
24Selfreferential theoriesJournal of Symbolic Logic. forthcoming.We study the structure of families of theories in the language of arithmetic extended to allow these families to refer to one another and to themselves. If a theory contains schemata expressing its own truth and expressing a specific Turing index for itself, and contains some other mild axioms, then that theory is untrue. We exhibit some families of true selfreferential theories that barely avoid this forbidden pattern.

174AGI and the KnightDarwin Law: why idealized AGI reproduction requires collaborationIn International Conference on Artificial General Intelligence, Springer. forthcoming.Can an AGI create a more intelligent AGI? Under idealized assumptions, for a certain theoretical type of intelligence, our answer is: “Not without outside help”. This is a paper on the mathematical structure of AGI populations when parent AGIs create child AGIs. We argue that such populations satisfy a certain biological law. Motivated by observations of sexual reproduction in seeminglyasexual species, the KnightDarwin Law states that it is impossible for one organism to asexually produce anot…Read more

278After generalizing the Archimedean property of real numbers in such a way as to make it adaptable to nonnumeric structures, we demonstrate that the real numbers cannot be used to accurately measure nonArchimedean structures. We argue that, since an agent with Artificial General Intelligence (AGI) should have no problem engaging in tasks that inherently involve nonArchimedean rewards, and since traditional reinforcement learning rewards are real numbers, therefore traditional reinforcement lea…Read more

64Can you find an xyequation that, when graphed, writes itself on the plane? This idea became internetfamous when a Wikipedia article on Tupper’s selfreferential formula went viral in 2012. Under scrutiny, the question has two flaws: it is meaningless (it depends on fonts) and it is trivial. We fix these flaws by formalizing the problem.

97Measuring the intelligence of an idealized mechanical knowing agentLecture Notes in Computer Science. forthcoming.We define a notion of the intelligence level of an idealized mechanical knowing agent. This is motivated by efforts within artificial intelligence research to define realnumber intelligence levels of compli cated intelligent systems. Our agents are more idealized, which allows us to define a much simpler measure of intelligence level for them. In short, we define the intelligence level of a mechanical knowing agent to be the supremum of the computable ordinals that have codes the agent knows t…Read more

159LeggHutter universal intelligence implies classical music is better than pop music for intellectual trainingThe Reasoner 13 (11): 7172. 2019.In their thoughtprovoking paper, Legg and Hutter consider a certain abstrac tion of an intelligent agent, and define a universal intelligence measure, which assigns every such agent a numerical intelligence rating. We will briefly summarize Legg and Hutter’s paper, and then give a tongueincheek argument that if one’s goal is to become more intelligent by cultivating music appreciation, then it is bet ter to use classical music (such as Bach, Mozart, and Beethoven) than to use more recent po…Read more

920Intelligence via ultrafilters: structural properties of some intelligence comparators of deterministic LeggHutter agentsJournal of Artificial General Intelligence 10 (1): 2445. 2019.Legg and Hutter, as well as subsequent authors, considered intelligent agents through the lens of interaction with rewardgiving environments, attempting to assign numeric intelligence measures to such agents, with the guiding principle that a more intelligent agent should gain higher rewards from environments in some aggregate sense. In this paper, we consider a related question: rather than measure numeric intelligence of one Legg Hutter agent, how can we compare the relative intelligence of …Read more

928Mathematical shortcomings in a simulated universeThe Reasoner 12 (9): 7172. 2018.I present an argument that for any computersimulated civilization we design, the mathematical knowledge recorded by that civilization has one of two limitations. It is untrustworthy, or it is weaker than our own mathematical knowledge. This is paradoxical because it seems that nothing prevents us from building in all sorts of advantages for the inhabitants of said simulation.

8394A type of simulation which some experimental evidence suggests we don't live inThe Reasoner 12 (7): 5656. 2018.Do we live in a computer simulation? I will present an argument that the results of a certain experiment constitute empirical evidence that we do not live in, at least, one type of simulation. The type of simulation ruled out is very specific. Perhaps that is the price one must pay to make any kind of Popperian progress.

33Arithmetical algorithms for elementary patternsArchive for Mathematical Logic 54 (12): 113132. 2015.Elementary patterns of resemblance notate ordinals up to the ordinal of Pi^1_1CA_0. We provide ordinal multiplication and exponentiation algorithms using these notations.

60The FirstOrder Syntax of Variadic FunctionsNotre Dame Journal of Formal Logic 54 (1): 4759. 2013.We extend firstorder logic to include variadic function symbols, and prove a substitution lemma. Two applications are given: one to bounded quantifier elimination and one to the definability of certain Borel sets.

76Formulas for Computable and NonComputable FunctionsRoseHulman Undergraduate Mathematics Journal 7 (2). 2006.

61Infinite graphs in systematic biology, with an application to the species problemActa Biotheoretica 61 (2): 181201. 2013.We argue that C. Darwin and more recently W. Hennig worked at times under the simplifying assumption of an eternal biosphere. So motivated, we explicitly consider the consequences which follow mathematically from this assumption, and the infinite graphs it leads to. This assumption admits certain clusters of organisms which have some ideal theoretical properties of species, shining some light onto the species problem. We prove a dualization of a law of T.A. Knight and C. Darwin, and sketch a dec…Read more

273An axiomatic version of Fitch’s paradoxSynthese 190 (12): 20152020. 2013.A variation of Fitch’s paradox is given, where no special rules of inference are assumed, only axioms. These axioms follow from the familiar assumptions which involve rules of inference. We show (by constructing a model) that by allowing that possibly the knower doesn’t know his own soundness (while still requiring he be sound), Fitch’s paradox is avoided. Provided one is willing to admit that sound knowers may be ignorant of their own soundness, this might offer a way out of the paradox

216Biologically Unavoidable SequencesElectronic Journal of Combinatorics 20 (1): 113. 2013.A biologically unavoidable sequence is an infinite gender sequence which occurs in every gendered, infinite genealogical network satisfying certain tame conditions. We show that every eventually periodic sequence is biologically unavoidable (this generalizes König's Lemma), and we exhibit some biologically avoidable sequences. Finally we give an application of unavoidable sequences to cellular automata.

234This sentence does not contain the symbol XThe Reasoner 7 (9): 108. 2013.A suprise may occur if we use a similar strategy to the Liar's paradox to mathematically formalize "This sentence does not contain the symbol X".

462A Machine That Knows Its Own CodeStudia Logica 102 (3): 567576. 2014.We construct a machine that knows its own code, at the price of not knowing its own factivity

267FastCollapsing TheoriesStudia Logica (1): 121. 2013.Reinhardt’s conjecture, a formalization of the statement that a truthful knowing machine can know its own truthfulness and mechanicalness, was proved by Carlson using sophisticated structural results about the ordinals and transfinite induction just beyond the first epsilon number. We prove a weaker version of the conjecture, by elementary methods and transfinite induction up to a smaller ordinal

214Guessing, MindChanging, and the Second Ambiguous ClassNotre Dame Journal of Formal Logic 57 (2): 209220. 2016.In his dissertation, Wadge defined a notion of guessability on subsets of the Baire space and gave two characterizations of guessable sets. A set is guessable if and only if it is in the second ambiguous class, if and only if it is eventually annihilated by a certain remainder. We simplify this remainder and give a new proof of the latter equivalence. We then introduce a notion of guessing with an ordinal limit on how often one can change one’s mind. We show that for every ordinal $\alpha$, a gu…Read more
Columbus, Ohio, United States of America
Areas of Interest
Epistemology 
Logic and Philosophy of Logic 
Philosophy of Mathematics 