Infinite number of elements in some sets but no infinitely large number

Pka,

I believe that nothing can be proved by using the word infinite unless you have an axiom (or more than one axiom) that creates infinity in your system. If you don't have axioms, you need to define unbounded sets that are finite at any point in time. You also need to define procedures that go through time.

I think that working without axioms is likely to be more acceptable to engineers and computer scientists than to mathematicians of this era.

I intend to show how calculus and complex numbers do not intrinsically depend on axioms.

On the Internet, it is hard to find a website defining math terms that does not conflate unbounded sets with infinite sets. This shows that there is little understanding of unbounded sets, and little understanding of the limits of what can be done with definitions.

The widespread belief that calculus depends on infinity is a myth.

Jim Adrian
 
The terms axiom and postulate are each synonyms of the term assumption. The aim of this effort is to define a mathematics that includes calculus and complex numbers, and does not use assumptions as the foundation of that mathematics.

Rational Mathematics is a system of scientific definitions in which rational numbers play a dominant role. The number line consists of rational numbers together with processes that produce rational numbers. It does not include transcendental numbers or other irrational numbers except as names for processes. This system contains no concept of an infinitesimal.

Students in every branch of engineering have been forced to acquire the ability to calculate quantities by means of a mathematics that is based on axioms. Since about 1600, these axioms have been imbued with a decidedly ethereal bent - especially as it affects descriptions of very large and very small quantities.

As an undergraduate, you probably were not told that infinity, if it is to exist in math at all, must be added as an axiom. Infinity can never otherwise be established. This does not mean that derivatives cannot be defined without axioms.

To this day, there is little recognition of the role, or potential role, of unbounded sets. You can easily verify that many websites defining mathematical terms fail to distinguish between infinite sets and unbounded sets. As a result of this, and related judgements, many math teachers perpetuate the myth that calculus cannot be developed without infinity, and they do so without proof.

The foundations of mathematics are much older and much more sensible than the math foundations of the modern era.

Definition - A number is the name of an amount or a quantity.

A scientific definition must name the term being defined and describe the meaning of that term without reference to the term itself, and without reference to any term not previously established, either as scientifically defined or as an experiential part of the common language. Terms like the and thing are undefined, but their meaning is required to be widely agreed in the culture at large. If ambiguity is possible due to multiple meanings, the meaning used for mathematical purposes must be identified. Within an intended context, the term being defined and the description of that term must be interchangeable. The experiential terms can be called empirical terms, of which there are several of interest to mathematicians. Attempts to define empirical terms always results in circularity. Notice that this is true of amount and quantity. A scientific definition is also called a formal definition.

Formal definitions cannot be the starting point of math because they rely on preexisting terms (words or phrases previously established in the common language). Here are some well-established terms that create mathematics:

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 reason and communicate with such perceptions as soon as they learn which words other people use to refer to them. We all have the 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. The terms any, some, more than, less than, at least, no more than, at most, and few refer to perceptions, not to formal definitions. These and related meanings are the starting point for mathematical definitions. These terms refer to empirical observations.

Throughout this writing, what the term thing refers to is intended to be thoroughly indefinite. An idea, an object, a sentence, a mark, an action, a description, a time, a location, or anything else can be named a thing.

There is a centuries-old reluctance among mathematicians to use the passage of time as an element in mathematical reasoning. However, this development recognizes our experience with time and includes terms such as before and afteras empirical terms referring to perceptions. Their meanings are too basic to be described by more basic terms. In an attempted formal definition, they are circular. Nonetheless, these terms can be used in formal definitions because they refer to reality in ways that are beyond dispute.

The order of time is thoroughly conspicuous. Events of any nameable kind come to pass in an order, with some events happening before others, and with any particular event coming next in order relative to the event preceding it. The terms order and next are too basic to be formally defined by more basic terms.

The same idea of order occurs in our perception of space. Rocks placed in a line give us next and immediately previous or immediately neighboring rocks. We often refer to and order of succession, or an order of importance.

Defining mathematical entities as having order, or having an order, or having been ordered, becomes more straight forward if these undefinable terms that we know so well are acknowledged.

Definition - An element is a thing contained in a collection.

Definition - T is a collection of things if and only if T is a collection containing at least a single thing; and, the element or elements in T are of any description except that no collection may contain itself.

The undefined term collection is used here in the sense that it may contain a single thing or more things. It is not require to contain at least a pair of things, as the term is sometimes used. Also, a collection may not contain itself. It may be an element in another collection, but it cannot be an element contained in itself.

Definition - A set is either a collection, or a named entity or space that does not contain elements. In the latter case, the set is said to be empty.


Jim Adrian
 
Is it your intention to destroy mathematics- or do you simply not understand what mathmatics is? Those "axioms", "postulates", and, if you wish, "assumptions" are what make mathematics so powerful! Mathematics starts with "undefined terms" and has "axioms" or "postulates" that tell how those undefined terms work together. We then apply mathematics to a specific problem by assigning meaning, to those undefined terms and choosing the "axioms" or "postulates" that are appropriate for the problem. Then we know that all the "theorems" proved from those "axioms" or "postulates" are also true.
 
