Question

If we define (i.e., by definition) a set as an abstract collection of at least one element, and

If we say that by nature (i.e., as an axiom) all sets contain one element that is the set itself at the same time
then

There is no empty set, and
All sets contain themselves, and
There is no set of all sets that do not contain themselves, i.e., there is no Russell's paradox.
As an example, the set of all cats would have a cat member that is, at the same time, the set of all cats. Also, the set of all numbers would have a number that is, at the same time, the set of all numbers.

There would be a casting test operator. With it you could ask "Is this cat, at the same time, the set of all cats?". Or you could answer questions like "Is that cat a member of this cat?" with an answer like "Yes, because this cat is, at the same time, the set of all cats". Another one: "Does the set of all cats is a member of itself?" with answer "Yes, because one of the cats is, at the same time, the set of all cats".

Do you see any contradiction in a theory sketch like this?

Do you see any contradiction if we add a universal set as an axiom? i.e.:

For all elements, there exists a unique set that contains all of them.
I have no degree in philosophy and I am also not a native english speaker, however, I have been humble while asking this question. Please, be humble while answering it.

Edit: I accepted waf9000 answer because it is true that with traditional equality axioms this theory sketch is contradictory. I will keep experimenting with new equality axioms outside this question. Thanks a lot for your time and patience.

Answer 1

The theory sketch you provided contains contradictions and does not align with the definition and properties of sets in mathematics.

Step-by-step explanation:

Based on the theory sketch you provided, there are several contradictions that arise. First, the statement that all sets contain themselves contradicts the definition of a set as an abstract collection of elements. If a set contains itself, it would create an infinite loop and violate the concept of a set. Additionally, the statement that all sets contain the set of all sets that do not contain themselves leads to Russell's paradox, which states that a set cannot contain itself.

Adding a universal set as an axiom doesn't resolve these contradictions. The existence of a universal set that contains all elements would lead to the same contradictions mentioned above. It would create an infinite loop and violate the definition of a set.

In conclusion, the theory sketch you provided contains contradictions and does not align with the definition and properties of sets in mathematics.