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Zermelo-Fraenkel Set Theory
Zermelo-Fraenkel Set Theory
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Question 1 of 5
120s
intermediate (4/10)
conceptual
The axiom of foundation (regularity) says every nonempty set
A
contains an element
x
with
A
∩
x
=
∅
. What does it actually exclude?
Hide and think first
A.
It requires every set to have a definable membership predicate, which prevents non-constructive sets like the Vitali set from existing.
B.
It restricts the cardinality of the universe to be no larger than the first inaccessible cardinal, blocking proper-class issues from infecting set theory.
C.
It rules out infinite descending membership chains
…
∈
a
2
∈
a
1
∈
a
0
and self-membership
a
∈
a
, but does not affect any standard mathematical construction.
D.
It rules out uncountable sets that fail to be well-ordered, ensuring the well-ordering theorem follows from the other axioms without needing choice.
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