Mathematics

We investigate the lower bound of the consistency strength of \mathsf{CZF} with Full Separation \mathsf{Sep} and a Reinhardt set, a constructive analogue of Reinhardt cardinals. We show that \mathsf{CZF+Sep} with a Reinhardt set interprets \mathsf{ZF^-} with a cofinal elementary embedding j:V\prec V . We also see that \mathsf{CZF+Sep} with a Reinhardt set interprets \mathsf{ZF^-} with a model of \mathsf{ZF+WA_0} , the Wholeness axiom for bounded formulas.

Our earlier publications showed semantic tableau admits partial exceptions to the Second Incompleteness Theorem where a formalism recognizes its self consistency and views multiplication as a 3-way relation (rather than as a total function). We now show these boundary-case evasions will collapse if the Law of the Excluded Middle is treated by tableau as a schema of logical axioms (instead of as derived theorems).

We show that the weakest versions of Foreman's minimal generic hugeness axioms cannot hold simultaneously on adjacent cardinals. Moreover, conventional forcing techniques cannot produce a model of one of these axioms.

We introduce bounded category forcing axioms for well-behaved classes ? . These are strong forms of bounded forcing axioms which completely decide the theory of some initial segment of the universe H λ + ? modulo forcing in ? , for some cardinal λ ? naturally associated to ? . These axioms naturally extend projective absoluteness for arbitrary set-forcing--in this situation λ ? =? --to classes ? with λ ? >? . Unlike projective absoluteness, these higher bounded category forcing axioms do not follow from large cardinal axioms, but can be forced under mild large cardinal assumptions on V . We also show the existence of many classes ? with λ ? = ? 1 , and giving rise to pairwise incompatible theories for H ? 2 .

We develop the machinery of indiscernible subspaces in continuous theories of expansions of Banach spaces, showing that any such theory has an indiscernible subspace and therefore an indiscernible set. We extend a result of Shelah and Usvyatsov by showing that a sequence of realizations of a (possibly unstable) minimal wide type p is a Morley sequence in p if and only if it is the orthonormal basis of an indiscernible subspace in p . We also give an example showing that minimal wide types do not generally have type-definable indiscernible subspaces (answering a question of Shelah and Usvyatsov), as well as an example showing that our result fails for non-minimal wide types, even in ω -stable theories.

We prove that if G=(V,E) is an ω -stable (respectively, superstable) graph with χ(G)> ℵ 0 (respectively, 2 ℵ 0 ) then G contains all the finite subgraphs of the shift graph Sh n (ω) for some n . We prove a variant of this theorem for graphs interpretable in stationary stable theories. Furthermore, if G is ω -stable with U(G)≤2 we prove that n≤2 suffices.

It is often claimed that analysis with infinitesimals requires more substantial use of the Axiom of Choice than traditional elementary analysis. The claim is based on the observation that the hyperreals entail the existence of nonprincipal ultrafilters over N, a strong version of the Axiom of Choice, while the real numbers can be constructed in ZF. The axiomatic approach to nonstandard methods refutes this objection. We formulate a theory SPOT in the st- ∈ -language which suffices to carry out infinitesimal arguments, and prove that SPOT is a conservative extension of ZF. Thus the methods of Calculus with infinitesimals are just as effective as those of traditional Calculus. The conclusion extends to large parts of ordinary mathematics and beyond. We also develop a stronger axiomatic system SCOT, conservative over ZF+ADC, which is suitable for handling such features as an infinitesimal approach to the Lebesgue measure. Proofs of the conservativity results combine and extend the methods of forcing developed by Enayat and Spector.

If we replace first order logic by second order logic in the original definition of Gödel's inner model L , we obtain HOD. In this paper we consider inner models that arise if we replace first order logic by a logic that has some, but not all, of the strength of second order logic. Typical examples are the extensions of first order logic by generalized quantifiers, such as the Magidor-Malitz quantifier, the cofinality quantifier, or stationary logic. Our first set of results show that both L and HOD manifest some amount of {\em formalism freeness} in the sense that they are not very sensitive to the choice of the underlying logic. Our second set of results shows that the cofinality quantifier gives rise to a new robust inner model between L and HOD. We show, among other things, that assuming a proper class of Woodin cardinals the regular cardinals > ℵ 1 of V are weakly compact in the inner model arising from the cofinality quantifier and the theory of that model is (set) forcing absolute and independent of the cofinality in question. We do not know whether this model satisfies the Continuum Hypothesis, assuming large cardinals, but we can show, assuming three Woodin cardinals and a measurable above them, that if the construction is relativized to a real, then on a cone of reals the Continuum Hypothesis is true in the relativized model.

We introduce a new inner model C(aa) arising from stationary logic. We show that assuming a proper class of Woodin cardinals, or alternatively M M ++ , the regular uncountable cardinals of V are measurable in the inner model C(aa) , the theory of C(aa) is (set) forcing absolute, and C(aa) satisfies CH. We introduce an auxiliary concept that we call club determinacy, which simplifies the construction of C(aa) greatly but may have also independent interest. Based on club determinacy, we introduce the concept of aa-mouse which we use to prove CH and other properties of the inner model C(aa) .

We introduce a notion of integration using ultrafilters and explore its relation to classical integrals.