Let's have a set A = {φ , {φ} , {{φ}},....} surely |A| = ∞ ,
Now →
1. A ∈ A .
2. A is countable
A could be found with this recursive Grammar →
A = φ | 2^A
Russel gave one paradox like → A = {x | x ∉ x} now does A∈ A?
If A ∈ A then contradict the property of A if A ∉ A it should be inside A by set building property of A .
So when to build such set operation we have to design the Axioms behind the set algebra such that is can't enter to a paradox . If an algebra enter in a paradox it is not well defined .
Sometime we use prove by contradiction . To prove A = 1 (true) we prove !A = 0 (false) . But it does not give us always good result . !A = 0 means A = 1 or A is a paradox . We must have to show that A = 1 in direct method , induction or by construction .
To define anything or to build some logical universe We must follow the exclusion principle . That is we will not include anything in our logical universe which is not necessary to build our universe under the Given observations / tautology .
Now →
1. A ∈ A .
2. A is countable
A could be found with this recursive Grammar →
A = φ | 2^A
Russel gave one paradox like → A = {x | x ∉ x} now does A∈ A?
If A ∈ A then contradict the property of A if A ∉ A it should be inside A by set building property of A .
So when to build such set operation we have to design the Axioms behind the set algebra such that is can't enter to a paradox . If an algebra enter in a paradox it is not well defined .
Sometime we use prove by contradiction . To prove A = 1 (true) we prove !A = 0 (false) . But it does not give us always good result . !A = 0 means A = 1 or A is a paradox . We must have to show that A = 1 in direct method , induction or by construction .
To define anything or to build some logical universe We must follow the exclusion principle . That is we will not include anything in our logical universe which is not necessary to build our universe under the Given observations / tautology .
No comments:
Post a Comment