Question

Is that true to say: For every set A defined by A={ A : ∅∉A } then the Cardinality of A equals to |A|-1 ?

Answer 1

A few thoughts...

Explanation:

First let us ask what the meaning of the expression `{ A : O/ !in A }` is.

It seems to be saying something like "The set of sets `A` such that the empty set `O/` is not an element of `A`".

From what collection of sets are we allowed to choose candidate elements for `{ A : O/ !in A }` ?

Here are a few ideas:

• The "set of all sets". The idea of the "set of all sets" leads to an inconsistent system. For example, we would have problems deciding whether `{ A : A !in A }` is an element of itself or not.

• The "class of all sets". If we recognise that the collection of all sets is too big to count as a set, we can then happily write `{ A : O/ !in A}`, meaning the collection of all sets that do not contain the empty set as a member, but this is too big to be a set too and is therefore a proper class (which has no defined cardinality).

• What if we restrict the choices to some set `Omega` of sets? Then we might more properly write: `{ A in Omega : O/ !in A }`. This is then a perfectly good set, but what can we say about its cardinality? We can say `abs({ A in Omega : O/ !in A }) <= abs(Omega)`. If this subset of `Omega` is also infinite then removing any finite number of elements from it will not change its cardinality. Hence:

  `abs({A in Omega : O/ !in A}) - 1 = abs({A in Omega : O/ !in A})`

That might look like a false assertion, but it's really just a property of infinite cardinals and their arithmetic.

Notes

I suspect the back story of this question relates to attempting to find paradoxes like `{ A : A !in A }` which was an historic example that drew attention to the need for formal set theory.