I am trying to convert some pretty dense academic literature into a way that anyone could understand. I'm stuck on this one way in which they describe how a message works. Or how a language is used. By the way the reason that this is important is because it relates to how we would perceivably interpret an extraterrestrial message.

They describe it in a very mathematical and likely confusing way for someone not familiar with the type of jargon they use. My goal is to be able to communicate this to the layperson or anyone. But it is important to keep intact the fact that there is a relationship between an ordered set and another set (which can either be ordered or unordered... But used in the context would be used in an unordered fashion, for example our alphabet is ordered but we don't use that order to write English). This relationship between these sets allows us to communicate information.

I need to communicate to whomever I'm teaching (assume that they are 16-year-olds on YouTube) this inherent way in which a message or a language works and emphasize it as it is the fundamental concept from which I build what I'm trying to really teach people.

The problem is I know that if I were to use the way that they describe it academically, no one would understand. So I need someone to help me figure out how to write this in a way that gets the same ideas and fundamental concepts across but is described in a digestible form.

I'm a bit stuck here.

My question:

How would you describe this?
Is the way the academic literature describes it confusing to you? Do you think the way that I described it which is towards the bottom makes sense in that you understand it?

Any other ways to describe this that you think are better are what I'm looking for...and general feedback. I

I asked this question in this section of stack exchange because it's about the core syntax of how language works and also because, well, you all are the english language communication experts as far as I can tell.

Academic Way Described 1:

"One of the sets should be preliminarily ordered uniquelyto provide numbered slots for the elements of the opposite setto be “written” against. Then, a mapping is projected such that, upon application to the ordered set, it produces a desiredmessage in the opposite set. (In formal language, one of thesets is first mapped bijectively to the set of natural numbersof the same cardinality, and a message in this case representsthe composition of two mappings.) Note that this method isgeneral and applies to any pair of sets of unordered elements,so long as at least for one of the sets there exists a bijectiveand unique mapping to the set of natural numbers."

Academic Way Described 2:

"I define a context free code for words to consist of an ordered set of symbols for every word, or some symbols and some words give insufficient information concerning the adjacent symbols to determine them uniquely out of the unordered set for the word. That is to say, the same symbol can be used in a variety of context of left and right adjacent symbols, and the ordering of the symbols and award carries information not found in the conjunction of the unordered set of symbols with the sequential dependency rules.

This general formulation of the relationship between certain ordered and unordered sets seems to be of some value by itself. However, the general formulation was designed primarily to apply to a particularly simple example of a context sensitive coding defined on a context free coding for the vocabulary of a real language like English, and it is this particular example which is of primary interest here."

Information Processing in The Nervous System: Proceedings of a Symposium held at the State University of New York at Buffalo 21st–24th October, 1968. K.N. Leibovic. Springer Science & Business Media, Jun 29, 2013.

My way of describing this:

A "message" is a mapping between 2 sets of things/symbols whereby one of the sets has to be ordered so that each thing or symbol from the other set can be placed in a position allowing it to have context and therefore can be used to communicate information. The other set can be ordered or unordered. It would more so be thought of as being unordered though.

In its simplest form, one of the sets is ordered by means of a number. As, in the usual context, a number is required for something to be in in "order." However you could order a set of symbols anyway you want as long as it remains consistent and then using the means by which you ordered the first set map a symbol from the 2nd unordered set.

Let's just think of the first set as being whole numbers starting at one though. Numbers provide an easy way to understand how one set is ordered.

So let's say you have a word. The word is "antidisestablishmentarianism." This has 28 letters.

The way I am communicating that word is because the first set which is a listing of numbers from left to right... (or actually it could be a right to left but that would make reading very confusing... although keep in mind that some languages right up to down or down to up or in a variety of different ordered ways).

In English, to construct a word specifically we order things from the left the right.


Set 1 is the ordered set (1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11...28). Other words are (1,2,3....number of letters in that word). () indicate ordered set while {} indicate unordered set.

Set 2 is all the symbols in the English alphabet which, although we order the alphabet in an alphabetical way, we are using it in an unordered way when we communicate with English.

Ordered set 1 = position Unordered set 2 = symbols available to put in that position

If set 2 were the ordered set, and set 1 was also still ordered...I'll explain what would happen down at the bottom of this post. I'll also explain the inverse of the normal way we do things.

Most importantly though, there must be a universally accepted way in how set 2 (letters) is mapped to set 1 (numbers in an order left to right). For our purposes, that understanding is ENGLISH. ENGLISH is like a code we use to communicate information in the form of a universally known mapping between ordered set 1 and our unordered set 2 (or our unordered use of "ordered set 2...the alphabet is making this confusing...I hate the alphabet actually as it clouds my explanation of this). For other languages, this mapping is different.

