FutureBeacon.com



Cognitive Foundations
of
Counting and Calculating


by

James Adrian




Perceptions, Terms, and Definitions


When young children learn the name of the sun, for a while, they might think that the name sun and the sun itself are the same thing. Children brought up in a bilingual family learn otherwise very quickly. Bilingual or not, children soon discover that anything having a name, may have more than a single name.

The meanings of very many names and other sorts of words are learned through experience. Blue is the color of the sky and the color of that paint that everybody calls blue.

Many meanings are learned through experience. We learn how and when words are used by people. After a considerable amount of this kind of learning, another option appears. People can tell us what they mean by a word - even if that word is new to us. Dictionaries do a lot of that.

By and large, dictionaries contain descriptions that remind us of meanings and associate some meanings with others.   Sometimes, but not always, a dictionary will state a formal definition.   Providing only formal definitions is not the purpose of most dictionaries.   They typically explain the most common words in use whether they can be assigned a formal definition or not.

A formal definition is constructed from meanings that are previously known to the reader or listener. They describe an exact meaning.

Here is an example of a definition:

Definition - A term is a word or a phrase.

Any statement that is offered as a definition must have all of the following properties:

A proposed definition must name the term being defined and provide a description of that term.

Here is an example: A hydrocarbon (defined term) is a compound consisting only of carbon and hydrogen (description).

Other sentence structures are possible, but both the defined term and its description must be somewhere in the statement.   A statement may be comprised of more than a single sentence.

Other than the term being defined, the meaning of each term used in a definition must be known before that definition is stated.

The term being defined must not appear in the description of that term.

Including the term being defined in the description of that term is circular. Consider the following:

A good car is a car you like.

The trick here is that the term being defined is good car and not car. In order for this definition to be a valid formal definition, the term car must have been previously defined.

Whenever possible, each term used in a description must refer to only a single meaning.

In English, there are many terms that have meanings that depend upon their context, but in a mathematical or scientific context, words and phrases, especially new ones, should each be assigned to a single meaning. At this point in history, it would be awkward to rename the number 1 (one) to make it different from the various and widespread meanings of that word. The selection of new terms in a mathematical development should strive to avoid making meanings dependent upon their context, but sometimes, it is unavoidable.

Within an intended context, the term being defined and the description of that term must be interchangeable.

It must be clear that the term and its description have exactly the same meaning.

Definition - A term is well defined if and only if it is described by a statement that has all of the properties required of a formal definition.

If the definitions we make are formal and precise, they are born of other sensible definitions but also born of undefined terms (empirical terms).   There would be no formal definitions if no undefined terms were in use.  We get our undefined terms from perceptions.   Fortunately, some of these perceptions are shared widely and recognized as true of the world.   Only these undefined terms are useful as undefined terms in formal definitions in a science and math.

Often, undefined terms must me explained to distinguish them from alternate or common meanings not intended for use in a particular definition.

There are many undefined terms that nobody bothers to mention in the development of technical jargon.   "If," "the," "we," "were," "to," "accept," "this," "as," and "true" are all undefined terms in most formal systems because there is great confidence that the meanings of these terms are know unambiguously to every person reading or hearing them.



Basic Perceptions and Terms


Here is a description of the term statement:

Sentences may contain more than a single statement.   A statement may be written as a clause in a compound sentence, like this:   "Birds fly and fish swim."   This entire sentence is a statement.   It contains statements which could be written this way: "Birds fly."   "Fish swim."   All of the sentences in quotation marks in this paragraph are statements.

On the other hand, a statement, such as an account of events or any description, may contain more than one sentence. Consider this example:

A substance (let's call it S) can be called a soup only if the following statements are true:
S is comprised only of food items and water.
S is intended for human consumption.
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Clearly, the terms statement and sentence are not synonymous. Several other sentences might have been included in the statement made above.

Mathematics is a direct consequence of our most basic perceptions.   Formal definitions cannot be the starting point for math because they rely on pre-existing terms.   Consider these definitions:

Definition - A name is a word, phrase, character, symbol, or mark that is used to refer to something other than itself.
Definition - A numeral is a single-character symbol used singly or in combinations only to form names.

