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HARRINGTON'S PRINCIPLE IN HIGHER ORDER ARITHMETIC
2015
Journal of Symbolic Logic (JSL)
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We also study strengthenings of Harrington's Principle over 2nd and 3rd order arithmetic. ...
Let Z 2, Z 3, and Z 4 denote 2nd, 3rd, and 4th order arithmetic, respectively. ...
The strength of Harrington's Principle over higher order arithmetic 3.1. The strength of Z 2 + Harrington's Principle. Theorem 3.1. Z 2 + HP is equiconsistent with ZF C. Proof. ...
doi:10.1017/jsl.2014.31
fatcat:gebr7evwmzaytoeqzhv3fk6se4
Some reflections on the relationship between logical incompleteness and concrete incompleteness
[article]
2024
arXiv
pre-print
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Secondly, concrete incompleteness is a higher order phenomenon over logical incompleteness. ...
In this paper, we aim to conceptually examine the relationship between logical incompleteness and concrete incompleteness which both study the incompleteness phenomenon. We argue for two main theses. ...
of zero sharp), and show that this concrete theorem saying that Harrington's principle implies the existence of zero sharp is expressible in Second Order Arithmetic, neither provable in Second Order Arithmetic ...
arXiv:2401.12531v3
fatcat:q5wevxbdvbfljavytsm5u5qvqy
On the depth of Gödel's incompleteness theorem
[article]
2021
arXiv
pre-print
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In this paper, we use Gödel's incompleteness theorem as a case study for investigating mathematical depth. ...
We take for granted the widespread judgment by mathematical logicians that Gödel's incompleteness theorem is deep, and focus on the philosophical question of what its depth consists in. ...
Friedman's work extends the research on concrete incompleteness from first order arithmetic to higher order arithmetic. ...
arXiv:2008.13142v2
fatcat:fcwyssrobfe5tath5lxzmb2b4e
Determining the factors that affect the quality of test prints at flexographic printing
2020
Eastern-European Journal of Enterprise Technologies
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Often, there are problems related to the correct reproduction of elements in the image with different complexity of the pattern, which affects the stability of the printing process, as well as obtaining ...
The determined arithmetic mean values of the deviation of the Ra profile, the depth of roughness Rz, the changes in the peak area Sp and holes Str of the surface confirm the effect of the chalked coating ...
That is why it is an important process to obtain the test prints at proofing equipment in order to pre-define their densitometric parameters and ensure the quality of the future run of printed products ...
doi:10.15587/1729-4061.2020.200360
fatcat:wlwawhnatreohprdoayvmkiika
Force a set model of Z_3 + Harrington's Principle
[article]
2014
arXiv
pre-print
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Let Z_3 denote 3^rd order arithmetic. Let Harrington's Principle, HP, denote the statement that there is a real x such that every x--admissible ordinal is a cardinal in L. ...
In this paper, assuming there exists a remarkable cardinal with a weakly inaccessible cardinal above it, we force a set model of Z_3 + HP via set forcing without reshaping. ...
This paper is based on part of the author's Ph.D. thesis written in 2012 at the National University of Singapore under the supervision of Chong Chi Tat and W.Hugh Woodin. ...
arXiv:1402.4659v2
fatcat:so2cxhx3srardfznxskcgzdph4
Educational Writings
1918
The Elementary school journal
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being the governing principle. ...
Let us sketch briefly the development of these books in order to be able to point out clearly the new tendencies in the design of courses in spelling. I. ...
It seems clear, however, that as a principle for the sound grading of words in spelling we should be insisting upon one which will make use of gradation of words in order of difficulty. ...
doi:10.1086/454626
fatcat:vs4hkxfvp5btvildalp5amvafi
Approach to development of mentoring potential in technical universities through the example of the Ural Region of the Russian Federation
2018
Istrazivanja i projektovanja za privredu
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Using Harrington's scale, qualitative levels of the potential of a young professional's mentor in higher education institutions were determined based on quantitative scores. ...
In this regard, introduction of effective methods for developing a mentoring mechanism in the higher education system is updated. ...
the research were defi ned based on the principles of multidimensionality and comprehensiveness. ...
doi:10.5937/jaes16-19487
fatcat:6lrvomgmg5d6daahqmdwub6mty
Gödel's Functional ("Dialectica") Interpretation
[chapter]
1998
Studies in Logic and the Foundations of Mathematics
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Analysis and higher type extensions The language of second-order arithmetic extends the language of Peano arithmetic with variables that range over sets of numbers, and a binary membership relation x ∈ ...
In the context of second-order systems, we omit the subscripts from the (AC ) and (DC ) principles. ...
