Lowenheim Skolem Theorem

This dissertation aims at characterizing the profound change occurred in Mathematics at the period crossing the end of the 19th to the dawn of the twentieth century wherein various tensions about Arithmetic Formalisation emerged around different schools within Mathematics. Harking back to Leibniz's dream of Lingua Universalis, Frege's Begriffschrift, and Russell and Whitehead's Principia Mathematica, those tensions are depicted as summoned by Cantor's work on Set Theory.

The challenge of formalization as proposed by Hilbert in the year of 1900, set forth the path for the schools of mathematical thought to deploy programs and tools to tackle it. In the thread of such endeavors, two formal works emerged that would change the mathematical landscape and horizons. The first, Gödel's Incompleteness Theorem, a bright star in the firmament of logics-mathematics; and the Löwenheim-Skolem Theorem of Non-Categoricity, less well known but full of consequences to the epistemological effort to certified mathematics as the queen of sciences.

By describing both Theorems and drawing their meaning and scope, this dissertation claims that, although tracing inherent limits to formalization outcomes in logic-mathematics, they nevertheless offered the key to understand Mathematics' unique creative nature and ceaselessly evolutive path within its strict logical and methodological foundations.

I make a Quinean case that Quine's ontological relativity marked a wrong turn in his philosophy. I think indeed that his fundamental commitments point towards the classical view of ontology that was worked out in the most detail in Word and Object.

In  this  article,  we  will  show  that  uncomputability  is  a  relative property  not  only  of  oracle  Turing  machines,  but  also  of  subrecursive classes. We will define the concept of a Turing submachine, and a recursive relative  version  for  the  Busy Beaver  function which we  will  call  Busy Beaver Plus function. Therefore, we will prove that the computable Busy Beaver Plus function defined on any Turing submachine is not computable by any program running on this submachine. We will thereby demonstrate the existence of a “paradox” of computability a la Skolem: a function is computable when “seen from the outside” the subsystem, but uncomputable when “seen from within” the same subsystem. Finally, we will raise the possibility of defining universal submachines,  and a hierarchy of negative Turing degrees.

Metabiology is a mathematical theory mainly based on algorithmic information theory and allows us to study an open-ended evolution of programs, that is, to study how fast the organisms/programs become more complex or more creative without never stagnate at a maximum level of complexity or creativity. We take, as starting point, the first chaitinian metabiological models in which the organisms are computable systems, Nature (or environment) is a hyper-computable system, mutations are randomly generated computable systems (that is, random algorithmic mutations) and the fitnesses are measured by how big is the output of the organisms. In this context, this paper focuses on two questions: Assuming that organisms are uncomputable systems, should metabiology become unsuitable or unsound? Or, on the other hand, assuming that Nature itself is a computable system, should metabiology become unsuitable or unsound? Thus, the main purpose is to demonstrate that metabiology walks towards a general theory of open-ended evolutionary systems, whether these systems are sub-computable, computable or hyper-computable. As a secondary purpose we show that scientific and philosophical consequences of our mathematical results spread across fundamental questions about biology and artificial life, as well as, about physics and artificial intelligence. To satisfy these two objectives, we build two new metabiological models for two new different configurations of organisms and nature. First, we demonstrate it is possible to build a metabiological evolution in which nature is a computable system and, hence, it makes the organisms and mutations to be sub-computable systems. To show this we introduce the recursive relative uncomputability (a computable/recursive version of the relative uncomputability that occurs among different Turing degrees). We do with Turing, Radó and Chaitin the same Skolem did with Cantor by making a relativization of the Busy-Beaver´s uncomputability in relation to the Omega number, so that this new uncomputability
can exist between a computer and any of his sub-computers. As a consequence, we introduce the “paradox” of uncomputability which says that exists at least one function that is not computable if we “look from the inside”, but that is computable if we “look from the outside”. Further, we correlate this pseudo-paradox with problems of human creativity and artificial intelligence and with problems of consciousness, artificial life and the computability of the Universe. Second, we build a metabiological model for the evolution of hyper-programs in which the organisms are hyper-computable systems of any finite order and nature is a hyper-program of ordinal order omega. In this last model we demonstrate it is possible an evolution of organisms/hyper-programs, with just “blind” random algorithmic mutations (it means, that do not take into consideration previous organisms/hyper-programs), which progressively leads the organisms/hyper-programs to be able of solving any problem in the arithmetical hierarchy. Allied to this, we discuss about computability or uncomputability of “living beings” and upon the possibility of mathematically studying metabiological evolutionary systems that can surpass, as much as we want, the “frontier” of the computable.