What the OP is trying to say is that many of the axioms of basic mathematics are generalizations from common experience. Completely trivial thought.
 
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JeffM,

That is not what I am saying. I am saying that all assumptions are unnecessary, and the history of math indicates that they are misleading and destructive. Even if you disagree, what is the harm is seeing what can be done with definitions employing the empirical terms that are related to quantity? Hallsofivy's comments are enormously uniformed. They are probably simply defensive of his time investment in his educations, which I regard as his victimhood. I am the one who did it your way for decades. To jump to conclusions about my way after a few minutes consideration is ridiculous. You would do better to ask questions.

Jim Adrian
 
Definition - An element is a thing contained in a collection.

Definition - T is a collection of things if and only if T is a collection containing at least a single thing; and, the element or elements in T are of any description except that no collection may contain itself.

The undefined term collection is used here in the sense that it may contain a single thing or more things. It is not require to contain at least a pair of things, as the term is sometimes used. Also, a collection may not contain itself. It may be an element in another collection, but it cannot be an element contained in itself.

Definition - A set is either a collection, or a named entity or space that does not contain elements. In the latter case, the set is said to be empty.

Definition - S is a collection of things if and only if S is a set; and, if E is an element in S, then E may be of any description.

There is no need to disallow phases such as a collection of trees or a set of marbles or a set of events or a set of locations. Such phrases can be used to create definitions and identify inferences. Any prohibition against such meanings is a needless attempt to make mathematics unnecessarily abstract.

Definition - A set K in C is a set such that each element in set K is also an element in a set C.

Definition - C is a subset of D if and only if C and D are sets and every element in C is also in D.

Definition - C is a proper subset of D if and only if C and D are sets; and, every element in C is also in D; and, there is at least a single element in D that is not in C.

Definition - The union of set T and set U is the set S of elements that are each either in set T or set U.

Definition - The intersection of set T and set U is the set S of elements that are each both in set T or set U; and, set S may be found to be empty.

Definition - An element E is removed from a named set S if and only if E is in set S, and S is then redefined to exclude E.

Definition - An element E is inserted in named set S if and only if E is not in set S, and S is then redefined to include E.

Definition - An element E is copied from set S to set T if and only if E is an element in set S; and, E is inserted in set T.

Definition - An element E is moved from set S to set T if and only if E is copied from set S to set T, and, E is then removed from set S.

Definition - One is the name of the amount or quantity that is associated with a single thing; or, one is the number associated with a single thing.

Definition - Two is the name of the amount or quantity that is associated with a pair things; or, two is the number associated with a pair of things.

Definition - Zero is the name of the amount or quantity that is associated with the absence of things; or, zero is the number associated with the absence of things.

Definition - Set J is a set of names if and only if every element in J is a name.

Definition - Element q in set J is a subscript of K if and only if each of the following statements is true:

K is a name and J is a set of names.

K together with element q in J form a name distinct from K and distinct from element q in J; and, this name is pronounce K sub q.

The name formed by q and K is written Kq.

K may be said to be subscripted; and, K may be said to be subscripted by q.

Kq may be said to be a name formed by subscripting.

If Kq isa name formed by subscripting, the q is a subscript.
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Any name formed by subscripting may be the name of an amount or quantity and therefore a number.

If R is a name and {t, a, h} is a set of names then {Rt, Ra, Rh} is a set of names formed by subscripting.

Definition - A numeral is a name that is a mark or an ordered set of marks not formed by subscripting.

A numeral could be used as a name for anything. It need not be the name of a number. Here are examples of numerals:

The numeral 0 is a name for the number zero.

The numeral 1 is a name for the number one.

The numeral 2 is a name for the number two.

The numeral 00 is a name for the number zero.

The numeral 01 is a name for the number 1.

The numeral 10 is a name for the number 2.

The numeral 2 is the subscript of name G2.

A subscript is part of a name. The subscripted name may be the name of a number. The number being named in not the being named by the subscipt alone. A numeral may be used as a subscript.

Some things may have more than one name. The numeral 10 may be associated with the number two and may serve as a name for the number two. Likewise, the numeral 2 may be associated with the number two and may serve as a name for the number two.

Definition - Event v is next in O if and only if O is a set of named events in time; and, g and v are elements in O; and, g occurs, or is to occur, before v occurs; and, v occurs, or is to occur, after g occurs; and, the is no event in O that occurs, or is to occur, both after g and before v.

The definition of next in O applies only to events and not to elements of other kinds. There are many kinds of events, and the definition applies to all events. For instance, an action if an event.

Definition - P is a procedure if and only if P is a sequence in time beginning with an initial condition J described by a statement K that is named and distinguished with a label; and, a set A of actions is each described by a statement in set S of statements such that each statement in S describes an action that is named and distinguished with a label; and, each statement in S identifies the statement in S that describes the action in A that is to be performed next in A; and, some statement in S describes an action in A that is to be the last action performed by P.