If the words and phrases used to define these terms must themselves be defined, we have no starting point.   The process must start with terms that we understand without having constructed a formal definition for them.   To create mathematics, there must be mathematical terms at the start that we understand precisely by some means other than by defining them.   Fortunately, there are several important terms that we come to understand through experience alone:

You can tell the difference between a single thing and a pair of things as surely as you can tell night from day or red from blue.   Children communicate those perceptions and reason with them as soon as they find out which words other people use to refer to them.   We all have an ability to perceive a single thing, a pair of things, and many things.   These terms are given meaning through experience in the world.   Terms this basic cannot be defined by more basic terms.   They are called undefined terms.   They may also be called empirical terms.   For example, the term thing cannot be assigned a formal definition.   Attempts will always be circular.   The terms any, some, more than, less than, at least, no more than, at most and few refer to perceptions, not to definitions.   These and related meanings are the starting point for our mathematical definitions.   These terms are associated with their corresponding experiences empirically.

The term thing is intended to be thoroughly indefinite as to the nature or description of whatever is being referred to.   Thing often refers to an object, but a list of things might sometimes be a list of terms, attributes, or attitudes. A list of things that are not typically thought of as objects may be referred to as a thing.   As used in this document, the terms item and thing will have exactly the same meaning.

The term one has more than one meaning as used in math. The undefined term is a synonym for single as in saying "one thing," meaning "single thing." The other meaning is one, or 1, as a number, which will be formally defined later. The phrase "one and only one" is often used, and in this phrase, the term is, of course, the synonym for "single." Normally, each term is given only one meaning, making it unnecessary to discern the meaning from the context; but this term is so universally known in these meanings, and their use is so widespread that substituting one with single would make some definitions awkward. This will need to be a rare exception to the well advised one term, one meaning policy.

Prior to counting and calculating, we learn some mathematically useful terms associated with perceptions, but we also come to understand certain fundamental features of the world:

To us, a pile of rocks is not merely a lot of rocks, it is an item named in the singular. We are capable of perceiving a rock pile as a single item that is distinct from any of the things that it contains.   In fact, we seem to insist on doing so.   Along with this kind of perception comes a compelling point of reasoning that I believe we all acknowledge as true:   No such thing may contain itself.

We could imagine a pile of rocks that is covered by additional rocks to form a new pile of rocks; but the two piles could never be said to be identically the same pile of rocks.

The term contain is used in the same sense that it is used in the sentence "The bottle will contain a boat."   In this document, the terms contained by, contained in, and in, are used as synonyms.

The passage of time is demonstrated by any succession of events.   Event A may occur before, after or simultaneous with event B.   The meaning of the terms associated with time are exceedingly well known.   Time is very often measured and characterized with the help of mathematics; but traditionally, time has not been used to help create math itself.   In my opinion, this should change.

Mathematical structures are routinely created by precise definitions.   There is no doubt that things like a series of events can also be precisely specified.   By defining specific actions, operations, and procedures, we can create and illustrate mathematical concepts that go through time.   Nowadays, most of the procedures in this world are computer programs.   I feel that defining structures that go through time is no longer prohibited by custom or contraindicated by any compelling reason.

The term theorem usually means a statement together with its proof, but this term is sometimes given the same meaning as the term conjecture, which means a statement that appears to be true, has not yet been proved or disproved, and is not intended to be assumed true or used to help prove other statements.   This meaning was used most famously in the case of Fermat's Last Theorem.   It was stated in 1637 but not proved until 1995.   Statements made here (other than historical ones) will not be called theorems unless they are stated together with their proof.

Anything may be known by more than a single name.   Languages differ.   Different academic disciplines, and different groups give their own chosen names to things.   The name of a thing does not change the thing.   This universal truth comes in handy:   It means that p and q can be shown to be equal to each other at anytime after p and q are defined.

I introduce a definition of the term equal:

Definition - A and B are equal, or A equals B if and only if A has the same meaning as B.

