1. The general idea behind Godel's proof 2. Tarski's theorem for arithmetic 3. The incompleteness of peano arithmetic with exponentation 4. Arithmetic without the exponential 5. Godel's proof based on consistency 6. Rosser systems 7. Shepherdson's Representation theorems 8. Definability and diagonalization 9. The unprovability of consistency 10. Some general remarks on provability and truth 11. Self-referential systems

http://www.gbv.de/dms/ilmenau/toc/111786401.PDF

Godel's Incompleteness Theorems

Purpose – This paper aims to provide a clarification of the meaning of the term integration.Design/methodology/approach – A taxonomy of integration definitions derived from the academic and trade literature is developed, analyzed, and documented.Findings – Integration is a word that is commonly used when discussing enterprise applications. The term integration is inserted in technical papers, e‐mail messages, correspondence, proposals, and even causal conversations. After many years of project work, and many misunderstandings and failed meetings and workshops, it can only be stated that the word has multiple and misunderstood meanings. For technical papers (research and trade), the term must be provided with context, or it is impossible to have a meaningful conversation. Next, multiple alternative definitions (that are valid in the literature for the appropriate context) are presented and explained in some detail.Research limitations/implications – The paper is not exhaustive, since new definitions of int...

What is integration

It is known that MSO logic for ordered unranked trees is undecidable if Presburger constraints are allowed at children of nodes. We show here that a decidable logic is obtained if we use a modal fixpoint logic instead. We present a characterization of this logic by means of deterministic Presburger tree automata and show how it can be used to express numerical document queries. Surprisingly, the complexity of satisfiability for the extended logic is asymptotically the same as for the original fixpoint logic. The non-emptiness for Presburger tree automata (PTA) is pspace-complete, which is moderate given that it is already pspace-hard to test whether the complement of a regular expression is non-empty. We also identify a subclass of PTAs with a tractable non-emptiness problem. Further, to decide whether a tree t satisfies a formula ϕ is polynomial in the size of ϕ and linear in the size of t.

/pdf/counting-in-trees-for-free-hoedosyjrl.pdf

Counting in Trees for Free

We consider certain families of automorphic representations over number fields arising from the principle of functoriality of Langlands. Let $$G$$
 be a reductive group over a number field $$F$$
 which admits discrete series representations at infinity. Let $$^{L}G=\widehat{G} \rtimes \mathrm{Gal}(\bar{F}/F)$$
 be the associated $$L$$
 -group and $$r:{}^L G\rightarrow \mathrm{GL}(d,\mathbb {C})$$
 a continuous homomorphism which is irreducible and does not factor through $$\mathrm{Gal}(\bar{F}/F)$$
 . The families under consideration consist of discrete automorphic representations of $$G(\mathbb {A}_F)$$
 of given weight and level and we let either the weight or the level grow to infinity. We establish a quantitative Plancherel and a quantitative Sato–Tate equidistribution theorem for the Satake parameters of these families. This generalizes earlier results in the subject, notably of Sarnak (Prog Math 70:321–331, 1987) and Serre (J Am Math Soc 10(1):75–102, 1997). As an application we study the distribution of the low-lying zeros of the associated family of $$L$$
 -functions $$L(s,\pi ,r)$$
 , assuming from the principle of functoriality that these $$L$$
 -functions are automorphic. We find that the distribution of the $$1$$
 -level densities coincides with the distribution of the $$1$$
 -level densities of eigenvalues of one of the unitary, symplectic and orthogonal ensembles, in accordance with the Katz–Sarnak heuristics. We provide a criterion based on the Frobenius–Schur indicator to determine this symmetry type. If $$r$$
 is not isomorphic to its dual $$r^\vee $$
 then the symmetry type is unitary. Otherwise there is a bilinear form on $$\mathbb {C}^d$$
 which realizes the isomorphism between $$r$$
 and $$r^\vee $$
 . If the bilinear form is symmetric (resp. alternating) then $$r$$
 is real (resp. quaternionic) and the symmetry type is symplectic (resp. orthogonal).

/pdf/sato-tate-theorem-for-families-and-low-lying-zeros-of-4lkd9ch4x8.pdf

Sato-Tate theorem for families and low-lying zeros of automorphic L-functions

In his book Logic and How It Gets That Way Jacquette (2010) presents ‘the formalization paradox’ which emerges from the attempt to formalize a sentence like ‘some monkey devours every craisins’, where craisins are imaginary non-existent fruits. From this paradox Jacquette concludes the expressive inadequacy of classical predicate quantificational logic. In this paper I analyse the three assumptions supposed in the emergence of the paradox, viz.: (i) colloquial expressions of the same logical form can and should be formally symbolized by applying the same symbolization schema); (ii) ‘Some monkeys devours every raisin’ is correctly translated as ∃x[Mx ∧ ∀y[Ry → Dxy]]; (iii) uninstantiated predicates can legitimately enter into (meaningful, true or false) predicatesquantificational symbolizations. I fully accept (iii), but reject both (i) and (ii). I argue, firstly, that (i) has at the first glance two possible interpretations, a trivial and a false one. So, I try to establish a third and more reasonable interpretation. Based on this interpretation I argue that ∃x[Mx ∧ ∀y[Ry → Dxy]] is not the adequate formalization of ‘some monkeys devours every raisin’. My basic claim is based on a generalisation of Russell’s theory of description: just like most sentences of natural language which contain definite descriptions are viewed as entailing existential force which must be made explicit in formalization, so do we also consider many, although not all, sentences which contain general terms. A criterion for deciding in each possible case if the sentence entails existential force will be presented and defended. In his book Logic and How It Gets That Way (2010, chap. 2) Jacquette presents ‘the formalization paradox’ which is supposed to emerge from the attempt to formalize some colloquial sentences into classical predicate quantificational logic. Based on this paradox Jacquette concludes the expressive inadequacy of this logic. I would like to thank to the audiences of the POL Colloquium in Curitiba, the Sixth International Symposium on Philosophy of Language and Metaphysics att UFF and the LanCog Group in Lisbon, in particular to Breno Hax, Dirk Greimann, and Ricardo Santos for many important suggestions. Vol. 4 No. 11 2017 IFCoLog Journal of Logics and their Applications

How Do We Know Things with Signs? A Model of Semiotic Intentionality

Presburger's essay on the completeness and decidability of arithmetic with integer addition but without multiplication is a milestone in the history of mathematical logic and formal metatheory. The proof is constructive, using Tarski-style quantifier elimination and a four-part recursive comprehension principle for axiomatic consequence characterization. Presburger's proof for the completeness of first order arithmetic with identity and addition but without multiplication, in light of the restrictive formal metatheorems of Godel, Church, and Rosser, takes the foundations of arithmetic in mathematical logic to the limits of completeness and decidability.

Mojżesz Presburger

Papers

On the completeness of a certain system of arithmetic of whole numbers in which addition occurs as the only operation