Podemos chamar o famoso “paradoxo” de Skolem de um pseudoparadoxo metalinguístico da denumerabilidade: de dentro do sistema um conjunto não é enumerável enquanto que de fora do sistema ele pode ser. O mesmo fenômeno aparece em relação à computabilidade, ou seja, existe um pseudoparadoxo metalinguístico da computabilidade? No qual “de dentro” algo é incomputável enquanto que “de fora” esse algo pode ser computável? É na metabiologia que esse fenômeno surge para ajudar a responder a pergunta sobre como ficaria a evolução dos organismos/programas sujeitos a mutações algorítmicas aleatórias numa natureza computável, visto que nos primeiros modelos demonstrados por Gregory Chaitin isso se dá numa natureza oracular, um hipercomputador.

Palavras-Chave: Metabiologia, Paradoxo de Skolem, Teoria da Computabilidade, Computabilidade da Vida, Hipótese do Universo Computável, Teoria da Informação Algorítmica.

Zahlen standen am Beginn des mathematischen Denkens und bilden bis heute einen syste-matischen Bezugspunkt im Aufbau und Verständnis der Mathematik. Für jede logische und philosophische Untersuchung der Mathematik spielt die Idee der Zahl eine zentrale Rolle. Im vorliegenden Buch wird anhand der natürlichen Zahlen, und später der reellen Zahlen, auf Fragen im Umfeld der Unendlichkeit der Zahlenreihe und der Eindeutigkeit des axiomatisch definierten Zahlbegriffs eingegangen, bis hin zur konkreten Einführung von Nichtstandardzah-len. An zentraler Stelle stehen Auseinandersetzungen um die Bedeutung und Tragweite der klassischen formalen Logik, der Mengenlehre und der modernen symbolischen (mathemati-schen) Logik als Darstellungsmittel mathematischer Zusammenhänge. Dies schliesst eine originelle philosophische Diskussion von Paradoxien und von verschiedenartigen Grenzen logischer und symbolischer Darstellbarkeit mit ein, die auch Paul Finslers Gedanken zur Grundlegung der Mathematik einbeziehen. Im Anhang findet sich ein Nachwort von Gregor Schneider, welches die vorliegenden Untersuchungen mit einigen modernen Tendenzen in der Philosophie der Mathematik konfrontiert, sowie ein bisher nicht auf Deutsch erschiener Aufsatz von Thoralf Skolem aus dem Jahre 1941 über die Grundlegung der Mengenlehre

Abstract Guillermo Rosado-Haddock:
In this short paper I am concerned with basically two especially important issues in Oswaldo Chateaubriand's Logical Forms II; namely, the dispute between first-and higher-order logic and his conception of logical truth and related notions, like logical property, logical state of affairs and logical falsehood. The first issue was also present in the first volume of the book, but the last is privative of the second volume. The extraordinary significance of both issues for philosophy is emphasized and, though there is a basic agreement with Chateaubriand's views, some critical remarks are interspersed.

Abstract response:
In my response to Guillermo Rosado-Haddock I discuss the two main issues raised in his paper. The first is that by allowing Henkin’s general models as a legitimate model-theoretic interpretation of second-order logic, I undermine my defense of second-order logic against Quine’s views concerning the primacy of first-order logic. The second is that my treatment of logical truth and logical properties does not take into account various systems of logic and properties of systems of logic such as the Löwenheim-Skolem property.

