Induction

[Guest]

I) Demonstration of the Principle of Non-Contradiction.

Let us imagine any set of perfect existence forms (hereafter called existence forms), with the largest natural number of forms which can occur in conjunction with a specific existence form - be these forms symmetric numbers or symmetric ideas - called W, and consider any subset of W, which contains the same specific form and call it A,

Principle of the smallest set of perfect existence forms containing a specific existence form

Supposing that the set W of forms of existence is bounded inferiorly, if given a subset A is such that it is a subset of W. Let us call "b" the smallest subset of A and "a" the smallest subset of W, where b>=a, that is a is the smallest or equal to the smallest subset of A.

Every formation a that belongs to W, which fulfils this condition is called the lower bound of A.

A lower bound of A, that belongs to W, that subset is called the minimum of A

If A is a subset of W, and bounded inferiorly, then A has a subset of existence forms with a specified existence form minimum.

First Principle of induction.

Let p(n) be an existence form that occurs on the sets of existence forms, greater than or equal to a subset a .



Suppose it is proved that.

-p(a) is true

- if k>a and p(k) is true, then p(k'), is also true.

Where k and k' are subsets of forms of existences , where K is a subset of k', different from and smaller than k'. And "a" is contained in "k", which in turn is contained in "k' "

Then p(n) is true for all n>a.

Being n the set that has the maximum number of existence forms, W . Since it is known that these forms of existence are limited in number
limited, since they occur in a limited space.

Demonstration.
A brief explanation of one of the properties of sets
Example:
Property: a set E is contained in a set R
We say E is in R, that is, every form of existence of E, is also in R

Let's suppose the second element of the sentence is false, that is, our W doesn't have at least one subset of A, and make an absurdity arise, that is, a contradiction

To do so, let us suppose

Let A in our example, and the set b that belongs to W, which is the aforementioned set, such that b>a and p(b) is false.

If we show that A does not exist, that is, it is empty (because as we said, the empty set does not exist), the principle of induction will be justified.
To do so, let us suppose that A is not empty.

Since A is bounded inferiorly, "a" is a lower bound, since a belongs to W and is <= b, and b belongs to A, so A has a minimal subset c . As we have seen p(a)is true, since a is <= b, so it cannot equal b, otherwise p(a) would be false, so the minimum c also cannot equal "a" ,since c belongs to A, it has to be > a, and then c'>=a, (remembering that c' is a subset of c, different and smaller than c). On the other hand, p(c') is true since it is also a lower bound of A, since c' can be =a and <b, so it is outside A, so taking into account the hypothesis that p(k') is true, since k=c' and p(k) is true if we call k'=c implies p(k')=p(c) is true. But this is absurd since c is in A. And every p(b) where b belongs to A is false.

Hence A is empty, and does not exist.

QED

Principle of non contradiction

If we take three subsets of W where a certain form of existence occurs, and we order them from smallest to largest, we can prove that necessarily, it will appear in the same form in any set which contains it

Let's take the subsets "e" , "x" and "r" of W, where "e" is the smallest subset, and such that "x">"e", "x" and "r" are subsets, such that "x" is different and smaller than "r". And "e" is contained in "x" and "x" is contained in "r".

We call p(n) a form of existence which occurs in a set of forms of existence greater than or equal to a subset "a".

And n>a

Now we call p(e)=p(a) where p(a) is true.

We suppose p(x)=p(k), where p(x) true,

Then p(r)=p(k') is also true.
Which is verified.
Then p(n) occurs in the same form in any set of existence forms in which it occurs , in other words the given form occurs in the same form from the smallest set that it appears to the largest "n", being. n = W, the last and most complete of these sets.
This is the demonstration of the principle of non-contradiction.
This form appears in the same form, i.e. its true form, and does not appear as false( other form) in any other set.

QED.