In place of (QF -AC ) this made use of the principle of ω-upper bounds (Lemma 9.3.1 above, clause 3). ...
doi:10.1016/s0049-237x(98)80020-7
fatcat:nzhyeybbwjecnhnzg47rea2tnq
Current research on Gödel's incompleteness theorems
[article]
2020
arXiv
pre-print
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2020 pre-print arXiv:2009.04887v1 fatcat:3jm3lq4vpnd2lknhwfodw6u4le
or Third-Order Arithmetic, but provable in Fourth-Order Arithmetic (i.e. the minimal system in higher-order arithmetic to prove this concrete theorem is Fourth-Order Arithmetic). ...
of zero sharp), and shows that this concrete theorem saying that Harrington's principle implies the existence of zero sharp is expressible in Second-Order Arithmetic, not provable in Second-Order Arithmetic ...
arXiv:2009.04887v2
fatcat:ws3hn7d7rjgbzmayaw2sfqkf6i
CURRENT RESEARCH ON GÖDEL'S INCOMPLETENESS THEOREMS
2021
Bulletin of Symbolic Logic
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formulated as the arithmetic formula which says that there exists an unprovable sentence in 2000 Mathematics Subject Classification. 03F40, 03F30, 03F25. ...
Gödel comments in a footnote of [46] that G2 is corollary of G1 (and in fact a formalized version of G1): if T is consistent, then the consistency of T is not provable in T where the consistency of T is ...
, not provable in Second-Order Arithmetic or Third-Order Arithmetic, but provable in Fourth-Order Arithmetic (i.e. the minimal system in higher-order arithmetic to prove this concrete theorem is Fourth-Order ...
doi:10.1017/bsl.2020.44
fatcat:uqklozcs3rapxc6t25fylacf7i
A NEW COMPUTATION OF THE Σ-ORDINAL OF KPω
2014
Journal of Symbolic Logic (JSL)
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It is also argued that the second-order theory is Σ1-conservative over KPω. ...
We also extend KP ω to a second-order theory with Δ 1-comprehension and strict- ${\rm{\Pi }}_1^1$ reflection and show that the Σ-ordinal of this theory is still the Bachmann-Howard ordinal. ...
Harrington's proof uses a forcing argument and has been reported in [22] (there is also a pure prooftheoretic proof of Harrington's conservation result, given in [10] ). ...
doi:10.1017/jsl.2013.31
fatcat:iqsc52qqhfbudosvw7iek34chq
Proof Theory in Philosophy of Mathematics
2010
Philosophy Compass
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A variety of projects in proof theory of relevance to the philosophy of mathematics are surveyed, including Gödel's incompleteness theorems, conservation results, independence results, ordinal analysis ...
Gödel's results apply in particular to first-order Peano Arithmetic (PA), a theory widely believed to implicitly define the elementary arithmetic of the natural numbers. ...
A third body of work aimed at partially realizing Hilbert's program is concerned with subsystems of second-order arithmetic, that is, arithmetic theories in which some of the quantifiers range over sets ...
doi:10.1111/j.1747-9991.2010.00282.x
fatcat:ioi4iu6k4bcfrgg4l36qf3fsum
On mathematical instrumentalism
2005
Journal of Symbolic Logic (JSL)
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On the other hand, theories studied in the Program of Reverse Mathematics that formalize powerful mathematical principles have only polynomial speed-up over IΣ1. ...
For example, we show that even though the fragment of Peanos Arithmetic known as IΣ1 is a conservative extension of the equational theory of Primitive Recursive Arithmetic (PRA). ...
It was conjectured in [9] that in order to get a similar result for W KL versus RCA, one could try to adapt Harrington's forcing proof of the conservativeness of W KL over RCA for arithmetical sentences ...
doi:10.2178/jsl/1122038914
fatcat:eetplmdmxvaxvgex5nrzkvkq7a
Information Systems Reliability in Traditional Entropy and Novel Hierarchy
2022
Cybernetics and Information Technologies
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The continuous progress of computing technologies increases the need for improved methods and tools for assessing the performance of information systems in terms of reliability, conformance, and quality ...
A higher value of 𝑝 𝑖 logically reflects stronger domination and more information in terms of order and competitive strength. ...
Harrington's preference function for psycho-physical classification [37] , as shown in Table 1 . ...
doi:10.2478/cait-2022-0024
fatcat:2wjojxu7vre3ba2gus5aw5vgfq
Method of Evaluation of Electronic Educational Resource Quality for Conformity Assessment (Certification)
2016
International Conference on Information and Communication Technologies in Education, Research, and Industrial Applications
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In the article the expediency of quality assessment and certification of electronic educational resources of the higher education institution is justified. ...
The order and scheme of internal certification of electronic educational resources are formed as a result of research. Certification includes the softwaretechnical and complex expertises. ...
, informatization of educational process in higher educational institutions. ...
dblp:conf/icteri/FedasyukHMKD16
fatcat:j6wgdtxtrvbjnpon7svbvwv7jm
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