On its own, the Löwenheim-Skolem theorem – as well as the ensuing Skolem Paradox – is of scant mathematical, logical, and even philosophical significance. It is of no consequence to practicing mathematicians, admits of extant resolutions at the logical level and is sufficiently vague to support all sorts of philosophical arguments. Nevertheless, much enquiry and ink has flowed on the matter, and we intend here to contribute to this muddy river, all the while trying to avoid waxing poetic or waning technical. In this essay, we purport to use the paradox as a paradigm by which we mean to evaluate the role of first-and second-order predicate logic in a post-foundational context. As there is no reason to prefer one kind of logic over another unless we specify to what purpose we intend to use it, we shall argue that if we want mathematical logic to axiomatize, describe and couch the language of informal mathematical practice, then second-order logic yields more intuitively appropriate models. As well, since the Skolem paradox is a problem purely for standard model-theoretic semantics, our secondary purpose will simply be to show through this example how modern model-theoretic results can illuminate our current understanding of what logic is, as well as of what we conceive the role of logic to be in the greater scheme of human understanding. Indeed it is our over-arching intention to ponder the inherent limitations and, paradoxically, the intrinsic openness of human thought. FOL is generally considered incapable of axiomatizing some of the most elementary branches of mathematics – including set-theory, which is crucial to nearly all branches of the discipline. Not surprisingly, what FOL cannot axiomatize is what it can't model: infinite structures. This is where LST can be considered the tipping-point, especially if one considers the fact that the language of infinity is crucial to our modern understanding of mathematics and physics. LST is an issue for model-theoretic semantics, and where FOL fails the test is precisely in the semantic section. The semantics involved in informal mathematical practice-even such well-understood notions as finitude, mathematical induction, minimal closure and well-founded relations like the predecessor and 'less than' relations (which may all be easily constructed with second-order formulas) – simply cannot be expressed in first-order axiomatizations. FOL neither resembles our mathematical structures nor seems useful to the working mathematician looking for guidance in far-off places like philosophy, logic and meta-mathematics. While SOL can no more provide a foundation for all of mathematics, it does shed more light on mathematical theory and practice than its first-order counterpart. The Löwenheim-Skolem theorem confronts us with the facile way in which we rely upon unspoken agreements as to our intended models. Indeed, the relativity results inherent to LST can lend credence to the idea that logic might just be a purely formal science. Given the LSTT, Skolem's paradox proves neither the existence nor the non-existence of non-denumerable sets as the first-order model actually allows sentences to be interpreted either in the denumerable or the non-denumerable domain. The interpreter's choice is then guided by other than purely formal considerations – such as one's prior commitment to the ontological existence or characteristics of non-denumerable sets. All that the LST actually states is that the models generated by FOL cannot characterize, cannot say anything about whatever the hypothetical structure of non-denumerable sets might be. In actuality, formal logic does rely on informal mathematical theories and practice to guide its investigations: it is those theories that we want to generate in our models. However, modern axiomatics are not merely meant to codify and crystallize static states of knowledge or disembodied mathematical notions floating in the sky above. Or, rather, if this is what some intend it to do, then modern

