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

JeffM,

Thank you for your message.

I am very certain that the results of Peano's axioms can be realized by a collection of definitions that assumes nothing. I am working on that section.

A collection of definitions may be long and tedious and yet have logical flaws.

This is what I am currently contemplating:

A number is a name for an amount or a quantity; therefore, a name for a quantity of things is a number.

A thing may have more than one name.

"One" thing is a single thing because "one" is a name associated with a single thing (previous definition).

A set containing exactly a single element contains one element.

S is a set containing one element.

C is a non-empty set of elements.

U is the union of S and C.

Q is the name of the quantity of elements in U.

Set R contains one more element than set P if and only if set K contains one element and R is the union of set K and set P.

Please allow me a little more time. Also, please consider evaluating the statements above. I am here for feedback.

Very sincerely,


Jim Adrian
 
topsquark said:
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
Click to expand...


Dan,

I am very sincere about this. I have been working on this for years. Currently, I have been comparing several ways to define ordered sets. When I invited discussion about that, I was really looking for input.

Since then, JeffM pointed to Peano's axioms. I fully expect to realize the results of Peano's axioms and I posted a provisional approach to the matter. I amassing for feedback and ideas. Questioning my motives is an unfortunate distraction. Defining mathematical meanings from well-establish empirical terms is real.

I assure you that there is no duplicity here.


Jim Adrian
 
jamesadrian said:
Dan,

I am very sincere about this. I have been working on this for years. Currently, I have been comparing several ways to define ordered sets. When I invited discussion about that, I was really looking for input.

Since then, JeffM pointed to Peano's axioms. I fully expect to realize the results of Peano's axioms and I posted a provisional approach to the matter. I amassing for feedback and ideas. Questioning my motives is an unfortunate distraction. Defining mathematical meanings from well-establish empirical terms is real.

I assure you that there is no duplicity here.


Jim Adrian
Click to expand...
With your new system:

  • Have you proven any conjecture that the old system failed to prove?

  • Have you disproven any conjecture that the old system proved to be true?

If you succeeded any any of these, please contact the mathematics department of Harvard or Cambridge or Oxford university.
 
jamesadrian said:
JeffM,

Thank you for your message.

I am very certain that the results of Peano's axioms can be realized by a collection of definitions that assumes nothing. I am working on that section.

A collection of definitions may be long and tedious and yet have logical flaws.

This is what I am currently contemplating:

A number is a name for an amount or a quantity; therefore, a name for a quantity of things is a number.

A thing may have more than one name.

"One" thing is a single thing because "one" is a name associated with a single thing (previous definition).

A set containing exactly a single element contains one element.

S is a set containing one element.

C is a non-empty set of elements.

U is the union of S and C.

Q is the name of the quantity of elements in U.

Set R contains one more element than set P if and only if set K contains one element and R is the union of set K and set P.

Please allow me a little more time. Also, please consider evaluating the statements above. I am here for feedback.

Very sincerely,


Jim Adrian
Click to expand...
Several points.

Subhotosh Khan has refined the point of my previous post. The Peano postulates are important because they are the basis for proving the arithmetic of the natural numbers. If you can avoid those postulates by using a set of definitions and still prove the validity of the arithmetic of the natural numbers, you will demonstrate the power of your definitions. So I agree with your response that, to show the power of your definitions, it is sufficient to show that you can replicate all the results derived from the Peano postulates without proving any of the postulates themselves. But Subhotosh Khan is correct that if your definitions are more difficult to work with and prove nothing more than the Peano postulates, then you have failed to prove the utility of your definitions.

I am a retired banker with academic training in western history and languages who tries to help kids understand basic mathematics. I may be of no help in critiquing a set of definitions to replace the axioms of set theory. But I look at your definitions, and at least some seem to me to be semantic tricks. For example, you use the undefined term “union of sets.” So you seem to be sneaking traditional set theory with its axioms into what is supposed to an axiom-free development of set theory.

This may be nit picking, but I do not believe that numbers are names. Of course numbers have names, but numbers to me are ideas: mental perceptions of similarities between things real or imagined.

