One to One

Transfinite theory employs the term "one-to-one correspondence," which is similar, but slightly different, from the term "one-to-one" relation used in the database industry. Since the terms are similar, there is a danger of confusion when employing the terms in the wrong context.

When a computer programmer talks about a one-to-one or one-to-many relation, she is referring to a defined relation between data sets. For example, a corporate database might have two tables called Company and Employee. An employee belongs to at most one company. A company can have more than one employee. 

In this case, there is a defined one-to-many relation between the Company and Employee tables. The figure to the right is called an Entity Relational Diagram (ERD). The boxes show that there are two tables. The first table is called Company. It has the primary key company_id. The company id uniquely identifies a row in the table.

The second box shows the employee table. The company_id column in this table denotes the company to which the employee is indentured. The line with the "crows feet" shows the cardinality of the relation.

This defined relation is independent of the number of rows in either table, and will continue to exist as the tables grow without bound. Since the relation is independent of the size of the sets, I believe it is safe to say that the defined relation would be preserved as the sets approach infinity. 

Transfinite theorists have a slightly different meaning when they speak of a one-to-one correspondence. The transfinite theorist is not interested in defined relations between finite sets, but in the abstract properties of the sets. If a transfinite theorist were given an infinite set of companies, and an infinite set of employees, they would ignore the defined relation, and look to see if they can establish any ordering of the sets that are equivalent. 

The transfinite theorist would say that if you began listing all companies and all employees, you would not run out of companies before employees; therefore the sets are the same size, and there is a one-to-one correspondence between the sets.

As our database grows to infinity, the database programmer would continue to say that there is a defined relation between the tables, while the transfinite theorist would say that there is a one to one correspondence between the tables. You can see the differences in these views in the following tables. 

Data Base View
One to One Relation
Company Employee
ID Company Name ID Employee Name
0 Apple 0
Steve Jobs
1 Microsoft 2
Bill Gate
Paul Allen
Jim Gray
2 Oracle 5 Larry Ellison
  ...   ...
Transfinite View
One to One Correspondence
Company Employee
ID Company Name ID Employee Name
0 Apple 0 Steve Jobs
1 Microsoft 1 Wozniac
2 Oracle 2 Bill Gates
3 IBM 3 Paul Allen
4 AOL 4 Jim Gray
5 McDonalds 5 Larry Ellison
  ...   ...
  FROM Company c,
       Employee e
 WHERE c.company_id = e.company_id
  FROM Company c
       Employee e
 WHERE c.company_id = e.employee_id

If you look at the SQL for the two different views, you will see that the database programmer joins the Company.compan_id to the Employee.company_id, while the transfinite theorist joins the Company.company_id to the Employee.employee_id. This gives us two radically different views of the data. As it happens, Bill Gates does not work for Oracle, and Larry Ellison does not work as a hamburger flipper for Micky Ds. ... well at least not yet.

I find this cross definition between database theory and transfinite theory quite confusing. The that one to many relations will turn into one-to-one correspondenses is the most counter intuitive aspect of theory. We will look at the problem again in the next essay on atoms.

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