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Rupert McCallum

Publications and source records attributed to Rupert McCallum.

17 recordsLinked to original sources
arXiv2026

Inconsistency of Reinhardt cardinals with $\mathsf{ZF}$

A proof will be presented that the existence of a non-trivial $\Sigma_1$-elementary embedding $j: V_{\lambda+3} \prec V_{\lambda+3}$ is inconsistent with $\textsf{ZF}$. Sections 1 and 2 shall review various important contributions from the literature, notably including \cite{Goldberg2020}, \cite{Schlutzenberg2020}, and \cite{Woodin2010}, the latter reference being where the crucial forcing construction is presented. Section 3 shall introduce some new large cardinal properties, of consistency strength intermediate between $\mathsf{I_3}$ and $\mathsf{I_2}$, and greater than $\mathsf{I_1}$, respectively. The proof of the inconsistency with $\mathsf{ZF}$ of the existence of a non-trivial $\Sigma_1$-elementary embedding $j:V_{\lambda+3} \prec V_{\lambda+3}$ shall be given in Section 4. The claims of Sections 2 and 4 are provable in $\textsf{ZF}$; those of Section 3, with the exception of the last two theorems, in $\textsf{ZFC}$.

math.LO↗
arXiv2021

New Large Cardinal Axioms and the Ultimate-L Program

We will consider a number of new large-cardinal properties, the $α$-tremendous cardinals for each limit ordinal $α>0$, the hyper-tremendous cardinals, the $α$-enormous cardinals for each limit ordinal $α>0$, and the hyper-enormous cardinals. For limit ordinals $α>0$, the $α$-tremendous cardinals and hyper-tremendous cardinals have consistency strength between I3 and I2. An $ω$-enormous cardinal has consistency strength greater than I0, and also all the large-cardinal axioms discussed in the second part of Hugh Woodin's paper on suitable extender models, not known to be inconsistent with ZFC and of greater consistency strength than I0. Ralf Schindler and Victoria Gitman have developed the notion of a virtual large-cardinal property, and a clear sense can be given to the notion of "virtually $ω$-enormous". A virtually $ω$-enormous cardinal can be shown to dominate a Ramsey cardinal. It can be shown that a cardinal $κ$ which is a critical point of an elementary embedding $j:V_{λ+2} \prec V_{λ+2}$, in a context not assuming choice, is necessarily a hyper-enormous cardinal. Building on this insight, we can obtain the result that the existence of such an elementary embedding is in fact outright inconsistent with ZF. The assertion that there is a proper class of $α$-enormous cardinals for every limit ordinal $α>0$ can be shown to imply a version of the Ultimate-L Conjecture.

math.LO↗
arXiv2020

Intrinsic justifications for large-cardinal axioms

William Tait and Peter Koellner have written on the topic of which reflection principles are intrinsically justified. Phillip Welch and Sam Roberts have recently sought to motivate much stronger reflection principles. This work also had some similarities with previous work of Victoria Marshall's on reflection principles, as may be seen from considerations which we will seek to articulate in what follows. The statement of our three philosophical theses shall be given in the Introduction. We will present a set of technical results in support of three philosophical theses which will be articulated at the end of the Introduction.

math.LO↗
arXiv2020

Brouwer fixed point theorem as a corollary of Lawvere

It is investigated in what sense the Brouwer fixed point theorem may be viewed as a corollary of the Lawvere fixed point theorem. A suitable generalisation of the Lawvere fixed point theorem is found and a means is identified by which the Brouwer fixed point theorem can be shown to be a corollary, once an appropriate continuous surjective mapping $A' \rightarrow X^{A''}$ has been constructed for each space $X$ in a certain class of "nice" spaces for each one of which the exponential topology on $X^{A''}$ exists, and here $A'$ and $A''$ have the same carrier set and the topology on $A'$ is finer than on $A''$. It is shown that there is a certain natural way of attempting to derive Brouwer as a corollary of Lawvere which is not possible, that is there is no space $A$ for which the exponential topology on $[0,1]^{A}$ exists and there is a continuous surjection $A \rightarrow [0,1]^{A}$. We then examine the range of contexts in which phenomena like those described in the first result occur, from a broadly model-theoretic perspective, with a view towards applications for the original motivation for the problem as a problem in decision theory for AI systems, suggested by the Machine Intelligence Research Institute.

math.LO↗
arXiv2020

A proof of P!=NP

We show that it is provable in PA that there is an arithmetically definable sequence $\{\phi_{n}:n \in \omega\}$ of $\Pi^{0}_{2}$-sentences, such that - PRA+$\{\phi_{n}:n \in \omega\}$ is $\Pi^{0}_{2}$-sound and $\Pi^{0}_{1}$-complete - the length of $\phi_{n}$ is bounded above by a polynomial function of $n$ with positive leading coefficient - PRA+$\phi_{n+1}$ always proves 1-consistency of PRA+$\phi_{n}$. One has that the growth in logical strength is in some sense "as fast as possible", manifested in the fact that the total general recursive functions whose totality is asserted by the true $\Pi^{0}_{2}$-sentences in the sequence are cofinal growth-rate-wise in the set of all total general recursive functions. We then develop an argument which makes use of a sequence of sentences constructed by an application of the diagonal lemma, which are generalisations in a broad sense of Hugh Woodin's "Tower of Hanoi" construction as outlined in his essay "Tower of Hanoi" in Chapter 18 of the anthology "Truth in Mathematics". The argument establishes the result that it is provable in PA that $P \neq NP$. We indicate how to pull the argument all the way down into SEFA.

math.LO↗
arXiv2019

Building lattices and zeta functions

We introduce the notion of a building lattice generalizing tree lattices. We give a Lefschetz formula and apply it to geometric zeta functions. We further generalize Bass's approach to Ihara zeta functions to the higher dimensional case of a building.