Finally, I go back to my original post. I am not sure that there is much difference between axioms and definitions: they are starting points that are to be accepted. Would it make much difference whether we call the notion of a successor a definition or an axiom? I suspect that your problem is that you perceive axioms to mean unjustifiable assertions. In fact, the historical roots of the axioms behind basic mathematics almost always have an inductive basis. The axioms are justified by experience, but they are not justified by deductive reasoning. Furthermore, axioms are frequently theorems given without proof. If I understand correctly, that is what Bourbaki was all about.
 
JeffM,

Thank you for your very constructive comments.

Before commenting on all of it, I have a question about the union of two sets:

According to https://www.math-only-math.com/union-of-sets.html

The union of two given sets is the smallest set which contains all the elements of both the sets.

My post this morning must have used the term in some way incorrectly. The definition I gave earlier was this:


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.

I might need to include that it is the smallest such set.

Please let me know what I did wrong.

Thank you for your help.


Jim Adrian
 
JeffM,

Perhaps there is an explanation. I previously defined a number of things. These definitions having to do with "sets" as I have defined them were in earlier posts:

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.

If Kq is a name, K may be said to be subscripted; and, K may be said to be subscripted by q.

If Kq is a name, Kq may be said to be a name formed by subscripting.

If Kq is a name formed by subscripting, the q is a subscript.
-----------------------------------------------------------------------


Jim Adrian
 
JeffM,

I believe that the definitions I just posted do not need the axioms of traditional set theory. They just need the undefined terms that are part of the common language that are too basic to be defined by more basic terms and cannot be defined without circularity. Of course, some of the definitions I have written use terms that I have defined earlier.

If there is something wrong with my use of the term "union" I do want to know about it.

Thank you for your help.


Jim Adrian
 
JeffM,

I was interested to read from you post that you to help kids understand basic mathematics. You might be interested in an article I wrote about mnemonics. It has made a dramatic difference in the careers of kids for the last 30 years. A couple of sisters who were not conspicuously outstanding students and who were entering high school though it was all sorts of fun to work on it together. I learned many years later that they both graduated from Vasar with honors and didn't pay to go there.

Future Beacon Memory Systems

Human memory, recall, mnemoics, memory systems
www.futurebeacon.com

Jim Adrian
 
I was merely looking at the list of definitions in that one post as you subsequently realized. My reaction to "union" was that there was no definition on that post.

I do not view myself as competent to determine whether a set of definitions is adequate for set theory. I am also nervous about relying on meanings from any natural language, each of which has many words with a broad field of meaning, which leads to a failure to generate rigorous arguments.

I will say this, however. I do not see the advantage of something that requires dozens of definitions just to get started if you can do all the work with a much smaller number of definitions and axioms. My suggestion is to get a good book on set theory and a good book on mathematical logic (which I am sure someone here can advise on) and see what theorems are developed in those books. You will need to prove them using your definitions. That is, if you cannot prove what has already been proved and is considered basic, no one is going to be interested. I admit that you do not have to prove someone else's axioms if you can achieve the results obtained by those axioms another way.

I must admit I do not see how you are going to prove something like

[MATH]a,\ b \in \mathbb N_{\ge 0} \implies a + b \in \mathbb N_{\ge0} \text { and } a + b = b + a \in \mathbb N_{\ge 0}[/MATH]
simply through definitions. But my failure to see it does not mean that it cannot be done. My wife tells me that I see almost nothing. But you cannot simply assert that it can be done. You have to do it.
 
JeffM,

The frame of mind I have fallen into is very different from the one I needed and employed as a math student.

At one point, wondering about axioms, I went to the Philosophy Department to find a logician. I met up with one who answered the question "What if an axiom is factually untrue?" He said that it would then be possible (if anybody really tried) to both prove and disprove every allowable statement in that system. I then went and talked with the Chairman of the Mathematics Department and asked the same question. He said that mathematicians reserve the right to explore any set of axioms to see where they lead.

During the following year I wrote a lot about the foundations of math. I was an older student who became friends with a math professor who was my age (about 27). Eventually, he told me that if I went around giving lectures about my approach to math, I would have a miserable life. I believed him. I went into microprocessor design.

I'm retired now. I believe that engineers and programmers will be more interested than mathematicians the math I want to help bring into existence. Trying out another way to compute is not of interest to living mathematicians. They have been introduced to math as a deep mental identity. Alternatives to the heroes they have are unthinkable for many of them, especially to professors. Their investment is too high.

