The set of all sets contradiction

Therefore, in summary, since there is no injection from P(A) to A, it can be concluded that |P(A)| > |A|.
  • #1
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Homework Statement



Let A be the set of all sets.

#1.) Show that P(A) is a subset of A.

#2.) Find a contradiction.


Homework Equations





The Attempt at a Solution



#1.) We know that P(A) is a set. Therefore it must be in A, since A is the set of ALL sets.

#2.) I cannot figure out what to do here. I do not have a theorem that says that |P(A)| > |A|. I do have a theorem that states that there is no injection from P(A) to A and also that there is no surjection from A to P(A).

Since there is no injection from P(A) to A, does this alone mean that |P(A)| > |A| ?
 
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  • #2
Yes, it does. If |C|<=|D| then there is an injection from C->D. If there is no injection then the opposite must hold so |C|>|D|.
 

1. What is the set of all sets contradiction?

The set of all sets contradiction, also known as Russell's paradox, is a mathematical paradox that arises when considering the set of all sets that do not contain themselves. This leads to a contradiction, as both including and not including this set in itself result in logical inconsistencies.

2. What are the implications of the set of all sets contradiction?

The set of all sets contradiction has significant implications in mathematics and philosophy. It challenges the foundations of set theory and raises questions about the nature of infinity and self-reference.

3. How was the set of all sets contradiction resolved?

In 1908, mathematician Bertrand Russell and philosopher Alfred North Whitehead introduced the theory of type theory, which avoids the set of all sets contradiction by restricting the way sets can be constructed. This theory forms the basis of modern set theory.

4. Can the set of all sets contradiction be applied to real-world situations?

The set of all sets contradiction is a mathematical paradox and does not have direct applications in the real world. However, it has important implications for the foundations of mathematics and can be used to explore abstract concepts such as infinity.

5. How does the set of all sets contradiction relate to other paradoxes?

The set of all sets contradiction is one of many paradoxes in mathematics, logic, and philosophy. It is closely related to other paradoxes such as the liar paradox and the barber paradox, which also involve self-reference and lead to logical inconsistencies.

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