172 research outputs found
Ideal presentations and numberings of some classes of effective quasi-Polish spaces
The well known ideal presentations of countably based domains were recently
extended to (effective) quasi-Polish spaces. Continuing these investigations,
we explore some classes of effective quasi-Polish spaces. In particular, we
prove an effective version of the domain-characterization of quasi-Polish
spaces, describe effective extensions of quasi-Polish topologies, discover
natural numberings of classes of effective quasi-Polish spaces, estimate the
complexity of the (effective) homeomorphism relation and of some classes of
spaces w.r.t. these numberings, and investigate degree spectra of continuous
domains
On Notions of Provability
In this thesis, we study notions of provability, i.e. formulas B(x,y) such that a formula
ϕ is provable in T if, and only if, there is m ∈ N such that T ⊢ B(⌜ϕ⌝,m) (m plays the
role of a parameter); the usual notion of provability, k-step provability (also known as
k-provability), s-symbols provability are examples of notions of provability.
We develop general results concerning notions of provability, but we also study in
detail concrete notions. We present partial results concerning the decidability of kprovability
for Peano Arithmetic (PA), and we study important problems concerning
k-provability, such as Kreisel’s Conjecture and Montagna’s Problem:
(∀n ∈ N.T ⊢k steps ϕ(n)) =⇒ T ⊢ ∀x.ϕ(x), [Kreisel’s Conjecture]
and
Does PA ⊢k steps PrPA(⌜ϕ⌝)→ϕ imply PA ⊢k steps ϕ? [Montagna’s Problem]
Incompleteness, Undefinability of Truth, and Recursion are different entities that
share important features; we study this in detail and we trace these entities to common
results.
We present numeral forms of completeness and consistency, numeral completeness
and numeral consistency, respectively; numeral completeness guarantees that, whenever
a Σb
1(S12
)-formula ϕ(⃗x ) is such that ⃗Q
⃗x .ϕ(⃗x ) is true (where ⃗Q
is any array of quantifiers),
then this very fact can be proved inside S12
, more precisely S12
⊢ ⃗Q
⃗x .Prτ (⌜ϕ(
•⃗
x )⌝). We
examine these two results from a mathematical point of view by presenting the minimal
conditions to state them and by finding consequences of them, and from a philosophical
point of view by relating them to Hilbert’s Program.
The derivability condition “provability implies provable provability” is one of the main
derivability conditions used to derive the Second Incompleteness Theorem and is known
to be very sensitive to the underlying theory one has at hand. We create a weak theory
G2 to study this condition; this is a theory for the complexity class FLINSPACE. We also
relate properties of G2 to equality between computational classes.O tema desta tese são noções de demonstração; estas últimas são fórmulas B(x,y) tais que
uma fórmula ϕ é demonstrável em T se, e só se, existe m ∈ N tal que T ⊢ B(⌜ϕ⌝,m) (m
desempenha o papel de um parâmetro). A noção usual de demonstração, demonstração
em k-linhas (demonstração-k), demonstração em s-símbolos são exemplos de noções de
demonstração.
Desenvolvemos resultados gerais sobre noções de demonstração, mas também estudamos
exemplos concretos. Damos a conhecer resultados parciais sobre a decidibilidade da
demonstração-k para a Aritmética de Peano (PA), e estudamos dois problemas conhecidos
desta área, a Conjectura de Kreisel e o Problema de Montagna:
(∀n ∈ N.T ⊢k steps ϕ(n)) =⇒ T ⊢ ∀x.ϕ(x), [Conjectura de Kreisel]
e
PA ⊢k steps PrPA(⌜ϕ⌝)→ϕ implica PA ⊢k steps ϕ? [Problema de Montagna]
A Incompletude, a Incapacidade de Definir Verdade, e Recursão são entidades que
têm em comum características relevantes; nós estudamos estas entidades em detalhe e
apresentamos resultados que são simultaneamente responsáveis pelas mesmas.
Além disso, apresentamos formas numerais de completude e consistência, a completude
numeral e a consistência numeral, respectivamente; a completude numeral assegura
que, quando uma fórmula-Σb
1(S12) ϕ(⃗x ) é tal que ⃗Q
⃗x .ϕ(⃗x ) é verdadeira, então este facto
pode ser verificado dentro de S12, mais precisamente S12
⊢ ⃗Q
⃗x .Prτ (⌜ϕ(
•⃗
x )⌝). Este dois resultados
são analisados de um ponto de vista matemático onde apresentamos as condições
mínimas para os demonstrar e apresentamos consequências dos mesmos, e de um ponto
de vista filosófico, onde relacionamos os mesmos com o Programa de Hilbert.