Here is an unordered set of our alphabet just to further demonstrate things.


Imagine it as a pool of symbols we can choose from as much as we want when making a word.

In the word "antidisestablishmentarianism," there are 28 uses of symbols from our alphabet.

Because I know English, I know which position each symbol from our alphabet is mapped to to communicate this word.

a n t i d i s e s t a b l i s h m e n t a r i a n i s m

because it is associated with the ordered set of natural numbers (this is just 1, 2, 3, 4, etc.) left to right like this:

1 = a

2 = n

3 = t

4 = i

5 = d

6 = i

7 = s

8 = e

9 = s

10 = t

11 = a

12 = b

13 = l

14 = i

15 = s

16 = h

17 = m

18 = e

19 = n

20 = t

21 = a

22 = r

23 = i

24 = a

25 = n

26 = i

27 = s

28 = m

You see?

What if both sets were unordered though? You could choose from a pool of unordered set 1 {1...9} and unordered set 2 {a..z}

How would we make a word?

Well, I guess I subjectively determine the position from set 1 or set 2?...but it's unordered so there's no position...really....Ok..I'll just randomly choose the position.

netpal alpetn ntpale planet nlpate anpetl

Hmm... There are 6! possible ways to write the word "planet." As each letter is only used once and only allowed to be used once. 6 * 5 * 4 * 3 * 2 * 1 = 720 ways.

Ok, great, we have 720 different ways to represent one one idea or concept. That's like having 720 different symbols that all represent one thing. The word "planet." Without explaining why Occam's razor applies here, let's go a little deeper.

In a way you are mapping and unordered set to an unordered set. A counterexample has been found, correct? Well, the notion of mapping your unordered set of the alphabet to an ordered set really just means that you are assigning a specific position to a symbol from the alphabet. You are assigning a specific "position" to a symbol from the alphabet here though. The position simply one-dimensional with a constraint in that it must take on one of 6 values.

It must be positioned in any of the combinations of letters that would represent the word planet. You would still have to assign a position to each word in your sentence unless the language allowed you to completely randomize that as well. And at that point if you are assigning positions to anything via an ordered set. Then why even use spaces between your words? At that point everything just becomes what would seem like a completely random string of letters.

Each one of these mappings to an ordered set from an unordered pool of symbols has different levels of hierarchy as well.

Also having so many different ways to represent one idea or concept is extremely inefficient. Ideally, you want a one word to represent one idea or concept. That way you maximize the amount of information that you can send in the most efficient manner.

It may be good idea to have a few different ways to represent the same idea or concept which perhaps slightly means something else but having every possible combination or permutation represent the exact same thing without a position ends up meaning nothing. I may as well just smash my head on my keyboard.

What about an ordered set with an ordered set? Numbers are ordered....the alphabet is ordered...

Now, back to how one set should be ordered and the other set unordered (unordered moreso meaning "a pool of symbols available for use."

If both sets were ordered to communicate with a language like this: (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26) (a,b,c,d,e,f,g,h,i,j ,k ,l ,m ,n ,o ,p ,q ,r ,s ,t ,u ,v ,w ,x ,y ,z )

They'd both have to denote position.

You'd have to go in order for both sets for each word...because they are both ordered. You also wouldn't know which set is the symbol and which denotes the position.

You'd be limited to words that look either like this: A sentence would look as such:

Abcdef abcdefgh abc abecdefgh abcdefg abcdefgh. or the inverse like this 123456 12345678 123 123456789 1234567 12345678.

You can't communicate much with that. The only information you have, really, is the length of each word. How would you describe or communicate..anything? Really, really long words?...what if you went past 9 digits...and you start having to use 2 and 3 digits for the numbered version? That'd be a pain. What if you went past the number of available letters in the alphabet? Use AA,AB,AC, etc.? At that point, you're simply communicating with the quantity of letters in the word and you're better off just use a quantity to represent a word.

You could use those quantities to represent each word in the language though.... 1 = kiss 107 = the 44 = chocolate 3 = bunny 17 please

What would a sentence look like?

1 107 44 3 17.

AGH! But then you're using the the ordering of those words in the sentence to express meaning. You're choosing to map this new number quantity representing a word (and a word is just a symbol of symbols) an unordered fashion (you took it from a set in an unordered way!) and put it in an ordered sequence! NOOOO!!!

Hmm.....is there ANY OTHER WAY TO DO THIS...TO GET AROUND HAVING TO USE AN ORDERED SET?....what if you used special patterns of your new language to represent things?

