[FOM] Extending Set Theory with Indiscernibles

[Ali Enayat]

Taranovsky's interesting recent note [Feb 27, 2005] on extensions of set 
theory has inspired me to briefly discuss some recent results that 
characterize the rather surprising strength of a natural system of set 
theory with a class of indiscernibles, here dubbed ZFCI.

ZFCI is a theory formulated in the language {epsilon, I(x), <}, where I(x) 
is a unary predicate [to distinguish the indiscernibles] and < is a binary 
relation [for a global well-ordering].  Intuitively speaking, ZFCI is an 
extension of ZFC that strongly negates Leibniz's dictum on the identity of 
indiscernibles by asserting "there are a proper class of indiscernibles".

The axioms of ZFCI are as follows:

0. The axioms of ZFC;

1. the sentence expressing "I is a proper class of ordinals";

2. a schema expressing that < is a global well-ordering;

3. the Replacement scheme for formulae using I and <; and

4. the Indiscernibility scheme, which is a scheme asserting that (I,<) is a 
class of order indiscernibles [in the usual sense of model theory] for 
formulae in the language {epsilon, <}.

It turns out that ZFCI goes well beyond ZFC since it proves the existence of 
n-Mahlo cardinals for each concrete natural number n. However, if 
consistent, it will not prove the statement "for each natural number n, 
there is an n-Mahlo cardinal".

Indeed, one can precisely describe the first order consequences of ZFCI in 
the usual language of set theory {epsilon}. In order to do so, let PHI be 
the set of sentences of following form (where n is a concrete natural 
number):

"there is an n-Mahlo cardinal kappa such that V(kappa) is a SIGMA_n 
elementary submodel of the universe".

It is not hard to see that ZFC + PHI  is equiconsistent with ZFC plus axioms 
of the form "there is an n-Mahlo cardinal" (again, where n is a concrete 
natural number n). The former theory of course proves the latter theory, but 
not vice versa.

Here is the first main result:

Theorem A. For any sentence S in the usual language of set theory {epsilon}, 
the following two conditions are equivalent:
(i)    ZFCI proves S;
(ii)   ZFC + PHI proves S.

It is worth pointing out that the situation for Peano arithmetic (or 
equivalently: the theory of finite sets) is quite different, i.e., if one 
formulates an analogous theory, PAI, extending PA, then the formalized 
version of Ramsey's theorem can be used to show that PAI proves precisely 
the same arithmetical sentences as PA itself. This adds more plausibility to 
the theory ZFCI since it shows that the indiscernibles allow ZFC to 
"catch-up" with PA.

One may also wish to *iterate* the idea of adding indiscernibles by adding 
countably many new unary predicates I_n (x) for each natural number n in 
order to formulate a theory ZFCI# extending ZFCI by adding axioms asserting 
that I_(n+1) is a proper class of indiscernibles for formulae in the 
language {epsilon, <}augmented with I_1, ., I_n.

As it turns out, this will not buy us any new theorem of set theory that 
ZFCI could not prove already, i.e., Theorem A can be improved to the 
following result which connects ZFCI and ZFCI! to other systems of set 
theory.

Theorem B. For any sentence S in the usual language of set theory {epsilon}, 
the following five conditions are equivalent:
(i)  ZFCI#  proves S;
(ii)  ZFCI proves S;
(iii) GBC + "the class of ordinals is weakly compact" proves S;
(iv) NFUA proves "S holds in CZ";
(v)   ZFC + PHI proves S.

Here GBC is the Godel-Bernays theory of classes; NFUA is the extension of 
the Quine-Jensen system of set theory NFU with a universal set obtained by 
adding the axioms of Choice, Infinity, and the axiom expressing "every 
Cantorian set is strongly Cantorian"; and CZ is the canonical model of ZFC 
that can be *interpreted* in models of NFUA.

The equivalence of (i), (ii) and (v) is will be soon available, but the 
equivalence of (iii), (iv), and (v) [which were inspired by the work of 
Solovay and Holmes] appear in my paper:

Automorphisms, Mahlo Cardinals, and NFU, Nonstandard Models of Arithmetic 
and Set Theory, Contemporary Mathematics (Enayat and Kossak, ed.), volume 
361, American Mathematical Society, 2004.

Also available on:  http://academic2.american.edu/~enayat/Aut.pdf

Best regards,

Ali Enayat