Cuneiform writing in Mesopotamia was primarily devoted to recording amounts of things being traded. Comparisons needed to be made. There is an ancient procedure for comparing the amounts of individually separate items. Originally, containers were used where the otherwise identical procedure defined below uses sets.

Definition - A one-to-one correspondence is a procedure specified by the following labeled statements describing actions to be performed:

Initial Condition - A pair of sets, j and k, each contain things, while another pair of sets, p and q, are empty.

Action 00 - Move one thing from set j to set p; and, continue by performing Action 01.

Action 01 - Move one thing from set k to set q; and, continue by performing Action 10.

Action 10 - If either set j or set k is empty, stop performing actions. Otherwise continue by performing Action 00
------------------------------------------------------------------------------------------

When this procedure stops performing actions, it is known that set p and set q contain the same quantity of things.


Jim Adrian
 
I notice that the subscripts are not rendered correctly. For instance, G2 should be G sub 2.


Jim Adrian
 
In ASCII, G[s u b]2[/ s u b], without the spaces, renders as G2 or, using "Latex", [t e x]G_2[/ t e x] renders as \(\displaystyle G_2\).
 
Hallsoflvy,

Thank you for this message. I copied and pasted this from an unadvertised page from my website where G sub 2 is rendered properly when my HTML code is G<sub>2</sub> . I notice that your code uses brackets and you do it in ASCII. I'm not sure how I could correct it on this page. I will try to edit my message.

Thank you for your help.

Jim Adrian
 
HallsofIvy said:
In ASCII, G[s u b]2[/ s u b], without the spaces, renders as G2 or, using "Latex", [t e x]G_2[/ t e x] renders as \(\displaystyle G_2\).
Click to expand...
Hey, G.I. I would not have us feed the Trolls if I had my way.
 
Pka,

I am truly amazed that a person with your education would react to the math I have written by calling me a troll.

Jim Adrian
 
jamesadrian said:
I am truly amazed that a person with your education would react to the math I have written by calling me a troll.
Click to expand...
Trolls put-up utter rubbish. You post utter rubbish . Therefore you are a Troll.
 
Pka,

It is the work of many years. It offends you, apparently because it does not require axioms and yet produces effective math. Why not criticize it on its merits? You would need to read it and think about it. Is that so terrible?

To define sets and procedures and the one-to-one correspondence is not nothing. I can go all the way because I know what the origin of math is, and I explain it. Do you think it's incorrect? I came here for feedback. How is it incorrect? Why is doing it without axioms rubbish?

Jim Adrian
 
Does anybody here know ways to define ordered sets or wish to discuss possible ways of defining ordered set?

Jim Adrian
 
Well, the Peano Postulates define an ordering plus provide the basis for numerous other theorems about numbers.

My suggestion is that you prove the eight Peano postulates as stated in wikipedia using your definitions and no axioms. If you can do that, I at least will grant that you have something significant to say.
 
jamesadrian said:
Does anybody here know ways to define ordered sets or wish to discuss possible ways of defining ordered set?
Click to expand...
Here is one standard axiomatic approach to the idea of order in the real numbers, \(\Re\).
Postulate the existence of a set \(\mathcal{P}\subset\Re\) having these two properties:
1) If \(x\in\Re\) then exactly one of the following is true:
_________ii) \(x\in\mathcal{P}\)________ii) \(-x\in\mathcal{P}\)________iii) \(x=0\).
2) The set \(\mathcal{P}\) is closed with respect to the basic operations of addition and multiplication.

Using the set \(\mathcal{P}\) we define \(a<b\) if and only if \(b-a\in\mathcal{P}\).

We would have before this point gone through the field axioms, proved that \(1\ne 0\) the two identities.
So that eliminates \(iii\) of the order axiom, leaving \(1\in\mathcal{P}\text{ or }-1\in\mathcal{P}\).
Suppose that \(-1\in\mathcal{P}\) But by the second requirement we have closure, so \(1=(-1)(-1)\in\mathcal{P}\).
We cannot have both. Thus \(1\in\mathcal{P}\) Therefore \(1\in\mathcal{P}\)
From that we get \(1-0=1\in\mathcal{P}\) meaning \(1>0\).
 
jamesadrian said:
Does anybody here know ways to define ordered sets or wish to discuss possible ways of defining ordered set?

Jim Adrian
Click to expand...
It seems to me that you already have the answer to your questions in mind. If you do then yes, you are a troll since you are clearly not trying to communicate but teach us your methods and only want to hand out tiny little facts that you propose. If you are indeed not a troll please stop doing this. Just tell us what you want to tell us!

My opinion as a Physicist is that, whereas no measurement has actually come up "infinite in size," infinity is a concept outside of Physics. Now, the concept does creep in because Physics relies heavily on Mathematics. But please be aware that infinity is a concept from outside of Physics. (I'm not trying to bash Mathematics, I'm just making a point.)

I really don't understand how you can "derive" Mathematics using no axioms or postulates. How do you even start?

-Dan
 
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