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# Why is the set of all sets a paradox?

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Why is the set of all sets a paradox?

Guest Aug 14, 2015

#1
+17746
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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
+17746
+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