Names, words, phrases, symbols or descriptions that have the same meaning are said to be equal and said to be equal to each other.   Terms that have the same meaning in a given context are interchangeable in that context.   For a pair of things to be equal they need not have the same name or be referred to by the same term or symbol.   If a pair of things are said to be equal, it is their meanings that are the same, not their spelling.   Numbers are not the only things that can be equal.   Here, there is no difference between the term equal and the term equivalent.

Definition - An equation is a statement that asserts that one written entity has the same meaning as another.

The equals sign written "=" is the symbol for equality. Here is an example:

A = C

The above statement is said this way: "A equals C."   Here are some other examples:

B = an item
a thing = an item
a pair of things = a single thing together with another single thing
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Any entity that appears on the left side of an equal sign can be substituted for the entity appearing on the right side of the equal sign.   The reverse is also true.

This leads to the easy proof of a statement that has been used as an axiom elsewhere:

Theorem   -   A = A.

Proof

Suppose B = A.   (Anything may be known by more than a single name.)
A = A.   (A may be substituted for B because A and B have the same meaning.)
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Because terms that are equal are interchangeable, A = B implies B = A.   Likewise, if A = B and B = C, then A = C.

Throughout this document, the mathematical meaning of the word but is exactly the same as the mathematical meaning of the word and.   The term but carries connotations like not withstanding previous considerations and the like.   It may be helpful to know that you can substitute and for but anywhere in a proof without altering the mathematical meaning of its statements.



A Few Rules of Inference


        If the truth of statement P implies the truth of statement Q, and if statement P is true, then statement Q is true.

        If the truth of statement P implies the truth of statement Q, and statement Q is false, then statement P is false.

        If statement P and statement Q cannot both be true, and statement P is true, then statement Q is false.

        If P implies Q, and Q implies R, then P implies R.

        If P implies Q, and Q implies P, then P is true if and only if Q is true.

        If all things of type T necessarily have attribute A, and a specific thing R is of type T, then R has attribute A.

Consistency itself requires these rules of inference to be acknowledged as valid in every language, in every court of law, and in every human field of endeavor.   Indeed, a sincere objection to any one of them is evidence of a mental defect.   These rules were not invented.   No mathematical system disputes them.   Why?   Any stated rule of inference is merely a recognition of the consistent use of our terms.   Other cases may be recognized.   It is not the rules that drive our reasoning. We have adapted to a consistent world.   Acknowledging and complying with this consistency is what specifies, compels and derives the rules by which we infer.

Other undefined observations, terms and perceptions will be discussed when we are closer to the place of their use in this document.



Containment


Definition - Item x is an element if and only if x is contained by another item, is contained in another item or is in another item.

Characterizing a thing as an element emphasizes its containment in something else, even if that something else is merely the world or the universe.

The term collection is widely understood, but another similar term is needed - a term that escapes the over-learned notion that a collection is a gathering or assemblage that necessarily involves at least a pair of things.   Simply saying that a collection may contain just a single item, or perhaps even nothing, generates a certain cognitive dissonance brought about through the contradicting and unlearning of years of experience.   It is always best to build new knowledge upon existing knowledge without rearranging learned associations and meanings that are already well established. The term collection needs to be replaced.

The term box could be considered because a box could be empty, or it could contain a single thing, or many things, or even another box; but a box is an object that might not easily be accepted as overlapping another box or containing unique objects in common with another box.   A term like region, space, place or area would solve these problems, but they carry extraneous associations having to do with grid coordinates, directions, and the like.   It would be better to find a term that is abstract in the sense that it is not required to be an object or place - a term that is devoid of its own peculiar associations.   Such a term is set.

The term set is well known to mathematicians, but it may not be part of your experience.   This problem is solved by utilizing familiar meanings as properties of the new meaning.

Definition - S is a set if and only if the following statements are true:
S does not contain itself.
S contains nothing, or S contains a single element, or S contains more than a single element.
The elements in S are each unique in S.
Each named element in S has a name that is unique among the elements named in S.
Elements in S may each have any description other than being S.
S may be defined as containing elements exclusively of a particular description.
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Notice that any set contains no duplicates.

Notation - The notation for sets is the listing in curly brackets of whatever it contains:
Set X = {Jim's chair, the Washington monument, Mary's gold pen}.

