Large Cardinals, Measures, and Elementary Embeddings: Examples and Counterexamples

In the displayed U, a coordinate equality set belongs to U exactly when f(i_0)=g(i_0). This proves directly that the quotient equivalence is equality at i_0 and evaluation is well-defined and injective. Every a in M has the explicitly defined constant function c_a in the product, so evaluation is surjective and sends [c_a] to a. Nonemptiness of M therefore gives a nonempty product and quotient, without any family of choices.

A constant symbol evaluates at i_0 to its original interpretation. For a function symbol F and representatives f_1,...,f_n, evaluation of its interpreted class is exactly $F^M(f_1(i_0),\dots ,f_n(i_0))$. This also shows independence of representatives in that interpreted symbol. A relation holds in the quotient exactly when its coordinate truth set contains i_0, that is, when it holds on the evaluated tuple in M. Thus evaluation preserves functions and preserves and reflects relations, including equality by step 1.1; it is an isomorphism. Zero-arity symbols give the same calculation with the empty tuple. This local verification uses the formulas of F1, not the choice-dependent general Los theorem.

F2 gives [c_alpha]=j(alpha)=alpha for alpha<kappa; F1 gives [id]=kappa. At every coordinate xi, the ordinal xi+1 is the set xi union {xi}, namely the successor of xi. F3 transfers this fixed defining formula, so the collapsed class of xi maps to xi+1 is the successor of [id], exactly kappa+1.

An infinite cardinal kappa is a limit ordinal: any infinite successor ordinal beta+1 is equinumerous with beta by shifting a countably infinite subset, so cannot be an initial ordinal. Therefore xi+1<kappa for every xi<kappa. The successor representative takes its values in kappa at all coordinates, so its class belongs to j(kappa) by coordinate membership. Step 1.1 identifies this member with kappa+1, proving the strict inequality.

A free ultrafilter on omega contains no finite set: if a finite union of singletons belonged to U, the ultrafilter complement decision and finite intersections would force one singleton to belong, making it principal. Hence each cofinite tail $B_m=\{n:n>m\}$ belongs to U. Their countable intersection is empty, explicitly witnessing failed countable completeness.

For n>m, $g_m(n)=n-m>0$ and $g_{m+1}(n)=n-m-1<g_m(n)$. Since these are finite von Neumann ordinals, this is exactly $g_{m+1}(n)\in g_m(n)$. F1 therefore gives $[g_{m+1}{]}_{U}E[g_m{]}_U$. Also every g_m(n) belongs to omega, so every displayed class belongs to [c_omega], the internal natural-number set. For n<=m both functions in the comparison are zero, so membership fails there; the truth set is exactly B_m, not just an unspecified large set.

Replacement forms the nonempty set $\{[g_m{]}_U:m\in \omega \}$. Each member has its next displayed class as an E-predecessor in this set, so the set has no E-minimal member and the relation is not well-founded. All functions and classes were explicitly defined; no choice of representatives and no additional AC were used.

Given an inaccessible, minimize the ordinals at or below it satisfying that property; Separation and ordinal well-ordering give a least one kappa. Let C be the club of infinite strong-limit cardinals below kappa from F2. No uncountable regular cardinal belongs to C, because such a member would be uncountable regular strong limit, an inaccessible below kappa. Thus the set of uncountable regular cardinals used in the page's definition of Mahloness misses the actual club C; omega is not in that set. AC is retained through F2's cardinal-size estimates; the least-ordinal selection itself uses only Separation and ordinal well-ordering.

Missing C means that set is nonstationary, so kappa is not Mahlo. If kappa were weakly compact, F1 would make it Mahlo, contradicting step 1.1. Therefore it is not weakly compact. AC is also retained through F1's cardinal estimates and pressing-down argument. The argument is conditional on the initial existence hypothesis and does not prove that hypothesis consistent.

Fix, externally, a purported finite ZFC derivation p of the sentence E asserting that an inaccessible exists. Only finitely many ZFC axiom instances occur as assumptions in p; call their conjunction A_p. Soundness F2 applied to this fixed finite derivation is a theorem of ZF saying that any nonempty set structure satisfying those instances satisfies E. No assertion about a truth predicate for V is involved.

Work now inside ZFC under E. Choose the least inaccessible kappa by ordinal minimization below one witness. F1 proves each of the finitely many instances in A_p relativized to V_kappa; combining these finite proofs and F3 makes its nonempty membership structure a model of A_p. Step 1.1 gives that V_kappa satisfies E. Thus some alpha in V_kappa is internally an inaccessible ordinal. Transitivity gives an actual ordinal alpha<kappa. Its internal cardinalhood is actual cardinalhood: any external bijection between alpha and a smaller ordinal has a graph of rank at most alpha plus finitely many successors, hence belongs to V_kappa, contradicting internal cardinalhood if it existed. F1's inaccessibility absoluteness now applies to this actual cardinal and makes alpha an actual inaccessible, contradicting leastness of kappa.

The preceding construction is a finite ZFC derivation of E implies contradiction, depending on the fixed finite proof p. Append it to p, which derives E, and infer a contradiction in ZFC. Therefore the existence of such p implies inconsistency of ZFC. Contraposition yields exactly the stated conditional nonprovability. This neither extracts a transitive model from Con(ZFC) nor invokes a uniform universe satisfaction relation.