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PI number contains an infinite number of 123 digit sequence, also it contains an infinite number of 12345678 digit sequence ans so on. One can take such sequence of 10^10000000 digits and the PI will contain an infinite number of this sequence too.

OK, we don't know a disjunctive sequence PI is or not, it doesn't matter. The most important thing that PI data are not repeated, so they're some sort of randomness.

Let’s name some sequence of 1's and 0's as structure. By mutating digits in a structure, one can produce an infinite number of variations of the structure and all of them will can be in the PI in the same way as the original structure. Moreover, some structure may contain substructures interfering with each other by some laws. Next, if some substructure has a memory reflecting some sequences of mutations and if it’s smart enough to understand the laws, it can predict the future with some probability and look into the past with some depth. The only thing it cannot do is to dive into its structure deeper than somewhat level and to overcome the bounds of the consisting structure because it will see nothing the truly randomness as an unprincipled sequence. As far as the memory of a structure has limits, it reflects the world states like a hash function allowing to interpret multiple mutated states of a structure as one state. The time in such world is connected with consisting laws and memory of smart structures. Without the laws the structure can’t imagine anything because these laws form this structure per se. As far as the laws are too complex to understand and they can go wrong with progressing of observer, the “mutatis mutandis” principle can be used to fit the mind into the just expanded universe. The worlds could be of many kinds. For example a world could be a structure consisting of all states of all substructures or a structure consisting of some state of all substructures. The last one obligates the substructures to move through time by imagining the next snapshot of the world’s structure and – thanks only to this – by moving automatically into an alternative substructure close enough to be the subsequent history of the given substructure. These transitions remain in the memory of the substructures and form the life. So, PI number contains an infinite number of such worlds since PI number contains an infinite number of random bits. So, the PI can contain such sequences and since it has infinite amount of data, their variations (mutations) could form our world.

As far as my question gathers many dislikes, I want to be clear:

  1. My goal is not to discuss a numeric property of PI itself because of Duhem–Quine thesis: if you don't convinced about a structure 01011110011111...111011 we could use 01011110011111...111101
  2. I understand that any kind of thinking of "sequence" involves many troubles because of any form of a subjective prospective. I intend to talk about "consisting", not "ordering": a sequence consists of memory & mind: our bodies consist of DNA sequences reflecting previous states (lossy encoding), a finite number of neurons (lossy encoding too). We can't track all changes or possibilities of states because of these limits: so, many worlds could be mapped into our consciousness as the same image.

Also, please remember that this is a question about possibility. If you don't like it, please simply ignore it. If you consider this question bad, let's close it.

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    Containing a numerical description of a world does not mean containing a world. the Lord of the Rings contains a description of Rivendell, it does not contain Rivendell. – armand Dec 25 '20 at 9:51
  • There's no numeric description by somebody in my example. I mean a sequence with non-zero probability with memory and ability to analyze previous states EXISTS in the PI data. It just EXISTS because it can be. – Dmitry Ovchinnikov Dec 25 '20 at 10:13
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    What would it mean for digits of pi to "analyze" other digits? – causative Dec 25 '20 at 10:17
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    I remain thoroughly unconvinced. – armand Dec 25 '20 at 11:20
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    The real question is whether a world can be "inside" a sequence of digits. That makes Pi a distraction - it's irrelevant because all sequences of digits occur in Pi eventually. – curiousdannii Dec 25 '20 at 14:00
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EDIT: I defined a particular property of number below and, not originally knowing the name of this property, decided to call it a "Hamlets number in base b." But @user4894 came to the rescue -- it's called a "disjunctive number in base b." So, find-replace "Hamlets" with "disjunctive."

First off, the property you're referring to is called normality, and it hasn't been proven that pi has it. A number is normal in base 10 if every possible sequence of n digits from 0-9 appears in its decimal expansion, and with the same density to boot (i.e. there aren't especially popular or unpopular sequences; you can find the technical definition on Wikipedia). While it's conjectured that pi is normal in base 10, and there is some evidence in that direction, it remains unproven. Even the weaker property that pi contains every sequence of n digits at least once (forget about the relative frequency of their occurrences) hasn't been proven; I don't think this property has a name so I'll call it a Hamlets number.

