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

jamesadrian said:
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
Click to expand...
Done
 
jamesadrian said:
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
Click to expand...

This is not a good definition because it does not assure that the single element in E in K is not in P. This one is better:

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, E; and E is not in P, and; R is the union of set K and set P.

I would appreciate any criticism of this definition.


Jim Adrian
 
jamesadrian said:
This is not a good definition because it does not assure that the single element in E in K is not in P. This one is better:

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, E; and E is not in P, and; R is the union of set K and set P.

I would appreciate any criticism of this definition.


Jim Adrian
Click to expand...
Most of this conversation is above my pay grade but your definition does not give a relationship between R, P, and K. I think you mean [math]P = R + K[/math], where [math]E \notin R[/math] and if we have K = E as a singleton set then [math]|R| = |P| + |E| = |P| + 1[/math].

However I don't believe that this statement is true if P is not finite.

-Dan
 
Dan,

You are absolutely right that the definition is not true if P is not finite.

I have not yet defined the operation of addition. I have thrown out this definition.

Here is a new one for your consideration:

Definition - M is the successor of N if and only if the following statements are true:

P is a set of elements.

K is a set containing one element.

The intersection of K and P is empty.

N is the number is element in P.

R is the union of P and K.

M is the number of elements in R.
-----------------------------------------


Jim Adrian
 
Definition - A is equal to B or A equals B if and only if either of the following statements is true:

A describes or names a quantity, and B describes or names the same quantity.

A describes or names a set of quantities, and B describes or names the same set of quantities.
-------------------------------------------------------------------------------------------------------------


Jim Adrian
 
Definition - A number is the name of an amount or a quatantity.

In axiomatic systems, a number is said to be much more than a name; but here, this term facilitates computation without including other features in its definition.


Jim Adrian
 
This includes a set that is empty.

Definition - M is the successor of N if and only if the following statements are true:

P is a set and P is empty; or, P is a set of elements.

K is a set containing one element.

The intersection of K and P is empty.

N is the number of elements in P.

R is the union of P and K.

M is the number of elements in R.
-----------------------------------------


Jim Adrian
 
Definition - Set S is a set of whole numbers if and only if each of the following statements is true:

S contains the number zero.

Every element in S is a number.

Each number is S other than zero is the successor of some number in S.

If M is in S, and M is the successor of N in S, then M is not the successor of any other number in S.
------------------------------------------------------------------------------------------------------------------

Jim Adrian
 
The next to last statement above should read "Each number in S other than zero is the successor of some number in S."

Jim Adrian
 
Definition - The sum of whole number j and whole number k is s if and only if the following statements are true:

J is a set; and, j is the number of elements in J.

K is a set: and, k is the number of elements in K.

The intersection of J and K is empty.

S is a set; and, S is the union of J and K; and, s is the number of elements in S.
------------------------------------------------------------------------

The sum of whole number j and whole number k may be written

j + k.

The sum of whole number j and whole number k is s may be written

s = j + k.


Jim Adrian
 
I would appreciate any criticism of any of the last seven or so definitions. They will form the basis of the rational numbers unless they are flawed.

Each of the defined terms used in the description section of each of these definitions has been previously established in this document:



Jim Adrian
 
Definition - A one-to-one correspondence is a procedure specified by the following labeled statements:

Start - A pair of sets, j and k, each contain things, while another pair of sets, p and q, are empty; and, actions begin with Action S.

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

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

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

When this procedure stops performing actions, it is known that set p and set q contain the same quantity of things; and, it is said that sets p and set q have a one-to-one correspondence.

Definition - N is the number of elements in S if and only if the follloiing statements are true:

Q is a set of whole numbers; and, N is in Q.

S is a proper subset of Q such that every element other than zero in Q is in S.

K is a set; and, the elements of K and S have a one-to-one correspondence.

N is the single element in Q that is not the successor of any element in Q.
------------------------------------------------------------------

N is the last element in Q. I'm sure that most of you are accustomed to infinite sets. Here, there are only finite sets defined. This system will provide the utility of unending sets or arbitrarily large sets by means of of unbounded sets. Unbounded sets depend on finite sets being defined previous to them.


Jim Adrian
 
JeffM,

Your messages to me have been constructive. You have not been rude or dismissive. I would value your suggestions as to what I should define next or soon. Please tell me about anything that seems not right.

Thank you for your help.


Jim Adrian
 
I have been looking at this on and off. Thinking about how I can respond.

