Is that true to say: For every set A defined by A={ A : ∅∉A } then the Cardinality of A equals to |A|-1 ?
1 Answer
A few thoughts...
Explanation:
First let us ask what the meaning of the expression
It seems to be saying something like "The set of sets
From what collection of sets are we allowed to choose candidate elements for
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