A condição de derivabilidade “demonstração implica demonstrabilidade da demonstração”
é uma das condições usadas para derivar o Segundo Teorema da Incompletude e
sabemos ser muito sensível à teoria de base escolhida. Nós criámos uma teoria fraca G2
para estudar esta condição; esta é uma teoria para a classe de complexidade FLINSPACE.
Também relacionámos propriedades de G2 com igualdades entre classes de complexidade
computacional
On Notions of Provability
In this thesis, we study notions of provability, i.e. formulas B(x,y) such that a formula ' is provable in T if, and only if, there is m 2 N such that T ` B(p'q,m) (m plays the role of a parameter); the usual notion of provability, k-step provability (also known as k-provability), s-symbols provability are examples of notions of provability.
We develop general results concerning notions of provability, but we also study in detail concrete notions. We present partial results concerning the decidability of k- provability for Peano Arithmetic (PA), and we study important problems concerning k-provability, such as Kreisel’s Conjecture and Montagna’s Problem:
(8n 2 N.T `k steps '(n)) =) T ` 8x.'(x), [Kreisel’s Conjecture] Does PA `k steps PrPA(p'q) ! ' imply PA `k steps '? [Montagna’s Problem]
Incompleteness, Undefinability of Truth, and Recursion are di↵erent entities that share important features; we study this in detail and we trace these entities to common results.
We present numeral forms of completeness and consistency, numeral completeness and numeral consistency, respectively; numeral completeness guarantees that, whenever a⌃b1(S12)-formula'(x~)issuchthatQ~x~.'(x~)istrue(whereQ~ isanyarrayofquantifiers),
then this very fact can be proved inside S12, more precisely S12 ` Q~ x~.Pr⌧(p'(x~• )q). We examine these two results from a mathematical point of view by presenting the minimal conditions to state them and by finding consequences of them, and from a philosophical point of view by relating them to Hilbert’s Program.
The derivability condition “provability implies provable provability” is one of the main derivability conditions used to derive the Second Incompleteness Theorem and is known to be very sensitive to the underlying theory one has at hand. We create a weak theory G2 to study this condition; this is a theory for the complexity class FLINSPACE. We also relate properties of G2 to equality between computational classes
Notions of Complexity Within Computable Structure Theory
This thesis covers multiple areas within computable structure theory, analyzing the complexities of certain aspects of computable structures with respect to different notions of definability.
In chapter 2 we use a new metatheorem of Antonio Montalb\'an's to simplify an otherwise difficult priority construction. We restrict our attention to linear orders, and ask if, given a computable linear order \A with degree of categoricity , it is possible to construct a computable isomorphic copy of \A such that the isomorphism achieves the degree of categoricity and furthermore, that we did not do this coding using a computable set of points chosen in advance. To ensure that there was no computable set of points that could be used to compute the isomorphism we are forced to diagonalize against all possible computable unary relations while we construct our isomorphic copy. This tension between trying to code information into the isomorphism and trying to avoid using computable coding locations, necessitates the use of a metatheorem. This work builds off of results obtained by Csima, Deveau, and Stevenson for the ordinals and , and extends it to for any computable successor ordinal .
In chapter 3, which is joint work with Alvir and Csima, we study the Scott complexity of countable reduced Abelian -groups. We provide Scott sentences for all such groups, and show some cases where this is an optimal upper bound on the Scott complexity. To show this optimality we obtain partial results towards characterizing the back-and-forth relations on these groups.
