Zermelo-Fraenkel Set Theory

CHAPTER 6. CARDINALS 47
Here is a proof that ZFC + “no strong inaccessible exists” is consistent (provided ZF is
consistent). For the logical background of this proof, see Section 7.2 (p. 54).
Suppose this theory is not consistent. That means: you can prove existence of an
inaccessible in ZFC. By Exercise 120.3/4, you can construct an inner model of ZFC that
has no inaccessible, contradicting the assumed existence of a proof that inaccessibles exist.
□
Since GCH is consistent, and, under it, strong and weak inaccessibles are the same,
non-existence of weak inaccessibles is consistent as well.
Let ZFCI be the theory: ZFC+“an inaccessible exists”.
Here is a proof that you cannot hope to prove ZFCI consistent by assuming only consistency
of ZFC:
Suppose you had such a proof, showing that (ZFC consistent ⇒ ZFCI consistent).
Whatever its details, you should at least be able to formalize it in the extremely
strong theory ZFCI. (In fact, trustworthy consistency proofs should only deal with concrete
objects: proofs and their combinatorial properties, and therefore they usually will be
formalizable already in ZF without Infinity Axiom.)
By Exercise 120, ZFCI proves that ZFC has a model, and hence ZFCI proves that ZFC
is consistent.
Since your hypothetical proof of the implication (ZFC consistent ⇒ ZFCI consistent)
has been formalized in ZFCI, it follows (by Modus Ponens) that ZFCI proves ZFCI
consistent.
Now Gödel’s second incompletability theorem says that (under suitable conditions that
are satisfied here) every theory that proves itself consistent must be, in fact, inconsistent.
Therefore, ZFCI is inconsistent.
But then, your hypothetical proof shows that, by contraposition, also ZFC must be
inconsistent!
□
Note the asymmetric behavior of ZFC with regard to the statement “an inaccessible
exists” and its negation: the latter one is easily shown to be consistent relative to ZF,
whereas consistency of ZF alone is insufficient to prove the former one to be consistent.
Compare this to e.g., AC and CH: these, as well as their negations, have been shown
consistent relative to ZF alone.
6.4 Cofinality and Regularity (ordinal version)
Here comes the ordinal version of these notions.
Definition 6.25 A subset B ⊂ A is cofinal in the linear ordering (A,