FOM: Does Shapiro's ZFC2 need a small cardinal axiom?

Robert Black Robert.Black at
Tue Apr 24 05:11:08 EDT 2001

>Does anyone know how to prove the existence of an
>empty set in Shapiro's ZFC2?
>(as defined in "Foundations without Foundationalism" 6.1 & 4.2)

I think it's easy using the axiom of infinity, but impossible without.

Easy using Infinity: If no set is empty then any set a_0 has a member a_1
with a member a_2 with a  .... Using Infinity and Replacement the a_i can
be collected into a set, and application of Foundation to this set gives a

Impossible without: Consider a model with a sequence of objects a_0, a_1,
a_2 ... which will behave rather like Urelemente except that each is to be
regarded as the singelton of the next. On top of these, at every finite
rank form every possible new finite nonempty set of material from below.
This gives us a structure in which no set is empty, Extensionality,
Foundation, Pairs, Unions, Powerset and second-order Replacement using
functions all hold but Infinity fails.


Robert Black
Dept of Philosophy
University of Nottingham
Nottingham NG7 2RD

tel. 0115-951 5845

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