122 research outputs found
The complexity of classification problems for models of arithmetic
We observe that the classification problem for countable models of arithmetic
is Borel complete. On the other hand, the classification problems for finitely
generated models of arithmetic and for recursively saturated models of
arithmetic are Borel; we investigate the precise complexity of each of these.
Finally, we show that the classification problem for pairs of recursively
saturated models and for automorphisms of a fixed recursively saturated model
are Borel complete.Comment: 15 page
Hilbert's "Verunglueckter Beweis," the first epsilon theorem, and consistency proofs
In the 1920s, Ackermann and von Neumann, in pursuit of Hilbert's Programme,
were working on consistency proofs for arithmetical systems. One proposed
method of giving such proofs is Hilbert's epsilon-substitution method. There
was, however, a second approach which was not reflected in the publications of
the Hilbert school in the 1920s, and which is a direct precursor of Hilbert's
first epsilon theorem and a certain 'general consistency result' due to
Bernays. An analysis of the form of this so-called 'failed proof' sheds further
light on an interpretation of Hilbert's Programme as an instrumentalist
enterprise with the aim of showing that whenever a `real' proposition can be
proved by 'ideal' means, it can also be proved by 'real', finitary means.Comment: 18 pages, final versio
Formalizing Mathematical Knowledge as a Biform Theory Graph: A Case Study
A biform theory is a combination of an axiomatic theory and an algorithmic
theory that supports the integration of reasoning and computation. These are
ideal for formalizing algorithms that manipulate mathematical expressions. A
theory graph is a network of theories connected by meaning-preserving theory
morphisms that map the formulas of one theory to the formulas of another
theory. Theory graphs are in turn well suited for formalizing mathematical
knowledge at the most convenient level of abstraction using the most convenient
vocabulary. We are interested in the problem of whether a body of mathematical
knowledge can be effectively formalized as a theory graph of biform theories.
As a test case, we look at the graph of theories encoding natural number
arithmetic. We used two different formalisms to do this, which we describe and
compare. The first is realized in , a version of Church's
type theory with quotation and evaluation, and the second is realized in Agda,
a dependently typed programming language.Comment: 43 pages; published without appendices in: H. Geuvers et al., eds,
Intelligent Computer Mathematics (CICM 2017), Lecture Notes in Computer
Science, Vol. 10383, pp. 9-24, Springer, 201
Pairs, sets and sequences in first-order theories
Asuransi sebagai aktivitas bisnis diharuskan memenuhi prinsip-prinsip hukum asuransi. Salah satu prinsip yang harus dipegang teguh adalah principle of utmost good faith, di samping prinsip yang lain. Prinsip ini berbunyi bahwa seorang tertanggung wajib memberi informasi secara jujur terhadap apa yang dipertanggungkan kepada penanggung. Dalam bisnis Islam, kejujuran merupakan prinsip yang harus dijunjung tinggi. Secara hukum, prinsip ini diatur dalam KUH Dagang. Persoalannya adalah apakah prinsip ini dianggap cukup dari sudut pandang hukum perjanjian syariah. Secara sekilas bahwa prinsip iktikad baik sempurna ini telah memenuhi asas perjanjian syariah, namun demikian tidak memiliki kriteria maksimal kejujuran. Ketiadaan kejujuran dalam bisnis asuransi akan berdampak pada batalnya perjanjian asuransi karena ada unsur cacat kehendak (‘uyub ar-ridla). Insurance as a business activity must fulfill principles of insurance law. One of the principles that must be hold on is the principle of utmost good faith. The principle says that an endured person must honestly give information of what should be given responsibility to the guarantor. In Islamic business, honesty is a principle that should be respected. From point of view of law, the principle is settled in commerce law. The problem is that whether the principle is represenative enough if it is viewed from law of syariah agreement. At glance, the principle has fulfilled the basic of syariah agreement, however, it does not have maximum criteria of honesty. Unavailability of honesty in insurance business will give effect of invalidate of insurance agreement, for there is a deformity of desire (‘uyub ar-ridla).</p
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