A metabiologia é uma teoria matemática baseada, principalmente, na teoria da informação algorítmica, nos permitindo estudar a evolução open-ended de programas, isto é, estudar quão rápido os organismos/programas se tornam mais complexos ou mais criativos, sem nunca estagnar num nível máximo de complexidade ou criatividade. Tomamos como base os modelos metabiológicos originais chaitinianos, nos quais os organismos são sistemas computáveis (i.e., sistemas emuláveis por um computador), a natureza (ou meio-ambiente) é um sistema hipercomputável (i.e., um sistema emulável apenas por algum hipercomputador), as mutações são sistemas computáveis aleatoriamente gerados (i.e., mutações algorítmicas aleatórias) e a aptidão dos organismos é medida estritamente por quão grande é o output desses organismos. Nesse âmbito, este trabalho se foca em responder duas questões. A metabiologia seria inapropriada se os organismos não forem sistemas computáveis? Ou, na direção oposta, ela seria inapropriada se a própria Natureza for um sistema computável? Portanto, o objetivo principal é mostrar que a metabiologia caminha em direção a uma teoria geral para evolução open-ended de qualquer sistema, seja ele subcomputável, computável ou hipercomputável. Como objetivo secundário, mostramos que as consequências científicas e filosóficas dos nossos resultados matemáticos podem se alastrar pelas questões fundamentais da biologia e vida artificial, como também, da física e da inteligência artificial. Para satisfazer esses dois objetivos, construímos dois novos modelos metabiológicos para duas configurações, diferentes entre si, de organismos e natureza. Primeiro, mostramos que é possível a evolução metabiológica de subprogramas, dentro da qual tomamos a natureza como sendo computável, o que fez com que os organismos e as mutações passassem a sistemas subcomputáveis. Para essa demonstração, apresentamos e utilizamos o fenômeno da incomputabilidade relativa recursiva (uma versão computável/recursiva da incomputabilidade relativa que ocorre entre as diferentes ordens de hipercomputadores dentro da hierarquia dos graus de Turing). Fazemos com Turing, Radó e Chaitin o que Skolem fez com Cantor: relativizamos a incomputabilidade da função Busy-Beaver atrelada ao número Ω para ela existir, de modo análogo, entre um computador e qualquer subcomputador do mesmo. Em decorrência, apresentamos o “paradoxo” da computabilidade, o qual diz que existe, pelo menos, uma função que não é computável se “olhada de dentro”, mas é computável se “olhada de fora”. Então, correlacionamos esse pseudoparadoxo com problemas da criatividade humana e inteligência artificial, da consciência, da vida artificial e da computabilidade do Universo. Segundo, construímos um modelo metabiológico para a evolução de hiperprogramas, no qual os organismos são sistemas hipercomputáveis de qualquer ordem finita (dentro da hierarquia de Turing) e a natureza é um hiperprograma de ordem ordinal ω. Nesse último modelo, mostramos que é possível uma evolução de organismos/hiperprogramas por meio de mutações algorítmicas aleatórias “cegas” (i.e., que não levam em conta os organismos/hiperprogramas anteriores) que vai levando os organismos/hiperprogramas a serem capazes de resolver qualquer problema na hierarquia aritmética. Aliado a isso, discutimos sobre a computabilidade ou incomputabilidade dos seres vivos e sobre a possibilidade de estudar matematicamente a evolução metabiológica de sistemas que podem ultrapassar, o quanto se queira, a “barreira” do computável.

This work is partially motivated by the recent articles related to the continuum hypothesis and the forcing axioms. Ever since Gödel's and Cohen's proofs that the ZFC cannot be used to disprove or approve continuum hypothesis, i.e., that it is independent of the ZFC axioms, there have been efforts to add at least one new axiom to ZFC that should illuminate the structure of infinite sets. The two main contenders for addition to ZFC are the inner-model axiom 'V-ultimate L' that builds a universe of sets from the ground up and Martin's axiom that allows to expand it outward in all directions. If Martin's axiom is correct, then the continuum hypothesis is false, and if the inner-model axiom is right, then the continuum hypothesis is true. There is also an orthogonal view assuming that there are myriad mathematical universes, some in which the continuum hypothesis is true and others in which it is false, but all equally worth of exploring. That model dependent view is underlined with the Löwenheim-Skolem theorem, no theory can have only nondenumerable models; if a theory has a nondenumerable model, it must have denumerably infinite ones as well. Accepting axiomatization in whatever shape means the notions 'denumerable' and 'nondenumerable' turn out to be relative, and it requires to accept the relativization of cardinals Skolem and Carnap first point it out. To show the impact of axiomatization, consider will be as an example a set created by a successive generation of algebraic pairs, each filled with a single transcendental number, and examined will be the impact of the axiom of choice on the determining the property of such set. It will be shown that the given example supports the Löwenheim-Skolem theorem. To underline the uncertainty implied by that theorem considered will be Wang's Σ model that generates a set in a constructive way, different from ZF. For the first time, provided will be an explicit example that satisfies Wang's model. In our example, a set of algebraic numbers will correspond to the 0-th Wang's layer, the 1-st layer will be a set of transcendental numbers generated by diagonal cuts applied to the 0 − th layers, and successive layers will be produced by diagonal cuts applied on the previously generated union of layers. It will be shown that this model also supports the Löwenheim-Skolem theorem.