But forget about pi, because it's been proven that most irrational numbers are normal numbers (technically, the set of non-normal numbers is measure 0). So, most irrational numbers contain every finite subsequence in their decimal expansions (and in relatively equal proportions). While I can't write out a particular example, here's a simple example of a Hamlets number that I'll call H:

0.0123456789
  00010203040506070809
  10111213141516171819
  20212223242526272829
  ...
  000001002003004005006007008009
  010011012013014015016017018019
  ...
  100101102103104105106107108109
  110111112113114115116117118119
  ...

This number is just an ordered sequence of all possible sequences of digits 0-9; it's very nearly just a list of all the natural numbers from 0 on up stuck together, except I had to stick in sequences with initial zeros. If you trace out the pattern, you'll see that eventually every finite sequence of digits will appear in this number.

But does that actually mean anything? For any possible question you state about the world that has an answer that's some sequence of digits, that answer can be found somewhere in H's decimal expansion -- but that's vacuous. As H is essentially just the set of all such sequences squashed together, it's completely equivalent to saying that "for any possible question you state about the world that has an answer that's some sequence of digits, that answer can be found in the set of all sequences of digits." The root of is that, given a particular question, or property of the world, there's no telling where in H the corresponding encoded sequence lies; it just is where it is. If pi is a Hamlets number, it would be much the same. Sure, you can find the number you're looking for somewhere, but since you don't know where to look you're not going to find it. It's just the same as the parable where if you had an infinite number of monkeys typing away randomly one of them will have produced the text of Hamlet, but you don't know which one and for every real copy of Hamlet there are infinities of garbled copies, copies with changed plot details, misspellings, completely different books, and of course, for the vast majority of them, complete gibberish.

My aim here wasn't to address the questions you posed about substructure in pi's digits "interacting" and having "memory". I hope I've just made it clear that there nothing mystical or special about pi, and that, if it is normal/Hamlets, its just one of infinitudes of such numbers, whose decimal expansions are actually, due being completely random, just boring noise.

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  • I didn't want to put an emphasis on numeric properties of the PI. I've used PI as an example because it's an irrational number, so its digits don't form any repeated sequence. My question was about AN ABILITY of any infinite sequence of non-repeating digits to include a sequences big enough to represent laws that form an universe. There is a non-zero probability that an irrational number's digits contain all bits of the Mona Lisa portrait but I tried to show that a dynamic world with laws & life could exist in the same way because it also could have a non-zero probability. – Dmitry Ovchinnikov Dec 25 '20 at 20:28
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    The property of containing every finite-length sequence of digits is called disjunctivity. Normality is containing each such sequence in the same proportion, in the limit. The idea that normality is disjunctivity might be the single most common mathematical misconception on the Internet. en.wikipedia.org/wiki/Disjunctive_sequence – user4894 Dec 25 '20 at 20:37
  • Let's take the following numbers: 100000 010000 001000 000100 000010 000001 It's a movement. But this movement could be observed by an external observer only. I meant such sequence with some kind of memory about states. If such sequence also has an ability to reflect upon itself, it's unnecessary to have such sequences connected: the next mutation moves the memory and consciousness into another pre-existing subsequence with the equal structure. – Dmitry Ovchinnikov Dec 25 '20 at 21:15
  • @user4894 Ah, thanks! I figured there was a name for this property. So, what I decided to call a "Hamlets number" in base b is better termed a disjunctive number in base b. I'll add a clarification in my answer. – jwimberley Dec 25 '20 at 21:56
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    @DmitryOvchinnikov You seem to have the misconception that irrational numbers 'unfold' somehow digit by digit, 'remembering' earlier sequences and 'producing' new ones. That's just not the case. The development digit-by-digit is only existent in our production of a representation of that magnitude, ie. due to our way of mathematically processing irrational numbers. It is not inherent to the number itself, which just is what it is. – Philip Klöcking Dec 26 '20 at 7:19
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The OP has clarified in a comment that he is not particularly interested in pi, but rather in any number that happens to contain every possible finite sequence of digits.

In this case the answer to the question is yes. There is a real number that encodes a description of the entire state of the universe, past, present, and future. Not only that; we can find such a number that is computable, meaning that there is a finite-length computer program that generates as many digits of the number as we like.

But wait, there's more! I will actually exhibit an explicit example of such a number in this post, a particularly simple one in fact. You have to promise not to tell anyone, such knowledge could be dangerous. You could in fact predict the future with perfect certainty, depending on some assumptions.

