist that if that’s possible, there should be exactly one, infinite, agent in the multiverse.
In any case, I agree with philip_b that this post is pretty hard to parse and harder to check for mathematical validity.
This only applies to some irrational numbers. (Though you might say it is an overwhelming majority of them.)
Sorry, found my mistake. For Solomonoffs induction the input string has to be computable. Therefore it is not from a uncountable set, only from an infinite one.
Confusion solved.
I am sorry if I sound a bit confused. I don’t speak English well and I am far more familiar with the mathematical terms in German. And I am confused by Solomoff induction.
My main point is the vast mathematical difference between “only” infinite (abzählbar unendlich) sets like the rational numbers or the computable numbers and the sets that are uncountable (überabzählbar) like the real numbers or the power set of the natural numbers.
If I have the set of real numbers between zero and one and “take” one number “randomly” I should “get” a non—computable number—in other words something that’s existence is more than doubtful. I certainly can not give a sample for that.
These numbers were not named “irrational” for nothing.
Perhaps this is only about the word particular in “particular infinite input string”. Something like “you can take it only if it exists”. But should it not be formulated like that? It feels to me like smuggling the whole concept of real numbers in by the backdoor.
What if you use instead the following sentence:
“More precisely, suppose that a particular input string x0 of unknown length is about to be fed into U.
However, you know nothing about x0 other than that each term of the string is either 0 or 1.”
Would that form into be a sort of weak Solomoff induction? Or would the thing than collapse into complete nonsense?
Please go, study math fundamentals properly, and then come back. What you wrote doesn’t make much sense.