•  24
    Self-referential theories
    Journal 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 self-referential theories that barely avoid this forbidden pattern.
  •  174
    AGI and the Knight-Darwin Law: why idealized AGI reproduction requires collaboration
    In 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 seemingly-asexual species, the Knight-Darwin Law states that it is impossible for one organism to asexually produce anot…Read more
  •  278
    After generalizing the Archimedean property of real numbers in such a way as to make it adaptable to non-numeric structures, we demonstrate that the real numbers cannot be used to accurately measure non-Archimedean structures. We argue that, since an agent with Artificial General Intelligence (AGI) should have no problem engaging in tasks that inherently involve non-Archimedean rewards, and since traditional reinforcement learning rewards are real numbers, therefore traditional reinforcement lea…Read more
  •  64
    Can you find an xy-equation that, when graphed, writes itself on the plane? This idea became internet-famous when a Wikipedia article on Tupper’s self-referential 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.
  •  97
    Measuring the intelligence of an idealized mechanical knowing agent
    Lecture 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 real-number 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
  •  159
    In their thought-provoking 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 tongue-in-cheek 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
  •  920
    Legg and Hutter, as well as subsequent authors, considered intelligent agents through the lens of interaction with reward-giving 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
  •  928
    Mathematical shortcomings in a simulated universe
    The Reasoner 12 (9): 71-72. 2018.
    I present an argument that for any computer-simulated 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.
  •  8394
    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.
  •  33
    Arithmetical algorithms for elementary patterns
    Archive for Mathematical Logic 54 (1-2): 113-132. 2015.
    Elementary patterns of resemblance notate ordinals up to the ordinal of Pi^1_1-CA_0. We provide ordinal multiplication and exponentiation algorithms using these notations.
  •  60
    The First-Order Syntax of Variadic Functions
    Notre Dame Journal of Formal Logic 54 (1): 47-59. 2013.
    We extend first-order 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.
  •  76
    Formulas for Computable and Non-Computable Functions
    Rose-Hulman Undergraduate Mathematics Journal 7 (2). 2006.
  •  61
    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
  •  273
    An axiomatic version of Fitch’s paradox
    Synthese 190 (12): 2015-2020. 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
  •  216
    Biologically Unavoidable Sequences
    Electronic Journal of Combinatorics 20 (1): 1-13. 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.
  •  234
    This sentence does not contain the symbol X
    The 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".
  •  267
    Fast-Collapsing Theories
    Studia Logica (1): 1-21. 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
  •  74
  •  214
    Guessing, Mind-Changing, and the Second Ambiguous Class
    Notre Dame Journal of Formal Logic 57 (2): 209-220. 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