[FOM] Continua/Finite Axiomatization

Harvey Friedman friedman at math.ohio-state.edu
Fri Feb 6 11:55:01 EST 2004

```The previous posting Characterization of Linear Continua, 2/4/04, 11:07AM
looks correct. I didn't yet look at Munkres that Carette suggests may have
some related results. I was not convinced by the posting by Frank, 2/4/04
8:21PM, providing a "counterexample".

In the finite axiomatization of class theory given in my Finite
Axiomatization of Class Theory 2/4/04 9:39PM, I followed the well known idea
of not stating separation, and instead relying on separation to be derived
from replacement. However, there is the well known difficulty that the empty
set is not, at least obviously, derivable.

This is easily fixed, by a tiny strengthening of replacement (axiom 7
below). Otherwise the following axiomatization is without change:

The atomic formulas are

1. x epsilon y.
2. x = y.
3. R(x,y).

The axioms are as follows.

1. Two sets are equal iff they have the same elements.
2. There is a set consisting of any two given sets.
3. There is a set consisting of the elements of elements of any given set.
4. There is a set consisting of the subsets (appropriately defined) of any
given set.
5. There is a nonempty set with no epsilon maximal element.
6. Every nonempty set has an epsilon minimal element.
7. (forall x in y)(therexists at most one z)(R(x,z)) implies (therexists
w)(forall z)(z in w iff (therexists x in y)(R(x,z))).
8. Every relation has a complement.
9. Any two relations have a union.
10. The epsilon relation on sets exists.
11. The equality relation on sets exists.
12. Every relation has a reverse (inverse).
13. Any two relations have a composition.

Harvey Friedman

```