In chapter 4, which is joint work with Csima and Rossegger, we study structures under enumeration reducibility when restricting oneself to only the positive information about a structure. We find that relations that can be relatively intrinsically enumerated from such information have a definability characterization using a new class of formulas. We then use these formulas to produce a structural jump within the enumeration degrees that admits jump inversion, and compare it to other notions of the structural jump. We finally show that interpretability of one structure in another using these formulas is equivalent to the existence of a positive enumerable functor between the classes of isomorphic copies of the structures
LIPIcs, Volume 274, ESA 2023, Complete Volume
LIPIcs, Volume 274, ESA 2023, Complete Volum
Three Essays on Later Wittgenstein's Philosophy of Mathematics: Reality, Determination, and Infinity
This dissertation provides a careful reading of the later Wittgenstein’s philosophy of mathematics centered around three major themes: reality, determination, and infinity. The reading offered gives pride of place to Wittgenstein’s therapeutic conception of philosophy. This conception views questions often taken as fundamental in the philosophy of mathematics with suspicion and attempts to diagnose the confusions which lead to them. In the first essay, I explain Wittgenstein’s approach to perennial issues regarding the alleged reality to which mathematical truths or propositions correspond. Wittgenstein diagnoses exotic pictures of mathematical reality as stemming from misleading analogies formed across empirical and mathematical propositions. The second essay explains Wittgenstein’s treatment of perplexity regarding the ability of a mathematical rule to determine its applications in advance. This too is found to depend on a misleading analogy, in this case across behavioral and mathematical senses of ‘determine’. The third and final essay discusses Wittgenstein’s general critical approach to “the infinite”. Philosophical perplexity about infinity is shown to depend on the verbal imagery regularly associated with proofs and their power to take hold of one’s imagination. Wittgenstein dampens the imaginative excesses often associated with “the infinite” by offering a sober redescription of Cantor’s famous method of diagonalization
Three Essays on Wittgenstein's Philosophy of Mathematics: Reality, Determination, & Infinity
This dissertation provides a careful reading of the later Wittgenstein’s philosophy of mathematics centered around three major themes: reality, determination, and infinity. The reading offered gives pride of place to Wittgenstein’s therapeutic conception of philosophy. This conception views questions often taken as fundamental in the philosophy of mathematics with suspicion and attempts to diagnose the confusions which lead to them. In the first essay, I explain Wittgenstein’s approach to perennial issues regarding the alleged reality to which mathematical truths or propositions correspond. Wittgenstein diagnoses exotic pictures of mathematical reality as stemming from misleading analogies formed across empirical and mathematical propositions. The second essay explains Wittgenstein’s treatment of perplexity regarding the ability of a mathematical rule to determine its applications in advance. This too is found to depend on a misleading analogy, in this case across behavioral and mathematical senses of ‘determine’. The third and final essay discusses Wittgenstein’s general critical approach to “the infinite”. Philosophical perplexity about infinity is shown to depend on the verbal imagery regularly associated with proofs and their power to take hold of one’s imagination. Wittgenstein dampens the imaginative excesses often associated with “the infinite” by offering a sober redescription of Cantor’s famous method of diagonalization.Doctor of Philosoph
Collected Papers (on Neutrosophic Theory and Its Applications in Algebra), Volume IX
This ninth volume of Collected Papers includes 87 papers comprising 982 pages on Neutrosophic Theory and its applications in Algebra, written between 2014-2022 by the author alone or in collaboration with the following 81 co-authors (alphabetically ordered) from 19 countries: E.O. Adeleke, A.A.A. Agboola, Ahmed B. Al-Nafee, Ahmed Mostafa Khalil, Akbar Rezaei, S.A. Akinleye, Ali Hassan, Mumtaz Ali, Rajab Ali Borzooei , Assia Bakali, Cenap Özel, Victor Christianto, Chunxin Bo, Rakhal Das, Bijan Davvaz, R. Dhavaseelan, B. Elavarasan, Fahad Alsharari, T. Gharibah, Hina Gulzar, Hashem Bordbar, Le Hoang Son, Emmanuel Ilojide, Tèmítópé Gbóláhàn Jaíyéolá, M. Karthika, Ilanthenral Kandasamy, W.B. Vasantha Kandasamy, Huma Khan, Madad Khan, Mohsin Khan, Hee Sik Kim, Seon Jeong Kim, Valeri Kromov, R. M. Latif, Madeleine Al-Tahan, Mehmat Ali Ozturk, Minghao Hu, S. Mirvakili, Mohammad Abobala, Mohammad Hamidi, Mohammed Abdel-Sattar, Mohammed A. Al Shumrani, Mohamed Talea, Muhammad Akram, Muhammad Aslam, Muhammad Aslam Malik, Muhammad Gulistan, Muhammad Shabir, G. Muhiuddin, Memudu Olaposi Olatinwo, Osman Anis, Choonkil Park, M. Parimala, Ping Li, K. Porselvi, D. Preethi, S. Rajareega, N. Rajesh, Udhayakumar Ramalingam, Riad K. Al-Hamido, Yaser Saber, Arsham Borumand Saeid, Saeid Jafari, Said Broumi, A.A. Salama, Ganeshsree Selvachandran, Songtao Shao, Seok-Zun Song, Tahsin Oner, M. Mohseni Takallo, Binod Chandra Tripathy, Tugce Katican, J. Vimala, Xiaohong Zhang, Xiaoyan Mao, Xiaoying Wu, Xingliang Liang, Xin Zhou, Yingcang Ma, Young Bae Jun, Juanjuan Zhang
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