The definition of the term element permits a set to be contained in another set as an element.   It is important to realize that if element q is in set B, and set B is in set A, element q is not thereby required to be in set A.   A less abstract model that can be used to clarify this is the idea of a list.   If item q is in list B, and list B is in list A, there still may be no entry in list A that is called q.   If q is to be in list A, q must be listed in list A.   If list B is listed in list A by name, the items in list B are not automatically listed in list A.

Every collection is also a set:

Definition - A collection is a set containing at least a pair of elements.

Of course, a set is not always a collection because some sets are empty, and some sets contain only a single thing.

Definition - An empty set is a set that contains nothing.

Notation - S = {} shall indicate that set S is empty.

Definition - S is a non-empty set if and only if S is a set, and S contains a single element or more than a single element.

Notice that, since a set may contain nothing, an empty set is a set.   An empty set contains nothing and does not contain anything.   Since no set may contain itself, any empty set may not contain itself.   If it did, it would contain something other than nothing.   It would contain a set.   This is different from the empty set of some other systems.

Also, nothing prevents us from discovering that a previously defined set is empty.   Because any set S may be defined as having elements of any description (other than being S), we cannot prevent sometimes discovering that S is empty and yet distinct from some other empty set.   Any particular set may be defined by what it may contain according to a criterion of element inclusion.   In the course of deducing what elements there are in a given set, the answer is sometimes none.

In this system, abstractness is not a value in itself.   A bank account that is empty is not regarded as identical to a region of space that contains nothing.   No useful purpose is served by insisting that they are both identical to a unique empty set.   This language is consistent with the notion that math is to help us quantitatively analyze real things.   Sets that contain all of the same elements do not necessarily have the same meaning.   Alice, George and Martha may be the only interior decorators in Meltown, but they could also be the first three people I met in the Fishing Club.   These sets are in some way similar, but they are also distinct.   Accordingly, when every element in set A is in set B, and every element in set B is in set A, A and B are coincident, but they are not necessarily equal:

Definition - Set A and set B are coincident if and only if every element in set A is in set B, and every element in set B is in set A.
Definition - Set A and set B are equal if and only if A and B have the same meaning.
Definition - Set A is coincident with set B if and only if every element in set A is in set B, and every element in set B is in set A.
Definition - Set S is a subset of set T if and only if each element in set S is also an element in set T, and S is not coincident with T.
Definition - Set S is a proper subset of set T if and only if S is a subset of T.
Definition - Set T is the superset of set S if and only if S is a subset of T.

In this system, a set S may not be called a subset of set T if S is coincident with T.   This means that in no sense may any set contain itself.   The term proper subset may be used as an emphatic indication that the subset and the superset are not coincident.

Definition - Set S is the intersection of set T and set R if and only if each element in S is an element in T, and each element in S is also an element in R, and S contains no other elements.
Definition - Set X is the union of set T and set R if and only if set X contains every element that is in set T and set X contains every element that is in set R, and X contains no other elements.
Definition - Set B is the complement of set A in set C if and only if B contains all of the elements in C that are not in A, and B does not contain A or C, and A does not contain B or C.



Axioms


An axiom is a statement that has these attributes:
It is assumed to be true.
It is incapable of being proved.
It seems useful in helping to prove other statements.
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If axioms are used at all, axioms must be limited to those statements that we cannot prove.   They are statements of logical relationships that we must perceive with great certainty.   If an axiom is ever found to be factually false, all statements that rely upon the truth of that axiom are then recognized as not proved.

Consider these statements:

Anything may be known by more than a single name.
Nothing may contain itself.
If B is contained in A, then A is not contained in B.
If event A occurs before event B, then event B does not occur before event A.
If event A occurs after event B, then event B does not occur after event A.
If event A occurs before event B, and event B occurs before event C, then event A occurs before event C.
If event A occurs after event B, and event B occurs after event C, then event A occurs after event C.
If event A occurs before event B, then event B occurs after event A.
A = A.
If A = B, then B = A.
If A = B, and B = C, then A = C.
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The statements listed above can be used as evidence in a proof, but they are not really axioms.   Let's take a closer look at each of the them:

Anything may be known by more than a single name.   If this is assumed to be true, p and q can be shown to be equal to each other at anytime after p and q are defined.   This is essential, but also obvious from the universally accepted meaning of the term name.   Its contradiction would be an impediment to proving statements, and it may be helpful to remember that it is true, but no system disputes it.   It is factually true prior to being assumed and therefore it cannot be an axiom.

