[FOM] Uses of Replacement
Harvey Friedman
friedman at math.ohio-state.edu
Thu Feb 23 00:40:44 EST 2006
Someone on the FOM has privately asked me for examples of (necessary) uses
of replacement. Here are some.
1. In set theory, starting with Cantor.
1a. The existence of a von Neumann ordinal that is a limit ordinal with a
limit ordinal below it.
1b. Every set has a transitive closure.
1c. The existence of the set of all hereditarily finite sets.
1d. The existence of an uncountable von Neumann ordinal.
1e. The construction of addition, or multiplication, or exponentiation on
the von Neumann ordinals.
1f. Every set is in one-one correspondence with a von Neumann ordinal.
1g. The construction of the cumulative hierarchy of sets. I.e., of the
V(alpha)'s.
1h. The usual proof of the existence of an algebraic closure of any field,
as a tower of length omega. With care, this can be done without replacement.
2. In the theory of Borel measurable functions on complete separable metric
spaces.
2a. Borel Determinacy, in:
D.A. Martin, Borel Determinacy, Annals of Mathematics, 102 (1975), 363-371.
H. Friedman, Higher set theory and mathematical practice, Annals of
Mathematical Logic, 2 (1971), 325-357.
2a. If E is a Borel set in the plane symmetric about the line y = x, then E
or R^2\E has a Borel selection.
On the Necessary Use of Abstract Set Theory, Advances in Math.,
Vol. 41, No. 3, September 1981, pp. 209-280.
2b. Certain other Borel theorems from paper in 2a above.
2c. Certain Borel selection theorems from
Selection for Borel relations, in: Logic Colloquium 01, ed. J. Krajicek,
Lecture Notes in Logic, volume 20, ASL, 2005, 151-169.
intimately related to work of the functional analysts in Paris: Debs and
Saint Raymond.
3. Grothendieck universes. Everything in 1,2 above is known to be NECESSARY
uses of replacement. Here it is doubtful that we have necessity, except if
the theorems actually mention universes.
SUBSCRIBERS - do you know other examples where the uses of replacement may
or may not be necessary?
Harvey Friedman
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