math.GR↗
arXiv2017

All large-cardinal axioms not known to be inconsistent with ZFC are justified

In other work we have outlined how, building on ideas of Welch and Roberts, one can motivate believing in the existence of supercompact cardinals. After making this observation we strove to formulate a justification for large-cardinal axioms of greater strength, and arrived at a motivation for a new large-cardinal property, which we define here and prove to be equivalent to the property of being a Vopěnka scheme cardinal. Making use of this result, one can also show that a theory $B_0(V_0)$ described in a previous paper of Victoria Marshall implies the existence of a Vopěnka scheme cardinal $κ$ such that $V_κ \prec V$ (and therefore, in particular, a proper class of extendible cardinals as well). Marshall left as an open question whether her theory $B_0(V_0)$, whose consistency is implied by the existence of an almost huge cardinal, implied the existence of supercompact or extendible cardinals. Here both questions are resolved positively. In the final section we give an account of how one could plausibly motivate every large-cardinal axiom not known to be inconsistent with choice while stopping short of the point of inconsistency with choice.

math.LO↗
arXiv2017

Which axioms of set theory are intrinsically justified?

We recently formulated a new large-cardinal axiom of strength intermediate between a totally indescribable cardinal and an $ω$-Erdős cardinal, positing the existence of what we called an "extremely reflective cardinal", and we showed that the property of being extremely reflective was in fact equivalent to the property of being remarkable, and we sought to argue that this axiom should be seen as intrinsically justified. This built on related earlier work in which the notion of an $α$-reflective cardinal was formulated. Then Welch and Roberts put forward a family of reflection principles, Welch's principle implying the existence of a proper class of Shelah cardinals and provably consistent relative to a superstrong cardinal, and Roberts' principle implying the existence of a proper class of 1-extendible cardinals and provably consistent relative to a 2-extendible cardinal. Roberts tentatively argued that his principle should be seen as intrinsically justified (at least on the assumption that a weaker form of reflection involving reflection of second-order formulas with a second-order parameter should be seen as intrinsically justified). This work overlapped with previous work of Victoria Marshall's on reflection principles. We analyze the relationship between reflection principles equivalent to those studied in my earlier work and stronger but similar reflection principles which are natural extensions of those of Welch and Roberts. We also show how a natural strengthening of Roberts' reflection principle yields the existence of supercompact cardinals, and in the process solve a question which Marshall left open, of whether her theory $B_0(V_0)$ is strong enough to imply the existence of supercompact cardinals. We also manage to resolve negatively her question of whether her theory $B_1(V_0)$ implies the existence of a huge cardinal.

math.LO↗
arXiv2016

Twisted Poincare Series and Zeta functions on finite quotients of buildings

In the case where $G=$SL$_{2}(F)$ for a non-archimedean local field $F$ and $Γ$ is a discrete torsion-free cocompact subgroup of $G$, there is a known relationship between the Ihara zeta function for the quotient of the Bruhat-Tits tree of $G$ by the action of $Γ$, and an alternating product of determinants of twisted Poincaré series for parabolic subgroups of the affine Weyl group of $G$. We show how this can be generalised to other split simple algebraic groups of rank two over $F$, and formulate a conjecture about how this might be generalised to groups of higher rank.

math.GR↗
arXiv2016

Height growth on semisimple groups

A condition is given, under which a general lattice point counting function is asymptotic to the corresponding ball volume growth function. This is then used to give height asymptotics in the style of the Batyrev-Manin Conjecture for certain intrinsically defined heights on semisimple groups.

math.NT↗
arXiv2014

Topological rigidity in totally disconnected locally compact groups

In \cite{Kramer11} Kramer proves for a large class of semisimple Lie groups that they admit just one locally compact $σ$-compact Hausdorff topology compatible with the group operations. We present two different methods of generalising this to the group of rational points of an absolutely quasi-simple algebraic group over a non-archimedean local field (the second method only achieves this on the additional hypothesis that the group is isotropic). The first method of argument involves demonstrating that, given any topological group $G$ which is totally disconnected, locally compact, $σ$-compact, locally topologically finitely generated, and has the property that no compact open subgroup has an infinite abelian continuous quotient, the group $G$ is topologically rigid in the previously described sense. Then the desired conclusion for the group of rational points of an absolutely quasi-simple algebraic group over a non-archimedean local field may be inferred as a special case. The other method of argument involves proving that any group of automorphisms of a regular locally finite building, which is closed in the compact-open topology and acts Weyl transitively on the building, has the topological rigidity property in question. This again yields the desired result in the case that the group is isotropic.

math.GR↗
arXiv2014

Free subgroups of special linear groups

We present a proof of the following claim. Suppose that $n$ is an integer such that $n>1$ and that $k$ is any field. Suppose that $g$ is an element of $\mathrm{SL}(n,k)$ of infinite order. Then the set $\{h\in\mathrm{SL}(n,k)\mid $ is a free group of rank two$\}$ is a Zariski dense subset of $\mathrm{SL}(n,\bar{k})$ where $\bar{k}$ is an algebraic closure of $k$.

math.GR↗
arXiv2012

A Local-to-Global Result for Topological Spherical Buildings

Suppose that Δ, Δ' are two buildings each arising from a semisimpe algebraic group over a field, a topological field in the former case, and that for both the buildings the Coxeter diagram has no isolated nodes. We give conditions under which a partially defined injective chamber map, whose domain is the subcomplex of Δ, generated by a nonempty open set of chambers, and whose codomain is Δ', is guaranteed to extend to a unique injective chamber map. Related to this result is a local version of the Borel-Tits theorem on abstract homomorphisms of simple algebraic groups.

math.MG↗