The biggest set (It's a riddle: Does such a set exist?)

[shahar]

The biggest set:

It a riddle:
Is it exist?

I have a hint: P(A)
I don't understand the hint.

 

[blamocur]

It a riddle:
Is it exist?

I have a hint: P(A)
I don't understand the hint.

I am guessing that P(A) stands for the power set, i.e., set of all subsets of A. I remember vaguely that there is a theorem that P(A) is always larger than A, which would mean that there is no biggest set.

 

[shahar]

I am guessing that P(A) stands for the power set, i.e., set of all subsets of A. I remember vaguely that there is a theorem that P(A) is always larger than A, which would mean that there is no biggest set.

Right. Thank you.

 

[shahar]

Does P(P(A)) > P(A) always?

 

[topsquark]

Does P(P(A)) > P(A) always?

This is a good statement to try to prove. All you need to do, though, is to prove that P(A) > A (and that, aside from the empty set there is no set such that P(A) = A.)

Yes. So there really is no biggest set, because as soon as you say, "A is the biggest set" you can always say, "But P(A) is bigger!"

-Dan

 

[blamocur]

aside from the empty set there is no set such that P(A) = A.

I'd argue that [imath]P(\empty) > \empty[/imath] since [imath]P(\empty)[/imath] has one element, which is the empty subset.

 

[blamocur]

I remember vaguely that there is a theorem that P(A) is always larger than A, which would mean that there is no biggest set.

I remember less vaguely now: https://en.wikipedia.org/wiki/Cantor's_diagonal_argument#General_sets

 

[Steven G]

Suppose A = {a₁. a₂, a₃,....}
If BCA and B = {a₃, a₅}, then we can denote B by 00101000..., where a 1 in the nth position means that a_n is in B.
Then P(A) is the set of all infinite strings of 0 and 1. Show that this set is uncountable.