Set theoretic "Puzzle" I made up.

[mvCristi]

I made up it, I didn't find anything about it, so, maybe, I'm the inventor.

The question is: How do different set theories handle the following definition?

U := {x : (x belongs_to A) and ( Not(A is_included_in Powerset( EmptySet)) )}

If possible in the set theory, require (A is_set). But it doesn't really matter.

The set U is interesting for some reasons.

 

[mvCristi]

I'm doing my PhD research in "Foundations of mathematics with the application in automated theorem proving". I denied the motif that Russel's Paradox is an obstacle. This is the reason why I consider "Godel incompliteness theorems" and "Tarski's undefinability of truth resoult" particular cases of a Math that is done wrong. I think Math is both complete and consistent, but done in the right way. I researched and I have my opinion. Yet, I'm not sure if it is decidable. Anyway...
I earned money from my PhD research, my everyday existence, but I have to pay them back if I do not succide. And everyone around me is skeptical I will succide. I came up with this "puzzle" just to attract attention. Second order logic + ZFC cannot handle it.

First, take a look at "Euler's identity": e^(i*pi)+1=0.
It contains only once important constants like e,i,pi,1,0; it contains only once addition (+), multiplication (*), exponantiation (^); it contains only once equality (=). Many mathematicians regard it as the most elegant and "beautiful" result of Mathematics.

Now, take a look at: (For_all(x) belongs_to A) and ( Not(A is_included_in Powerset( EmptySet)) ).
It contains: belongs_to, is_included_in, Powerset, EmptySet: these are fundamental constants in set theory. They appear only once.
Quantifiers: For_all: the standard quantifier appears only once.
Logical connectives:
and: with arrity 2
not: with arrity 1.
Free variables:
A appears twice: as the arrity of "and"
x appears once: as the arrity of "not".

Now let me make a citation attributed to Stephan Banach: "Good mathematicians see analogies. Great mathematicians see analogies between analogies."

I would enjoy a disscution obout the set U.

Thank you!

P.S. I didn't want a solution to the "puzzle", just a disscution about it.
P.P.S. Sorry for bothering you with my everyday life frustrations.

 

[CaptainBlack]

I made up it, I didn't find anything about it, so, maybe, I'm the inventor.

The question is: How do different set theories handle the following definition?

U := {x : (x belongs_to A) and ( Not(A is_included_in Powerset( EmptySet)) )}

If possible in the set theory, require (A is_set). But it doesn't really matter.

The set U is interesting for some reasons.

The power set of the empty set has one element the empty set itself. So Not(A is_included_in Powerset( EmptySet)) (equivalently: \(\lnot (A \in \mathcal{P}(\emptyset))\) )is equivalent to A is not the empty set.

So your statement is U=A where A is a non-empty set.CB

 

[mvCristi]

Thank you for your reply. You broke the ice.

I said not: not(A belongs_to Powerset( EmptySet))
I said: not( A is_included_in Powerset( EmptySet))

P.S. It's just a "puzzle". My real work begins with a much more simpler and Axiomatic Set Theory proposition.
P.P.S. Sorry that I do not know LateX. So, please, anybody, correctly edit the definition of "U" in my first post in LateX. My fault. :D

Later edit: I've just voted on this (my) poll with: No, your just idiot. Just to counteract the discrepances. :D

 

[CaptainBlack]

Thank you for your reply. You broke the ice.

I said not: not(A belongs_to Powerset( EmptySet))
I said: not( A is_included_in Powerset( EmptySet))

P.S. It's just a "puzzle". My real work begins with a much more simpler and Axiomatic Set Theory proposition.
P.P.S. Sorry that I do not know LateX. So, please, anybody, correctly edit the definition of "U" in my first post in LateX. My fault. :D

Later edit: I've just voted on this (my) poll with: No, your just idiot. Just to counteract the discrepances. :D

Ambiguous language, by included you mean is a subset? a proper subset? ...

Since the Power set of the empty set contains one element The Empty Set, it has two subsets: The Empty Set and the set whose only element is The Empty Set, so \( A\) is \(\emptyset\) or \( \{ \emptyset \} \)

CB

 

[mvCristi]

I just meant that (A is_set). But it doesn't really matters.
Not this is the main "issue" with U (EmptySet is_included in U: is very easily handled in ZFC, and others; can you tell me how?)
In fact the relation with {EmptySet} is important. The other, with EmptySet, was just part of the "puzzle" to make it more weird.
Can you ellaborate on this?

 

[CaptainBlack]

I just meant that (A is_set). But it doesn't really matters.
Not this is the main "issue" with U (EmptySet is_included in U: is very easily handled in ZFC; can you tell me how?)

I would suggest that you do some more work on the clarity of your writing, I'm afrain that makes no sense to me.

CB

 

[mvCristi]

I feel asheamed to state, but: I would suggest that you do some more work on the clarity of your understanding of the Foundations of Maths. Please, forgive me, but I'm not a native English speaker. Maybe, this is the source of the discrepancy.

However, I respect you, I appreciate you engaged in this disscution and I have to state that for me, there are other fields of math I have to understand better, fields in which you seem expert.