X = any "word" (represented by either it's quantity of letters like 11, abcdef, or 1234567891011 - you'd have issues due to the fact that there are only 10 different symbols representing digits btw) used here as bX where b = the quantity representation of the word (because I havn't defined a word for all quantites) if a bX is shown, it means any word. etc.

1X = Kiss 107X = the 44X = chocolate 3X = bunny 17X please bX = anything (b>10)X = any word represented by a quantity greater than 10. etc.

We could say, when 1X 1X occurs together it means "Smash" and 1X (b<20)X 1X all together means "Frogs" despite that it could mean "Kiss Bunny Kiss" or "Kiss Please Kiss", we're just mapping "Frogs" to that pattern instead.

Remember, we're trying to avoid having to use a strictly ordered set...which represents position.

When appearing in a sequence such as 107X 1X 17X 3X it means "I love hotdogs." We are mapping I love hot dogs to that sequence. However that sequence represents just multiple positions (An ordered set) of symbols. So you're doing the same thing again.

Being able to establish a position and then use a pool of letters or symbols from an unordered set (even if in other contexts that set is ordered...such as in the alphabet...invented by the devil) for each position allows the receiver of the message to understand which set implies the order/sequencing and which set implies the symbols available for use.

Without the prior culturally taught knowledge that English communicates via set 2 used in an UNORDERED fashion and mapped to set 1 (ie. the order of the alphabet means NOTHING when communicating), you wouldn't know which direction the mapping should go (is it from set 1 to set 2 OR set 2 to set 1?) to map each symbol to.

If you didn't know that you map letters to an ordered set of natural numbers then the letters in a word Would denote the ordering of the other set which would be numbers....would be numbers arranged via the order of the alphabet.

So, antidisestablishmentarianism.

Would be 114209491951920121291981351420118911491913. a = 1 n = 14 t = 20 etc.

We would probably need more symbols to represent numbers that are one digit besides 0 to 9 to avoid confusion.

That's why we must either assume or know which set represents the order and which set represents symbols mapped to each order...Having 1 ordered set and 1 unordered set clears this ambiguity and makes the notions of communication in this way work. In essence, the ordering of our alphabet only serves to confuse someone trying to understand it (assume they were like an alien). In our culture the ambiguity is only clear due to culturally defined context...."Hey bro, you spell things from left to write in the order of numbers using our letters."

From the above all communication is done via the mapping, even if the mapping becomes exotic, and not 1 to 1 (although this would begin to violate Occam's razor), of a symbol or other means such as speech to an ordered set which represents a dimension (either somewhere in space, or time).

closed as primarily opinion-based by MetaEd Aug 24 '17 at 22:00

Many good questions generate some degree of opinion based on expert experience, but answers to this question will tend to be almost entirely based on opinions, rather than facts, references, or specific expertise. If this question can be reworded to fit the rules in the help center, please edit the question.

  • 1
    Have you seen the Stackexchange Writers beta aimed at technical writers? – Weather Vane Aug 20 '17 at 19:21
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    Yes, but technical writing is aimed at the appropriate target. I didn't read all your post. It is too long; didn't read. No matter the target, you have to aim within their attention span, and remove all superfluous detail, such as the reference to Occam's razor. – Weather Vane Aug 20 '17 at 19:34
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    Before your question sinks without trace, I offer a suggestion. There is no way I would attempt to explain anything like this without reference to diagrams. So it's a waste of time trying to produce a perfect prose-only piece. After a lifetime of teaching technical subjects I have learned that a picture is worth a thousand words. – David Aug 20 '17 at 20:24
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    Why is this question open? It's (a) a request for writing advice, and (b) apparently requires an expert in information theory to answer. Maybe should be posted a more appropriate site. Mathematics maybe? – MetaEd Aug 24 '17 at 17:19
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    We can send it to Writers, but in any case, this question needs some serious editing for length and clarity. – Kit Z. Fox Aug 24 '17 at 17:36

Perhaps you could use dialling numbers on old analog dial phones as an analogy?

There's an unordered set of numbers and each one has a different dial tone associated with them. There's a separate ordered set of dial tones. Mapping the numbers from the unordered set to the ordered set via dial tones gives you an ordered set of numbers, i.e. a telephone number.

The result: if someone came up to you and sang the dial tones at you, you wouldn't understand a thing (because essentially these tones are just ordered place holders without any meaningful elements). Whereas if someone came up to you and said the same string out loud but with the numbers, you'd understand the sequence as a telephone number as well as additional information like which country the person is based in via the country code. The dial tone gives you the order, the numbers with dial tone features give you the unordered elements, and the mapping of dial tone to dial tone gives you the mapping process which relates elements from an unordered set to an ordered set resulting in a message that is comprehensible to you, i.e. resulting in your friend's telephone number.

