MATH CIRCLE - SET THEORY WEEK 4 We were working on two projects …
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We were working on two projects: (1) Define mathematics starting only with set theory! That is, use set theory as a foundation for ordinary math. …
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MATH CIRCLE - SET THEORY WEEK 4
SAM LICHTENSTEIN
We were working on two projects: (1) Define mathematics starting only with set theory!
That is, use set theory as a foundation for ordinary math. We proceeded to try to do this
by starting with ordinary counting numbers. After a little debate, we ultimately agreed that
we must define the numbers using only sets of sets, not sets of numbers. What should 0 be?
Eventually it was settled that it should be the null set {}, for lack of a better alternative,
although there was some disagreement.
Side note: is there a rule defining the empty set? Yes! We realized that we can define the
empty set as the set of all x such that x 6= x! (Except that first we have to say what we
mean by =. We already have two possible definitions for this, but in building up a formal
set theory, we have to choose one.)
In trying to define the counting numbers, one observation that was made is that the
operation of “adding one” is somehow fundamental: if we have 0 (and we do, as {}) then we
can get…
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