First, I do not know how to critique a definition. Definitions are free creations of the human mind. If I understood better where you are trying to go, I might be able to give an opinion on whether I think these definitions are likely to be helpful. My knowledge of set theory and the foundations of mathematics is sketchy so whether my opinions are helpful is a different question.

Second, I am quite uncertain which definitions you want considered. You have given a lot and revised some of them. "Last seven or so" is unfortunately a bit vague.

Third, it seems to me that some of your definitions are simply equating words and symbols. For example, somewhere you have a definition on what a = b means, and the definition merely states an equivalence between the symbol = and the words "is equal to" and "equals." Great. You are making your use of language clear, but there is no substantive content, nothing that helps lead to something else. You could dispense with that definition entirely, and just say that the meaning of "the quantity A equals the quantity B" is one of those simple things that cannot be defined in terms of simpler things; it can only be exemplified.

Fourth, my son, who's a software engineer, has made a hand-waving argument that everything can be reduced to algorithms. Possibly. It sounds at least superficially plausible. But the moment you recognize that you have to start with some undefined things, whether you use them in definitions or axioms or postulates or algorithms seems to be arguing about names rather than substance. This is why I suggested looking at the Peano postulates. Here they are (quoting wiki in condensed form)

1. 0 is a natural number.
2. For every natural number x, x = x.
3. For all natural numbers x and y, if x = y, then y = x.
4. For all natural numbers x, y and z, if x = y and y = z, then x = z.
5. For all a and b, if b is a natural number and a = b, then a is also a natural number.

The naturals are assumed to be closed under a single-valued "successor" function S.
6. For every natural number n, S(n) is a natural number.
7. For all natural numbers m and n, m = n if and only if S(m) = S(n).
8. For every natural number n, S(n) = 0 is false.

Peano's original formulation of the axioms used 1 instead of 0 as the "first" natural number. This choice is arbitrary, as these axioms do not endow the constant 0 with any additional properties. However, because 0 is the additive identity in arithmetic, most modern formulations of the Peano axioms start from 0.

Axioms 1, 6, 7, 8 define a unary representation of the intuitive notion of natural numbers: the number 1 can be defined as S(0), 2 as S(S(0)), etc. However, considering the notion of natural numbers as being defined by these axioms, axioms 1, 6, 7, 8 do not imply that the successor function generates all the natural numbers different from 0. Put differently, they do not guarantee that every natural number other than zero must succeed some other natural number.

The intuitive notion that each natural number can be obtained by applying successor sufficiently often to zero requires an additional axiom, which is sometimes called the axiom of induction.

9. If K is a set such that:
0 is in K, and for every natural number n, n being in K implies that S(n) is in K,
then K contains every natural number.
The induction axiom is sometimes stated in the following form:

The Peano axioms can be augmented with the operations of addition and multiplication and the usual total (linear) ordering on N. The respective functions and relations are constructed in set theory or second-order logic, and can be shown to be unique using the Peano axioms.

OK Me again. Quite frankly, whether we call 1 through 8 definitions or axioms seems to me to be completely inconsequential: a rose by any other name .... You can even state 9 as a definition. If you can come up with definitions that replicate Peano, you're in business.
 
jamesadrian said:
Definition - A one-to-one correspondence is a procedure specified by the following labeled statements:

Start - A pair of sets, j and k, each contain things, while another pair of sets, p and q, are empty; and, actions begin with Action S.

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

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

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

When this procedure stops performing actions, it is known that set p and set q contain the same quantity of things; and, it is said that sets p and set q have a one-to-one correspondence.

Definition - N is the number of elements in S if and only if the follloiing statements are true:

Q is a set of whole numbers; and, N is in Q.

S is a proper subset of Q such that every element other than zero in Q is in S.

K is a set; and, the elements of K and S have a one-to-one correspondence.

N is the single element in Q that is not the successor of any element in Q.
------------------------------------------------------------------

N is the last element in Q. I'm sure that most of you are accustomed to infinite sets. Here, there are only finite sets defined. This system will provide the utility of unending sets or arbitrarily large sets by means of of unbounded sets. Unbounded sets depend on finite sets being defined previous to them.
Jim Adrian
Click to expand...

The last statement in the definition is wrong. This is the correction:

Definition - N is the number of elements in S if and only if the following statements are true:

Q is a set of whole numbers; and, N is in Q.

S is a proper subset of Q such that every element other than zero in Q is in S.

K is a set; and, the elements of K and S have a one-to-one correspondence.

N is the single element in Q that has no successor in Q. **********
------------------------------------------------------------------


Jim Adrian
 
JeffM said:
I have been looking at this on and off. Thinking about how I can respond.