  • I sincerely appreciate the fact that you gave an answer instead of just writing stuff in the comments like everyone else. I very much respect that given the context of what I've seen thus far. I upvoted your answer. However it seems like you may have complicated it more than it needed to be. And also that you may have misunderstood things, which would be my fault. I believe that I should stress a position or a dimension very strongly (the order from left to right or the timing and order of how you say sentence as time moves forward). – Taal Aug 23 '17 at 22:30
  • The word -mapping- is key. When you map something you are finding the correct location for it. So...you may have mixed up a few things here. However, seeing how you attempted to answer this gives me a lot of useful information as to how I should likely phrase my explanation of the concept and what parts to emphasize. If you are saying each number on the telephone corresponds by a one-to-one relationship to a specified tone, then that would be 2 ordered sets in that one to one context. – Taal Aug 23 '17 at 22:31
  • Now they are used in an unordered way, when they are mapped to a position. The ordered set you need to correctly map those numbers to get your friends phone number is the position in which each number appears from left to right. So for position 1 it would be part of the country code (lets assume its US and it's just the # 1). Position two would begin the area code. Position three and then four would also be the area code. 1646.... Then position five until the end of the phone number would position the correct numbers in the correct sequence to get the phone number for your friend. – Taal Aug 23 '17 at 22:31
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    No, the set of numbers remains unordered. Each element has a feature associated with it (the dial tone). So there is one set of ordered dial tones and one set of unordered numerical elements with dial tone features. – Nworb Aug 24 '17 at 7:30
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    @Taal I'm not sure I understand exactly what you're asking for though. Are you talking about the way that words are mapped to linear sentences in a language like English? There are strong arguments that the linear order isn't meaningful in that situation (cf. constituent structure and Chomsky's "Can eagles that fly swim?"). However you could take a look at templatic morphology if you want the idea of a sort of slot machine that orders unordered meaningful parts of words. I agree that it would be helpful if you could maybe condense your question a bit so that it's a bit clearer. – Nworb Aug 24 '17 at 7:36

Euclid's Elements is the first thing that comes to mind for me and 16 year olds and aliens likely know something of geometry. Have you read the original (in translation)? It is founded on a small number of Definitions, Postulates and Common Notions. (and ancient Greek was unordered by the way; has no one pointed that out?)

From there Euclid goes on to build increasingly complex and sometimes exquisitely elegant reasoning with geometric shapes, relationships, ratios. It's much preferable to textbooks because Euclid and you (if you play along and do the Propositions yourself!) build up a common language together. I'd had Geometry in middle and high school and it was okay. But Euclid is a whole different deal. It's learning a system of thought, not just textbook "how to get answers for test". He literally wrote the book on moving from simple first principles to complicated concepts. His Geometry became the pre-req for Philosophy or any reasoning.

Descartes, Pascal, Newton, Einstein and pretty much everyone who was able to teach us something modeled on Euclid. And it's not just mathematics, it's everything. Physicists at NASA today sound just like Euclid when starting or rethinking a project (my experience). Start as simply as you can and build from there (saw Borucki of Kepler fame begin many a presentation with "this is a piece of chalk; I'm going to make a dot that will represent a point..."). It holds true for musicians, mechanics, programmers, biologists, etc. I've been amazed to see that what seemed like patronizing simplicity that seems so sloowww to begin with easily and naturally becomes exponentially more complex. While starting complex, even at low level, usually follows a linear path and takes a looonngggg time to get to conclusions only to have to back track and check assumptions.


Definitions are what is a line, point, angle, circle, congruent, etc. Good to look at the whole list.

  1. A straight line segment can be drawn joining any two points.
  2. Any straight line segment can be extended indefinitely in a straight line.
  3. Given any straight line segment, a circle can be drawn having the segment as radius and one endpoint as center.
  4. All right angles are congruent.
  5. If two lines are drawn which intersect a third in such a way that the sum of the inner angles on one side is less than two right angles, then the two lines inevitably must intersect each other on that side if extended far enough. (parallel postulate)
Common Notions
  1. Things which equal the same thing also equal one another.
  2. If equals are added to equals, then the wholes are equal.
  3. If equals are subtracted from equals, then the remainders are equal.
  4. Things which coincide with one another equal one another.
  5. The whole is greater than the part.

After reading your statement, and the other answers, I have come to the conclusion you are talking about:

Godel's Incompleteness Theorems

The first states that no system is complete in it's own context, and the second that no system can prove itself inconsistent.