The hard part is the physical assumptions. We will assume first that the universe is finite. We do know that the known universe contains only a finite number of atoms, around 10^78. As integers go, that's pretty small. It's a 1 followed by 78 zeros. It's not even a Googol, the number after which the company Google is named. It's smaller than Skewes's number, Graham's number, Tree(3), the Busy Beaver numbers, and a lot of other large integers that mathematicians and computer scientists care about. There really aren't very many atoms in the known universe.

We further have to assume that each atom has a specific position, velocity, energy, and momentum. We know from modern physics that there is uncertainty in these measurements, and I am going to gloss over these details. I think it's fair to say that if the position, momentum, etc. of each particle can each be constrained within a known error range, we will still have a finite amount of information that describes the state of the universe.

We can encode all this data in a finite string of digits representing a large integer or a real number between 0 and 1 if we put an implied decimal point in front of it. It makes no difference either way.

So modulo physical details, we will assume that there is a finite sequence of digits that describes the state of the universe to a given degree of precision at a given time. Assuming that the lifetime of the universe is finite, we can divide it into seconds, or milliseconds, or microseconds, whatever you like, and thereby encode the entire state of the universe at each moment of time.

In the end we still end up with a long but finite sequence of digits representing the past, present, and future of the universe.

Now I will present a known infinite sequence (or real number, if you put a radix point in front of it) that is guaranteed to contain this number as a contiguous subsequence.

A simplification is to convert everything to binary. We have already convinced ourselves that we have some decimal number that represents the state of the universe at every interval of time from the Big Bang to the End. The assumption of finiteness is of course necessary in all of this, because if there are infinitely many intervals to keep track of we can't hope to encode it all in a finite string of digits.

So now we have the universe encoded into some binary string, or bitstring. Call it S, just to give it a name.

And here is a sequence (or real number, with a binary point in front) that is guaranteed to contain that string. It's called the Champernowne constant. The link is to the Wiki article on disjunctive sequences, which has a definition of this constant more suitable to our purposes than the Wiki article on the Champernowne constant itself.

First we write down all the bitstrings of length 1, of which there are 2:

0 1

Then we append all the bitstrings of length 2, of which there are 4:

0 1 00 01 10 11

Then we append all the bitstrings of length 3, of which there are 8:

0 1 00 01 10 11 000 001 010 011 100 101 110 111

and we keep on going just like that. The 16 bitstrings of length 4, the 32 bitstrings of length 5, and so forth. By construction, every possible finite-length bitstring appears in this sequence, including our universal state sequence S.

Not only that, but generating this sequence would be an elementary programming exercise in Python or Java or C++ or any popular programming language. The algorithm is simple and we can generate as many bits of this sequence as we like, subject only to constraints of time, space, and energy.

So that's it. There is a bitstring that is guaranteed to contain a finite-length encoding of a description of the entire state of the universe, moment-by moment, subject to measurement uncertainty; as long as the amount of data is finite. The sequence is computable and easy to visualize. It's not random at all; it's the output of a simple algorithm.

Of course if someone objects to my physics, that's not important. We can do the encoding differently. We can encode the story of your life, or the story of everyone's life, every book ever written, every math theorem that could ever be, and so forth. Any information that can be encoded as a finite bitstring will do for this example.

If you want tomorrow morning's lottery numbers, they're in there somewhere. The only problem is that we can't know where. All the losing lottery numbers are in there too.

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  • Thank you. This is precisely I've tried to ask. – Dmitry Ovchinnikov Dec 27 '20 at 8:32
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PI number contains an infinite number of 123 digit sequence, also it contains an infinite number of 12345678 digit sequence ans so on.

What leads you to believe that?

One can take such sequence of 10^10000000 digits and the PI will contain an infinite number of this sequence too.

As far as I know, it is unknown whether there exists in pi, a sequence of 12 9's.

I think you are assuming something about pi that may not be true.

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  • OP has already said in a comment he doesn't care about pi. If we run the same argument with some disjunctive real number, then it's true that we can encode the state of the entire universe in some finite substring of any disjunctive number. It's not a meaningful result in any way, it's just a mathematically true fact about such numbers with no physical significance. The real numbers are just a mathematical abstraction in any event. – user4894 Dec 27 '20 at 5:50

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