First, I do not know how to critique a definition. Definitions are free creations of the human mind. If I understood better where you are trying to go, I might be able to give an opinion on whether I think these definitions are likely to be helpful. My knowledge of set theory and the foundations of mathematics is sketchy so whether my opinions are helpful is a different question.

Second, I am quite uncertain which definitions you want considered. You have given a lot and revised some of them. "Last seven or so" is unfortunately a bit vague.

Third, it seems to me that some of your definitions are simply equating words and symbols. For example, somewhere you have a definition on what a = b means, and the definition merely states an equivalence between the symbol = and the words "is equal to" and "equals." Great. You are making your use of language clear, but there is no substantive content, nothing that helps lead to something else. You could dispense with that definition entirely, and just say that the meaning of "the quantity A equals the quantity B" is one of those simple things that cannot be defined in terms of simpler things; it can only be exemplified.

Fourth, my son, who's a software engineer, has made a hand-waving argument that everything can be reduced to algorithms. Possibly. It sounds at least superficially plausible. But the moment you recognize that you have to start with some undefined things, whether you use them in definitions or axioms or postulates or algorithms seems to be arguing about names rather than substance. This is why I suggested looking at the Peano postulates. Here they are (quoting wiki in condensed form)

1. 0 is a natural number.
2. For every natural number x, x = x.
3. For all natural numbers x and y, if x = y, then y = x.
4. For all natural numbers x, y and z, if x = y and y = z, then x = z.
5. For all a and b, if b is a natural number and a = b, then a is also a natural number.

The naturals are assumed to be closed under a single-valued "successor" function S.
6. For every natural number n, S(n) is a natural number.
7. For all natural numbers m and n, m = n if and only if S(m) = S(n).
8. For every natural number n, S(n) = 0 is false.

Peano's original formulation of the axioms used 1 instead of 0 as the "first" natural number. This choice is arbitrary, as these axioms do not endow the constant 0 with any additional properties. However, because 0 is the additive identity in arithmetic, most modern formulations of the Peano axioms start from 0.

Axioms 1, 6, 7, 8 define a unary representation of the intuitive notion of natural numbers: the number 1 can be defined as S(0), 2 as S(S(0)), etc. However, considering the notion of natural numbers as being defined by these axioms, axioms 1, 6, 7, 8 do not imply that the successor function generates all the natural numbers different from 0. Put differently, they do not guarantee that every natural number other than zero must succeed some other natural number.

The intuitive notion that each natural number can be obtained by applying successor sufficiently often to zero requires an additional axiom, which is sometimes called the axiom of induction.

9. If K is a set such that:
0 is in K, and for every natural number n, n being in K implies that S(n) is in K,
then K contains every natural number.
The induction axiom is sometimes stated in the following form:

The Peano axioms can be augmented with the operations of addition and multiplication and the usual total (linear) ordering on N. The respective functions and relations are constructed in set theory or second-order logic, and can be shown to be unique using the Peano axioms.

OK Me again. Quite frankly, whether we call 1 through 8 definitions or axioms seems to me to be completely inconsequential: a rose by any other name .... You can even state 9 as a definition. If you can come up with definitions that replicate Peano, you're in business.
Click to expand...


JeffM,

Thank you very much for this response.

The root of the problem is that formal definitions have requirements that must first be fully understood in order to make sense of a system of definitions.

1. A scientific definition must name the term being defined and describe the meaning of that term without reference to the term itself.

2. There are terms in language that represent perceptions shared by the vast majority of people. The term thing is an example. These words cannot be assigned a formal definition. Attempts to do so are always circular. I provide a list early in the online article.

3. Some agreed words that are undefinable have more than one popular meaning. In such a case, they cannot be used in a formal definition without first identifying which meaning is to be used in definitions.

When you can detect that a definition is not a correct formal definition adhering to these features, the definitions I write will make more sense.

Of course, the axiom of induction is not a definition and cannot be included in the development that I am attempting.

I was a programmer myself. We can reduce all interesting math to procedures, if we want to, but only after the terms we use to define the procedures are themselves either well defined or of the kind that I have described above that cannot be assigned a formal definition.

It is possible to give a person enough empirical experience with the verbiage of other programmers in an intended programming language so that he eventually knows what it all means. We often see this assumption in bad books on programming. Definitions formed from fewer and more common terms is faster.

We would have seen more procedures in formal math if there had not been a prohibition on the part of most mathematicians against using time in mathematical reasoning; although this is going away. I mention this and include the words before and after and similar terms as being among the terms that cannot be assigned a formal definition.