A proof of the first incompleteness theorem involves assigning numbers to mathematical operators, and deriving the necessary contradiction.

So, what you are asking for is impossible.


Ancient Greek used a large number of cases based on mood, voice, person, and number. They subsets of those so basically start with 45 forms of nouns and verbs, adjectives, pronouns, adverbs and then a bunch of special rules for more. Words can be in any order so in a piece of text you find the object and any modifying words that match it (that can be anywhere in the sentence), verb and modifiers, subject and modifiers, then figure out where everything left over belongs. Sounds crazy but once you do it for a while, not so bad. But there were very few people literate in written Greek and much fewer that could write in it (an often overlooked thing that ability to read ≠ ability to write).

English has a problem with "I am" statements (among others). I am angry, I am his mom, I am Bob, I am going. Take the first one, "I am angry"; if we say "angry" we'll then know that someone is experiencing anger, just have to find out who; this form of "anger" has to modify another word but we are not given a clue as to number (single person or thing or could be 2, 3 or any number), does it modify subject or object; is it about past, present, future? Lots more cases.

So English could have angryther to mean it modifies the subject, past, plural and translate as "We got angry". You could even have We similarly modified so we know it's the subject and in the past by having the word wether so that no matter what the word order angryther goes with wether because they are both the same subject/past/plural.

Next we have dogjo mean dog as object/past/singular, and blackjo as same case for black. And let's do an article to agree as well, thejo (the) or ajo (a).

So no matter how the words are ordered there can only be one meaning. dogjo angryther blackjo wether thejo = We got angry at the black dog = thejo wether blackjo angryther dogjo.

And we do have a little bit of that in that angry is a form of anger that has to modify another word while anger can stand by itself.

But to the earlier point, when we say "I am angry" we don't mean it as "I = angry" but rather "I am having a feeling of anger toward something outside of me". (Response to "I am angry" is typically "At what?" whereas "I am experiencing anger" is commonly replied with "Why?".) I wonder if this is psychologically important? Do we English speakers feel the need to embody Anger itself rather than experience an emotion for perhaps a moment? Gets more ambiguous with "I am blue." Are you blue like a smurf, skin is blue? Are you sad? Is your name Blue? Are you cold? Exasperated, blue in the face? "I am hungry" in reality means "I am getting feedback from my stomach/mind that usually indicates my stomach is shrinking" which is a normal process for a stomach not getting ready to eat, but we go ahead and stuff it anyway because we can't differentiate with a concept like "I am hungry" as our prevalent baseline; it's either "I am" or "I am not" - no middle ground.

Many other languages get around this by saying they have a feeling or like Spanish use an inherently temporal word like Estoy (I am - for the moment, v.Estar) instead of Soy (I am - always, v.Ser).

And the more I think about this it seems English is a difficult language for expressing anything accurately without using complex sentences with ever increasing word order dependence. That is unless we have lots of words to mean planet that reflect our relation, emotional or intellectual context, meta information included in our choice.

And another major thing to consider is the concrete vs. conceptual/abstract thinkers. Concrete folks want, need a specific word for something, don't feel comfortable without it, somehow feel comfortable once the have the name for it even if they don't understand anything about it (car parts, deity, food). If they hear the word "car" in their heads it's words and images of specific cars. If abstract thinker hears "car" it's a device or part of one (train car) for people to get in to move from place to place. So if you formed a technically superior language, it wouldn't help if folks got different things in their heads. Hell, we've already got a superior language that people use a fraction of. And telling the truth is difficult and often unwanted though, ideally, that would be a useful thing in a language. And there are the terrible thinking loops such as when someone asks if you are married? I have a wife = she is mine= I own her. There are some good sources for the way we choose words, our preferences of "bad words" by number of letters, syllables, hard stops. Or glossy glass glistens.

  • More words in a language means there are more words for people to know and learn the definition of. Essentially, everyone's vocabulary would have to be enormous. However, instead of having a one meaning for each word, we instead "map" the words to an order...aka put them into a sequence. A sequence of words takes the smaller meanings of each word and then puts them together in a contextually dependent manner to make a new meaning. – Taal Aug 27 '17 at 4:28
  • Ironically, this part of stackexchange, where people ask for words which define "and they list out the definition they want" so much is evidence that our culture chose to balance the # of possible words with the ways we could arrange them to create meaning. The ability to position words around each other in so many ways allows us to communicate complex & specific meanings while having degeneracy: It means more than 1 word can mean the same thing...allowing for redundancy thus reduction in communication errors while also allowing the mixture and context of words to mean something different. – Taal Aug 27 '17 at 4:42

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