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Why is the set of all sets a paradox?

Guest Aug 14, 2015

Best Answer 

 #1
avatar+17745 
+10

The "set of all sets" would have to contain itself, along with all the other possible sets. 

For ease of writing, let's pretend that there are only three sets: X, Y, and Z.

Now the set of all sets is this set: {X, Y, Z}. Let's call this set S so that S = {X, Y, Z}.

But, S doesn't contain set S. Now the set of all sets is {S, X, Y, Z}.

Let's call this set T so that T = {S, X, Y, Z}.

But T doesn't contain set T. Etc. Etc. Etc. Do you see the problem? 

geno3141  Aug 15, 2015
 #1
avatar+17745 
+10
Best Answer

The "set of all sets" would have to contain itself, along with all the other possible sets. 

For ease of writing, let's pretend that there are only three sets: X, Y, and Z.

Now the set of all sets is this set: {X, Y, Z}. Let's call this set S so that S = {X, Y, Z}.

But, S doesn't contain set S. Now the set of all sets is {S, X, Y, Z}.

Let's call this set T so that T = {S, X, Y, Z}.

But T doesn't contain set T. Etc. Etc. Etc. Do you see the problem? 

geno3141  Aug 15, 2015

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