I wish to avoid any axioms because they are assumptions. I am offering a system of definitions as an alternative to axiomatic math, and not as a world-wide replacement for it. It is a choice. Some people, particularly in engineering, may find it more to their liking. It is obvious that most well-trained mathematicians dislike it. They have an actual reverence for axioms. More people would like it if both approaches were available in their youth.

I have not defined arithmetic yet. I certainly will.

For this project, the set of undefined terms that I use is the full extent of human intuition. No statements can be assumed.

If you try your hand at writing formal definitions, eventually, you will find them a useful addition to your modes of expression and thinking.

I hope this is clarifying.

Be well.


Jim Adrian
 
This was incorrect. This is the correction:

Definition - N is the number of elements in S if and only if the following statements are true:

Q is a set of whole numbers; and, N is the single element in Q that has no successor in Q.

K is a set of whole numbers.

K is a proper subset of Q such that every element other than zero in Q is in K.

S is a set; and, the elements of set K and set S have a one-to-one correspondence.
------------------------------------------------------------------------------------------------


Jim Adrian
 
Last edited:
jamesadrian said:
JeffM,

Thank you very much for this response.

The root of the problem is that formal definitions have requirements that must first be fully understood in order to make sense of a system of definitions.

1. A scientific definition must name the term being defined and describe the meaning of that term without reference to the term itself.

2. There are terms in language that represent perceptions shared by the vast majority of people. The term thing is an example. These words cannot be assigned a formal definition. Attempts to do so are always circular. I provide a list early in the online article.

3. Some agreed words that are undefinable have more than one popular meaning. In such a case, they cannot be used in a formal definition without first identifying which meaning is to be used in definitions.

When you can detect that a definition is not a correct formal definition adhering to these features, the definitions I write will make more sense.

Of course, the axiom of induction is not a definition and cannot be included in the development that I am attempting.

I was a programmer myself. We can reduce all interesting math to procedures, if we want to, but only after the terms we use to define the procedures are themselves either well defined or of the kind that I have described above that cannot be assigned a formal definition.

It is possible to give a person enough empirical experience with the verbiage of other programmers in an intended programming language so that he eventually knows what it all means. We often see this assumption in bad books on programming. Definitions formed from fewer and more common terms is faster.

We would have seen more procedures in formal math if there had not been a prohibition on the part of most mathematicians against using time in mathematical reasoning; although this is going away. I mention this and include the words before and after and similar terms as being among the terms that cannot be assigned a formal definition.

I wish to avoid any axioms because they are assumptions. I am offering a system of definitions as an alternative to axiomatic math, and not as a world-wide replacement for it. It is a choice. Some people, particularly in engineering, may find it more to their liking. It is obvious that most well-trained mathematicians dislike it. They have an actual reverence for axioms. More people would like it if both approaches were available in their youth.

I have not defined arithmetic yet. I certainly will.

For this project, the set of undefined terms that I use is the full extent of human intuition. No statements can be assumed.

If you try your hand at writing formal definitions, eventually, you will find them a useful addition to your modes of expression and thinking.

I hope this is clarifying.

Be well.


Jim Adrian
Click to expand...
But I absolutely can define Peano's induction axiom as a definition

[MATH]\text {The set of natural numbers } \mathbb N \text { is that set of numbers that contains 0 and that contains}\\ S(x) \text { if } \mathbb N \text { contains x.}[/MATH]I have turned what is traditionally called an axiom into a definition. You really should not casually dismiss other people's comments if you expect others to try to take yours seriously. Moreover, the moment you begin to talk about "intuitive" and "scientific" in the same text, there are people like me who are going to wonder whether you are not talking utter balderdash. I greatly doubt that you took seriously one of the first things I said: namely the basic axioms of mathematics are inductive conclusions from experience.
 
JeffM,

I'm sorry that you feel I casually dismissed your comments. My answer, which was rather long, reflected what I thought was most important. The frame of mind is very different. It is hard to estimate how different. I don't know what you mean about referring to "intuitive" and "scientific" in the same context. The definition of the whole numbers and the sum of two whole numbers that I have written have been posted. They don't look much like Peano's Axioms, but I would like to see all of those axioms converted into definitions. Do you think that other mathematicians would find them desirable? Whether they would or not, I would applaud the effort. The order of definitions is important. I don't understand "contains S(x) if the natural numbers contains x. I need at least a definition of the natural numbers first. The list of properties is not enough. Perhaps that why they are presented as axioms. Nothing I write is intended to slight you in any way. I appreciate having my attention drawn to something that I have ignored. The world is very complicated, as you know.

Be well.

Jim Adrian
 
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