Nothing may contain itself.   This statement is true of all things and it follows from the meaning of the term contain.   The statement can be used as evidence in a proof, but so can many other statements that are true by definition or by the agreed meaning of undefined terms.   It is not an axiom.   The widespread understanding of the term as it is used in sentences such as "the box does not contain the letter" is the only meaning of this term that is used in this system.   The set of all things that are not human contains itself and is therefore not a possible set.   Like the sound of one hand clapping, it owes its apparent meaning to the fact that language goes through time.

If B is contained in A, then A is not contained in B.   This statement can be used to help prove other statements, but it follows from the meaning of the term contain.

If event A occurs before event B, then event B does not occur before event A.
If event A occurs after event B, then event B does not occur after event A.
If event A occurs before event B, and event B occurs before event C, then event A occurs before event C.
If event A occurs after event B, and event B occurs after event C, then event A occurs after event C.
If event A occurs before event B, then event B occurs after event A.


These statements introduce time and remind us that such statements can be used to help prove other statements, but they follow from the meaning of the terms before and after.   Since they are true by virtue of the meaning of the terms use to express them, they are actually true and cannot be assumed true.   They are not axioms.   No axiom would be meaningful if it were a statement that could not be otherwise.

A = A.
If A = B, then B = A.
If A = B, and B = C, then A = C.


These statements follow from the meaning of the term equal.   A and B are equal, or A equals B if and only if A has the same meaning as B.   With this definition, proving all three of these statements of equality is straightforward, and so they cannot be used as axioms.

There are no axioms.   The statements considered above could not be otherwise.   They are actually true and not assumed.   They are true by virtue of the definitions and the agreed meaning of the undefined terms that are used to state them.   They may be used as evidence in any proof.   They are important in this system but may not be provable in other systems.

In general, paradoxes are avoided by eliminating axioms.   The argument is whether anything of value to math is left.   I argue that everything of real value is indeed left.   I make the case by showing that there are other ways to obtain the results that others have said require axioms.



Order


In this development, numbers require the prior existence of ordered sets (such as the alphabet).   Therefore, numbers, or terms such as less than or greater than cannot be used to help construct a definition of the term ordered set.   Instead, this term is defined by starting with the undefined term association.

An association can be prompted by information that we receive from the environment, or from a dream, or from a memory, or from a thought.   An association may be made inadvertently or intentionally in the process of thinking.   We are free to associate anything with anything else, and for any reason.   Multiple meanings of this term are widely understood, but in order to be used in the formal definitions of this mathematical development, a single meaning must be chosen.   Consider this statement:

Lake Ontario is always associated with water, but water is not always associated with Lake Ontario.

This kind of association is different from the association of a pair of things with each other symmetrically.   Some associations only go in a single direction, or tend toward a single direction.   We need a meaning that is abstracted from any particular example (such as the Lake Ontario example) in order to focus on the relationship and not on any accidental characteristics.   Here is the specific meaning that will be used in this writing:

A is associated with B whenever it is explicitly stated that A is associated with B; and whenever A is associated with B, B is not thereby associated with A. In such a case, B is associated with A if only if that fact is also explicitly stated.

This is not a definition.   It is a legitimate undefined term having a specific meaning with which we have extensive experience.   This is the meaning that we understood when we leaned the alphabet.   Not many of us can quickly recite the alphabet backwards.   None of us learned the alphabet as a system of definitions leading to the establishment of an ordered set.   We simply associated each letter with the next in this precise and very limited sense. It doesn't need to be a two-way association, and it isn't. Learning the alphabet in the familiar direction by rote does not form the complementary associations in the opposite direction.

Notation - A → B means that A is associated with B.   B ← A also means that A is associated with B.   Up arrows and down arrows are used in the same way.   If the arrow points away from A and toward B, then it means that A is associated with B.   (And it does not tell us that B is associated with A.)

