When young children learn the name of the sun, for a while, they might think that the 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

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:

Any statement that is offered as a formal definition (as in science, math, law, and other disciplines) must have all of the following properties:

Here is an example: A

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.

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

A

The trick here is that the term being defined is

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.

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

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.

Here is a description of the term

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

S is comprised only of food items and water.

S is intended for human consumption.

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Clearly, the terms

Mathematics is a direct consequence of our most basic perceptions.

You can tell the difference between a

The term

The term

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

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

The term

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

Names, words, phrases, symbols or descriptions that have the same

The

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:

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

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.

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

The term

The term

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.

Set X = {Jim's chair, the Washington monument, Mary's gold pen}.

The definition of the term

Every collection is also a set:

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

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

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

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

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.

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:

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

If A = B, then B = A.

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

These statements follow from the meaning of the term

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.

In this development, numbers require the prior existence of

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:

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:

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.

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

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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S is an ordered set containing element x.

No element in S is associated with x.

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S is an ordered set containing element z.

Element z in S is not associated with any element in S.

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

Why would I not accept this definition? It is because I cannot accept

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.