Some notions

[ShadmiDoron]

A person cannot be considered as a professional mathematician if he cannot distinguish between a mathematical object and its representation.

 

According to the Axiomatic Set theory, a non-empty Set is a collection of objects, where each object is clearly distinguish from the rest of the objects.

 

The property of being clearly distinguished exists in the foundations of the axioms that define any member on the real-line, no matter if it is a Natural, a Rational or an Irrational number.

 

In other words, there is an unbreakable connection between being clearly distinguished and the ability to define the quantity and location (order) that each member has along the real-line, no matter what representation method is used.

 

For example, by using the place value representation method we can represent the real-line's member PI in infinitely many ways that are different from each other by the their base value. No one of these representations is the mathematical object PI.

 

If we wish to prove something about the mathematical objects, we have to use an extra care not to mix up between them and any possible representation of them.

 

By using this insight, let us re-examine the famous Cantor's second diagonal argument.

 

As a fundamental approach, we have to use the minimal condition that exists in some collection of mathematical objects that is considered as a non-empty set. This minimal condition is exactly the property of each mathematical object, which enables to find its exact quantity and location (order) along the real-line.

 

In that case we actually can represent each real-line's member by a finite and unique string of symbols (as we did in the case of PI).

 

In that case (when using only the minimal condition as mentioned above) Cantor's second diagonal argument does not hold.

 

Furthermore, the necessity to use infinitely many symbols in order to represent a member like PI, actually tells us that we use an inaccurate representation method for some irrational number (the reason is very simple, because each base value in the place value method is a natural number > 1, and no irrational number can be represented by a finite string that is the result of a ratio between natural numbers).

 

Cantor ignored these simple facts, and as a result its argument is based on a representation method and not on the mathematical objects themselves, in this case.

 

Some claims that the non-repeated pattern of infinitely-many symbols is an inherent property of any irrational number. In that case the exact value of an irrational number cannot be satisfied, which contradicts the basic notion of the Set concept (every set's member is A priori clearly distinguished from the rest of the members).

 

When defining the membership relations between non-empty sets, we have to understand each definition that is used in some argument, before we use it.

 

For example, If a proper sub-set is any set that fully included in some set,

then each sub-set is a member of the set (which is trivial).

 

If we examine the mapping between a set and its proper sub-set, then if some member is written in the column that represents the proper sub-set's members, it must be written also in the column that represents the set's members.

 

Furthermore, since any number is not less then a quantity and an order along the real-line, we have no choice but to write down in the side that represents the set's members, every set's member, which is less or equal to the number that was written in the proper sub-set column. For example let us re-examine the mapping that exists between the set of Natural numbers and its proper subset of even numbers.

 

If the set of even numbers is a proper sub-set of the set of Natural numbers, then each even number is also a member of the set of Natural numbers.

 

In that case, if some even number is in the proper sub-set of the even numbers, it is also in the set of the Natural numbers, and we have no choice but to write down in the column that represents the Natural numbers, any single Natural number that is less or equal to the even number that was written in the side that represents the even numbers:

 

1 <--> 2

2 <-->

...

 

 

1 <--> 2

2 <--> 4

3 <-->

4 <-->

...

 

 

1 <--> 2

2 <--> 4

3 <--> 6

4 <-->

5 <-->

6 <-->

...

 

 

1 <--> 2

2 <--> 4

3 <--> 6

4 <--> 8

5 <--> 10

6 <--> 12

7 <-->

8 <-->

9 <-->

10 <-->

11 <-->

12 <-->

...

 

 

Since each Natural number is not less than a cardinal and an ordinal that is clearly distinguished from the rest of the Natural numbers, and since each member of a proper sub-set is also a member of the set, there cannot be a bijection between a set and its proper sub-set (as shown above).

 

In the case of the proper sub-set of even numbers and the set of Natural numbers, there exists a permanent ratio of 1 to 2 that prevents the bijection.

 

There can be a bijection between two non-finite sets only if they are disjoined in any recursion level.

 

Some claims that the property of a set is the result of the membership between members, so a set's property defined by its membership. But as I showed, there is a property which is not the result of the membership but it is actually its reason, and this reason is based on the A prioristic uniqueness of each set's member, which defines A priori the property of any non-empty set.