Definition - S is an ordered set if and only if the following statements are true:
S is a set containing at least a pair of elements.
A single element j in S, and only j in S, is such that no other element in S is associated with it.
A single element k in S, and only k in S, is not associated with any element in S.
Each element p in S, other than element k in S, is associated with a single other element in S, and only that single other element in S.
If element x is associated with element y in S, then no element other than x is associated with element y in S.
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The term next has a useful meaning in this context:

Definition - Element y is next in order from x if and only if the following statements are true:
S is an ordered set containing elements x and y.
Element x in S is associated with element y in S.
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Definition - S is an unordered set if an only if S is a set, and S is not an ordered set.
Definition - S is an unordered pair if an only if S is an unordered set, and S contains exactly a pair of elements.
Definition - S is an ordered pair if an only if S is an ordered set, and S contains exactly a pair of elements.
Definition - Element x in S is the starting element of S if and only if the following statements are true:
S is an ordered set containing element x.
No element in S is associated with x.
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Definition - Element z in S is the ending element of S if and only if the following statements are true:
S is an ordered set containing element z.
Element z in S is not associated with any element in S.
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Notation - In the case of an ordered set, the curly brackets that are normally used to enclose elements in a set are replaced by parentheses.   Unless otherwise specifically required, the starting element is written on the left and the ending element is written on the right.   Here is an example:

S = (t, u, v, w, x, y, z) = (t → u → v → w → x → y → z)

Definition - A is the English alphabet if and only if A is an ordered set and A = (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) or A = (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).

Recall that set A and set B are coincident if and only if every element in set A is in set B, and every element in set B is in set A, whereas set A and set B are equal if and only if A and B have the same meaning.   If set A is an ordered set and set B is not, they are not equal but they are coincident.   It makes little sense to defined A and B as equal if they differ in such a consequential characteristic.

The definition of an ordered pair that is most conventional among mathematicians is mentioned in a history of the subject made available on line at this Wikipedia article.   Here is a very abbreviated summary of that history:

    Norbert Wiener proposed the first set theoretical definition of the ordered pair in 1914:

    (a, b) = {{{a}, R}, {{b}}}


    Also in 1914, Felix Hausdorff proposed this definition:

    (a, b) = {{a, 1}, {b, 2}}    "where 1 and 2 are two distinct objects different from a and b"


    In 1921, Kazimierz Kuratowski offered the now-conventional definition of the ordered pair (a, b):

    (a, b) = {{a}, {a, b}}


Authors such as David M. Burton in his Introduction to Modern Abstract Algebra (1967) utilize the Kuratowski definition and point out that a and b are not elements of the ordered pair but components of it.   The elements of the ordered pair are {a} and {a, b}. Ordered sets having more than a pair of components are recognized as pairs of pairs.   In this example, {a, b} lists the components of the ordered pair and {a} names the first in the order of the pair of components.

Why would I not accept this definition?   It is because I cannot accept first as an undefined term in a system that is to clarify the meaning of numbers.   Using the definition (a, b) = {{a}, {a, b}} to help define first would be circular if Burton's account of the meaning of the equation were all we had to go on.   I would rather use undefined terms like "association" and "item" and "collection" and "thing" and "single" to define terms having meanings that are more obviously mathematical.

While there may be another way to explain the equation, I find unacceptable any definition that requires the reader to "understand" or "get" some notion that is not formed by the rules of meaning in the introduction that I have explained earlier in this document (formal definition and unique specification of undefined terms).   Kuratowski's equation would need to be built up from terms explained in whole sentences in the common language (in this case English); although I must point out that, to my knowledge, and unlike the development here, Kuratowski did not offer his definition as a precursor to numbers.

In this system, an ordered set such as X = (a, b, c, d) contains elements a, b, c, and d.   These elements are not called "components."   One is free to draw attention to subsets of X.   Being a set is part of the definition of an ordered set.   X is a set; and a set such as Y = {c, a} satisfies the definition of a subset of X.   Every element in Y is also in X, and Y is not coincident with X.