General Computer Science 320201 GenCS I & II Lecture ... - Kwarc

Idea: We need a notion of “counting”, associating every member of a set with a unary natural
number.
Problem: How do we “associate elements of sets with each other”?
(wait for bijective functions)
c○: Michael Kohlhase 47
But before we delve in to the notion of relations and functions that we need to associate set
members and counding let us now look at large sets, and see where this gets us.
Sets can be Mind-boggling
sets seem so simple, but are really quite powerful (no restriction on the elements)
There are very large sets, e.g. “the set S of all sets”
contains the ∅,
for each object O we have {O}, {{O}}, {O, {O}}, . . . ∈ S,
contains all unions, intersections, power sets,
contains itself: S ∈ S (scary!)
Let’s make S less scary
A less scary S?
c○: Michael Kohlhase 48
Idea: how about the “set S ′ of all sets that do not contain themselves”
Question: is S ′ ∈ S ′ ? (were we successful?)
suppose it is, then then we must have S ′ ∈ S ′ , since we have explicitly taken out the sets
that contain themselves
suppose it is not, then have S ′ ∈ S ′ , since all other sets are elements.
In either case, we have S ′ ∈ S ′ iff S ′ ∈ S ′ , which is a contradiction!
(Russell’s Antinomy [Bertrand Russell ’03])
Does MathTalk help?: no: S ′ := {m | m ∈ m}
MathTalk allows statements that lead to contradictions, but are legal wrt. “vocabulary”
and “grammar”.
We have to be more careful when constructing sets! (axiomatic set theory)
for now: stay away from large sets. (stay naive)
c○: Michael Kohlhase 49
Even though we have seen that naive set theory is inconsistent, we will use it for this course.
But we will take care to stay away from the kind of large sets that we needed to constuct the
paradoxon.
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