I am trying to show that people have knowledge of words that have quantitative meanings that they use without mistake or flaw, such as a single thing or a pair of things. Terms such as these are too basic to be defined by more basic terms and cannot be defined without circularity. When they are used in a definition of a mathematical idea that goes further, such a definition can be logically correct and useful in the construction of numbers, calculus, and complex numbers. I know this for a fact. My current version of the effort is motivated by an attempt to make the whole thing more clear. Had hoped that somebody would want to join me. I think it is more clear if you start without a degree or a psychological investment in formal math. Scientific definitions can do the whole job. You are right in saying that it is up to me. I am not likely to find any qualified partners in this perceived crime against axioms.

The lost of infinity is no lost to engineers. Unbounded sets are all they ever needed to make it to the moon.


Jim Adrian
 
JeffM,

Many mathematicians have insisted that a number is much more than a name of an amount or a quantity of things.

A system that does not use axioms and does not subscribe to any beliefs cannot be held to this idea. You need to see whether the definition allows the system of definitions to produce a means of calculating. If it does, this definition of the term number is part of a system of calculating that does not diminish or endorse anything about axiomatic systems. Saying that numbers are more than an amount or quantity does not support or diminish or even mean anything in a system that does not use axioms.

Throughout my 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 referred to as a thing.


Jim Adrian
 
Following the set-related definitions and the definition of one, I have added this definition:

Definition - Set R contains one more element than set P contains if and only if K is a set; and, K contains exactly one element; and; R is the union of set K and set P.

I would appreciate any criticism of this definition.


Jim Adrian
 
JeffM,

Here is the very start of definitions:

Definition - A term is a word or a phrase.

Definition - A name is a term that refers to a thing.

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

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

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. The term collection is synonymous with the term collection of things, where things are of any description. 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.


These definitions are preceded by a list of undefined terms. Because the undefined term collection has more than one widely understood meaning, one of those meanings are is chosen and identified when the term is first used in a definition.

There is a document online that contains what seems to be settled. It is searched for typos and appended every day:


Much more has been written, but this file contains only what looks final to me.


Jim Adrian
 
JeffM,

The title is not meant to imply that other kinds of math are not rational. As these first two paragraphs explain, the title comes from the fact that rational numbers play a central role in the development:


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.


Jim Adrian
 
Unfortunately, there is a 30 minute time limit for replacing an are with an is. Might it be changed to a day?

Jim Adrian
 
First, I will not argue about definitions. Anyone is free to make up definitions any way they see fit, provided that the definition is used consistently. Whether a set of definitions are useful is a different question entirely.

Second, I do not believe in the physical reality of real numbers, complex numbers, continuity, etc. They are purely creatures of the human imagination that turn out to be highly useful. I find standard analysis to be ugly, and it turned me off math for years. I believe Platonism and Kantianism are nonsense on stilts.

Third, none of this has anything to do with your assertion that mathematics can be derived without assumptions. You want to base mathematics on rational numbers. Prove the facts of arithmetic relating to rational numbers without any assumptions. Mathematicians like proof rather than rhetorical games based on negative connotations associated with the word “assumption.”
 
Last edited:
jamesadrian said:
Unfortunately, there is a 30 minute time limit for replacing an are with an is. Might it be changed to a day?

Jim Adrian
Click to expand...
The moderators will make late edits upon request if they deem them material.
 
jamesadrian said:
Unfortunately, there is a 30 minute time limit for replacing an are with an is. Might it be changed to a day?

Jim Adrian
Click to expand...
I have the power to edit (undefined power). Please coy the response # and exactly what needs edited - pin-pointing the location of the offending "word".
 
Subhotosh Khan said:
I have the power to edit (undefined power). Please coy the response # and exactly what needs edited - pin-pointing the location of the offending "word".
Click to expand...
JeffM,


Thank you for your help. The word are below is the one that should be is,
---------------------------------------------------
Here is the very start of definitions:

Definition - A term is a word or a phrase.

Definition - A name is a term that refers to a thing.

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

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

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. The term collection is synonymous with the term collection of things, where things are of any description. 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.


These definitions are preceded by a list of undefined terms. Because the undefined term collection has more than one widely understood meaning, one of those meanings [ are ] chosen and identified when the term is first used in a definition.

There is a document online that contains what seems to be settled. It is searched for typos and appended every day:

Rational Mathematics

Much more has been written, but this file contains only what looks final to me.


Jim Adrian
 
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