We examine recursive monotonic functions on the Lindenbaum algebra of EA. We prove that no such function sends every consistent ϕ to a sentence with deductive strength strictly between ϕ and pϕ^Conpϕqq. We generalize this result to iterates of consistency into the effective transfinite. We then prove that for any recursive monotonic function f , if there is an iterate of Con that bounds f everywhere, then f must be somewhere equal to an iterate of Con.Definition 3.2. Given an elementary presentation xα, ăy of a recursive wellordering and a sentence ϕ, we use Gödel's fixed point lemma to define sentences Con ‹ pϕ, βq for β ă α as follows.EA $ Con ‹ pϕ, βq Ø @γ ă β, Conpϕ^Con ‹ pϕ, γqq.We use the notation Con β pϕq for Con ‹ pϕ, βq.Remark 3.3. Note that, since it is elementarily calculable whether a number represents zero or a successor or a limit, the following clauses are provable in EA.‚ Con 0 pϕq Ø J ‚ Con γ`1 pϕq Ø Conpϕ^Con γ pϕqq ‚ Con λ pϕq Ø @γ ă λ, Con γ pϕq for λ a limit.Note that this hierarchy is proper for true ϕ by Gödel's second incompleteness theorem. We need to prove that for transfinite α, Con α is monotonic over the Lindenbaum algebra of EA. Before proving this claim we recall Schmerl's [12] technique of reflexive transfinite induction. Note that "Prpϕq" means that ϕ is provable in EA. Proposition 3.4. (Schmerl) Suppose that ă is an elementary linear order and thatEA $ @αpPrp@β ă α, Apβqq Ñ Apαqq. Then EA $ @αApαq.Proof. From EA $ @αpPrp@β ă α, Apβqq Ñ Apαqq we infer EA $ Prp@αApαqq Ñ @αPrp@β ă α, Apβqq Ñ @αApαq.Löb's theorem, i.e., if EA $ Prpζq Ñ ζ, then EA $ ζ, then yields EA $ @αApαq. ❑ Proposition 3.5. If ϕ $ ψ, then Con α pϕq $ Con α pψq.Proof. Let Apβq denote the claim that Con β pϕq $ Con β pψq.We want to prove that Apαq, without placing any restrictions on α. We prove the equivalent claim that EA $ Apαq. By Proposition 3.4, it suffices to show that EA $ @αpPrp@β ă α, Apβqq Ñ Apαqq.Reason within EA. Suppose that Prp@β ă α, Apβqq, which is to say that Prp@β ă α, pCon β pϕq $ Con β pψqqq.Since Con α pϕq contains EA, we infer that Con α pϕq $ @β ă αpCon β pϕq $ Con β pψqq, which is just to say that Con α pϕq $`@β ă α, EA $ pCon β pϕq Ñ Con β pψqq˘.Since Con α pϕq proves that for all β ă α, EA & Con β pϕq we infer that Con α pϕq $ @β ă αpEA & Con β pψqq.EA proves its own Σ 0 1 completeness, i.e., EA proves that if EA does not prove a Σ 0 1 statement ζ, then ζ is false. Thus, Con α pϕq $ @β ă αpCon β pψqq.

It is well-known that natural axiomatic theories are well-ordered by consistency strength. However, it is possible to construct descending chains of artificial theories with respect to consistency strength. We provide an explanation of this well-orderedness phenomenon by studying a coarsening of the consistency strength order, namely, the $\Pi ^1_1$ reflection strength order. We prove that there are no descending sequences of $\Pi ^1_1$ sound extensions of $\mathsf {ACA}_0$ in this ordering. Accordingly, we can attach a rank in this order, which we call reflection rank, to any $\Pi ^1_1$ sound extension of $\mathsf {ACA}_0$ . We prove that for any $\Pi ^1_1$ sound theory T extending $\mathsf {ACA}_0^+$ , the reflection rank of T equals the $\Pi ^1_1$ proof-theoretic ordinal of T. We also prove that the $\Pi ^1_1$ proof-theoretic ordinal of $\alpha $ iterated $\Pi ^1_1$ reflection is $\varepsilon _\alpha $ . Finally, we use our results to provide straightforward well-foundedness proofs of ordinal notation systems based on reflection principles.

In mathematical logic there are two seemingly distinct kinds of principles called “reflection principles.” Semantic reflection principles assert that if a formula holds in the whole universe, then it holds in a set-sized model. Syntactic reflection principles assert that every provable sentence from some complexity class is true. In this paper, we study connections between these two kinds of reflection principles in the setting of second-order arithmetic. We prove that, for a large swathe of theories, [Formula: see text]-model reflection is equivalent to the claim that arbitrary iterations of uniform [Formula: see text] reflection along countable well-orderings are [Formula: see text]-sound. This result yields uniform ordinal analyzes of theories with strength between [Formula: see text] and [Formula: see text]. The main technical novelty of our analysis is the introduction of the notion of the proof-theoretic dilator of a theory [Formula: see text], which is the operator on countable ordinals that maps the order-type of [Formula: see text] to the proof-theoretic ordinal of [Formula: see text]. We obtain precise results about the growth of proof-theoretic dilators as a function of provable [Formula: see text]-model reflection. This approach enables us to simultaneously obtain not only [Formula: see text], [Formula: see text] and [Formula: see text] ordinals but also reverse-mathematical theorems for well-ordering principles.

It is a well known empirical observation that natural axiomatic theories are pre-well-ordered by consistency strength. For any natural theory T , the next strongest natural theory is T`Con T . We formulate and prove a statement to the effect that the consistency operator is the weakest natural way to uniformly extend axiomatic theories.

Influence theory is a foundational theory of physics that is not based on traditional empirically defined concepts, such as positions in space and time, mass, energy, or momentum. Instead, the aim is to derive these concepts, and their empirically determined relationships, from a more primitive model. It is postulated that there exist things, which are call particles, that influence one another in a discrete and directed fashion resulting in a partially ordered set of influence events. The problem of consistent quantification of the influence events is considered. Observers are modeled as particle chains (observer chains) as if an observer were able to track a particle and quantify the influence events that the particle experiences. From these quantified influence events, consistent quantification of the universe of events based on the observer chains is studied. Herein, the kinematics and dynamics of particles from the perspective of influence theory are both reviewed and further developed.