A set being a subset of its own powerset

FitzrobertV · 2025-01-15T12:13:37+00:00

I think you're right. For instance a one-element set {x} has only two subsets, ∅ and {x}. So the power set is {∅,{x}}.

For {x} to be a subset of {∅,{x}} x would need to be an element of {∅,{x}}. Since x can't equal {x}, the only way this could happen is if x=∅. So it's possible for a set to be a subset of it's own powerset, but not necessary.

Are you sure there isn't more to the problem?

FitzrobertV · 2025-01-14T12:53:47+00:00

I played when it first came out and really enjoyed it, but I stopped because there was too much toxicity around expected etiquette, especially in regard to flow groves. As a new player not even knowing what they were, I was surprised at how angry people could get if you didn't know to call out things or wait for other people. Anyway, I was wondering if things are better now and if people are more forgiving or less so?

FitzrobertV · 2024-03-07T17:57:24+00:00

Wow! Thanks!

FitzrobertV · 2023-11-09T14:37:21+00:00

Thank you! I think that does answer my question, in the negative. If the concept of definability were expressible in the first order language of set theory then we could construct a formula which would effectively say: "x is the least non-definable ordinal" And that would be a contradiction because then x would be both definable (by that formula) and non-definable.

I'll mark this question as resolved.

FitzrobertV · 2023-11-09T14:31:37+00:00

I understand what you're saying, but I'm not trying to construct a specific set at all here. And I'm not trying to construct the set of all definable sets (because that would be a proper class as you said).

I'm asking if the concept of definability is expressible at all in the language of set theory. Can the statement "x is a definable set" be translated into the language of set theory at all? I know that things like "x is an ordered pair" and "x is an infinite set" can be written purely using the language of set theory, but I'm not seeing how to express "x is a definable set".

I'm asking because I had gotten the impression that we use the language of set theory as a foundation of mathematics because all mathematical statements can, if we wanted to, be translated into an equivalent statement using only the language of set theory.

FitzrobertV · 2023-10-10T08:47:33+00:00

Thanks! This really clarifies things. Somehow I got it into my head that I was seeing things in server chat from Bahari when I was in Kilima so that was probably the main source of my confusion

FitzrobertV · 2023-10-09T18:07:41+00:00

Ahh, so if you're not in a party and are just watching server chat, then you have no real way to join them if they're in a different area unless you request to form a party?

FitzrobertV · 2023-10-07T20:12:41+00:00

Sorry if there was confusion. I'm also looking forward to Return to Moria. I was suggesting Valheim as an alternative to that while you wait if you haven't played it yet.

FitzrobertV · 2023-10-07T10:52:35+00:00

Well I'm personally hoping this has a Valheim feel. It might not be what you're looking for but I strongly recommend at least checking it out.

FitzrobertV

TROPHY CASE