On its own, the Löwenheim-Skolem theorem – as well as the ensuing Skolem Paradox – is of scant mathematical, logical, and even philosophical significance. It is of no consequence to practicing mathematicians, admits of extant resolutions at the logical level and is sufficiently vague to support all sorts of philosophical arguments. Nevertheless, much enquiry and ink has flowed on the matter, and we intend here to contribute to this muddy river, all the while trying to avoid waxing poetic or waning technical. In this essay, we purport to use the paradox as a paradigm by which we mean to evaluate the role of first-and second-order predicate logic in a post-foundational context. As there is no reason to prefer one kind of logic over another unless we specify to what purpose we intend to use it, we shall argue that if we want mathematical logic to axiomatize, describe and couch the language of informal mathematical practice, then second-order logic yields more intuitively appropriate models. As well, since the Skolem paradox is a problem purely for standard model-theoretic semantics, our secondary purpose will simply be to show through this example how modern model-theoretic results can illuminate our current understanding of what logic is, as well as of what we conceive the role of logic to be in the greater scheme of human understanding. Indeed it is our over-arching intention to ponder the inherent limitations and, paradoxically, the intrinsic openness of human thought. FOL is generally considered incapable of axiomatizing some of the most elementary branches of mathematics – including set-theory, which is crucial to nearly all branches of the discipline. Not surprisingly, what FOL cannot axiomatize is what it can't model: infinite structures. This is where LST can be considered the tipping-point, especially if one considers the fact that the language of infinity is crucial to our modern understanding of mathematics and physics. LST is an issue for model-theoretic semantics, and where FOL fails the test is precisely in the semantic section. The semantics involved in informal mathematical practice-even such well-understood notions as finitude, mathematical induction, minimal closure and well-founded relations like the predecessor and 'less than' relations (which may all be easily constructed with second-order formulas) – simply cannot be expressed in first-order axiomatizations. FOL neither resembles our mathematical structures nor seems useful to the working mathematician looking for guidance in far-off places like philosophy, logic and meta-mathematics. While SOL can no more provide a foundation for all of mathematics, it does shed more light on mathematical theory and practice than its first-order counterpart. The Löwenheim-Skolem theorem confronts us with the facile way in which we rely upon unspoken agreements as to our intended models. Indeed, the relativity results inherent to LST can lend credence to the idea that logic might just be a purely formal science. Given the LSTT, Skolem's paradox proves neither the existence nor the non-existence of non-denumerable sets as the first-order model actually allows sentences to be interpreted either in the denumerable or the non-denumerable domain. The interpreter's choice is then guided by other than purely formal considerations – such as one's prior commitment to the ontological existence or characteristics of non-denumerable sets. All that the LST actually states is that the models generated by FOL cannot characterize, cannot say anything about whatever the hypothetical structure of non-denumerable sets might be. In actuality, formal logic does rely on informal mathematical theories and practice to guide its investigations: it is those theories that we want to generate in our models. However, modern axiomatics are not merely meant to codify and crystallize static states of knowledge or disembodied mathematical notions floating in the sky above. Or, rather, if this is what some intend it to do, then modern

Does 'all' mean all? The question might seem to embody the kind of mischief-making that gives philosophers a bad name. Of course 'all' means all, one is tempted to respond in exasperation, what else could it possibly mean? The argument of this paper is that this bad-tempered response is basically correct.