How is the Russell paradox a paradox? Don't all sets contain

>>8262050
contain as an element, not as a subset

>>8262056
What's the difference?
{{X}} contains {X} as an element and {X} is also a subset, right? Or is {{X}} the subset? Both?

>>8262133
it contains {X} as an element.
The subsets of {{X}} are the empty set and {{X}}. In other words, {X} is not a subset of {{X}}, because X is not in {{X}}.

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>>8262050
Have you ever considered the possibility that you don't actually understand the scientific terms you're using? That maybe all of the advanced scientific explanations you've ever heard have just been metaphors, oversimplified to the point of removing all useful information? And that, therefore, you not only lack the knowledge to answer the questions you have, but also to know what questions are even vaguely coherent?

I would strongly recommend looking into this possibility.

{a,b} is a set. Its subsets are {}, {a}, {b}, and {a,b}. None of these subsets are elements - the elements are a and b, and a =/= {a}, because one is a, and the other is a set containing a. In particular, {a,b} is not an element, so the set does not contain itself.

In fact it is not even possible for a finite set contain itself.

>>8262139
>>8262183

ok makes sense

but why is it necessary for a =/= {a}?

>>8262169
Hey guys, check out this ass hat.

>>8262196

It is simply the difference between elements and sets. Whatever a is, it is just that object. {a} is a set containing that object.

Of course you might wonder what if A is a set containing itself? Well, this is not well-defined: you cannot tell me what A is other than "it contains itself". In particular, we don't know what the other members of A are. So we can suppose it has no others, so A is empty, except for itself. Try to build it: start with empty set {}. This isn't A because it doesn't contain itself, so make one which contains the emptyset, {{}}. This does not contain itself.. try again {{{}}}. Still not right.. and this will evidently never end. And this is in the case in which we suppose A is as empty as possible!

So having an object which is equal to a set containing that object is a bitch, and descends into nonsense. This is why it is best to distinguish between objects, and sets which contain objects, and to not conflate the two.

>>8262196
It's a convention. Set Theories are theories written down in formal logic and, by it's axiomatizations, ought to capture some notion of "set".

Theories were a set can be inside of itself are
https://en.wikipedia.org/wiki/Non-well-founded_set_theory
but very fringe

>>8262196
>>8262214


I'm going to add, this distinction is also how the natural numbers are constructed in set theory. We identify 0 with the empty set, and then for any number X, the next number is X U {X}.

So "1" = 0 U {0} = {} U {{}} = {{}}
"2" = 1 U {1} = {{}} U {{{}}} = {{},{{}}}
"3" = {{},{{}},{{{}}}}

and so on. If X=={X}, then we'd never get past zero.

>>8262229
If
X = {X}
then
(X U {X}) = X

If
s(X):=X U {X}
then this reads
s(X)=X

Not more and not less.

The doesn't work with the Peano axioms, of course.

But it's not like we'd have to use {} or "start" anywhere to discuss X = {X}.

And yeah, there is no such set in ZFC and the proof is only a few lines away from the axioms.

>>8262229
Ah okay, I remember this. In that case I'm good with accepting that a =/= {a}.
(I just started reading The Principles of Mathematics today, how fucked am I (a physics student)?)

>>8262218
I'll check that out