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Henkin's completeness proof: forty years later

1991
*
Notre Dame Journal of Formal Logic
*

*Henkin's*result has come to be known as the Strong

*Completeness*Theorem for First-Order L. ... To commemorate the publication of

*Henkin's*paper, we offer here two new

*completeness*

*proofs*for first-order Z,. 4 The language considered by Henkin had an unspecified number of terms to start with, but ... The rule, proposed by Tarski in a lecture of 1927 and published seven

*years*

*later*in Tarski [13] , was called by him-we noted in Section I -the rule of infinite induction. ...

##
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Book Review: Mathematical logic. An introduction to model theory

1979
*
Bulletin of the American Mathematical Society
*

Again a

doi:10.1090/s0273-0979-1979-14718-9
fatcat:vbcllp2n3fbgpdz7xpepma5fzi
*completeness*theorem holds paralleling that above for propositional calculus. However,*Henkin's**proof*leads to a different, and more used, formulation. ... 'Tor a more rigorous [sic]*proof*of this [the*completeness*] theorem [for propositional calculus], see Henkin (1949) " (p. 102). ... At last there is at hand in this book a systematic and*complete*(though austere) presentation of the recent extensive developments in category theory leading to the notion of an elementary topos. ...##
###
Page 1201 of Mathematical Reviews Vol. , Issue 93c
[page]

1993
*
Mathematical Reviews
*

’s

*completeness**proof*:*forty**years**later*. ... set if S is term-*complete*and maximally consistent. ...##
###
Dugundji's Theorem Revisited

2014
*
Logica Universalis
*

useful Chagrov and Zakharyaschev's criterion of tabularity for modal logics is provided. 4 In [14]. 5 Although it was considered the original Dugundji's article [9], we preferred to follow the clearer

doi:10.1007/s11787-014-0106-4
fatcat:6nvr4pa3sbe4vnbqwcdaodtoye
*proof*... This question was resolved by Dugundji eight*years**later*: it was shown that not only Łukasiewicz's matrices, but no finite matrix can be a*complete*semantics for any system between S1 and S5. ...*Proof*: The*proof*is similar to the previous one, but now using Proposition 5.2. X = N if X is cofinite N − {0} otherwise . ...##
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The Proliferation of International Courts and Tribunals: International Adjudication in Ascendance

2000
*
Proceedings of the ASIL Annual Meeting
*

rendered

doi:10.1017/s027250370005549x
fatcat:djunnvt6ybg7tnhimnokywssdm
*forty*-five judgments. ... This divestiture is so*complete*that last*year*the United States was found to be in noncompliance with its Algiers Accords obligation simply by permitting private parties to file lawsuits in U.S. courts ...##
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A brief history of algebraic logic from neat embeddings to rainbow constructions
[article]

2015
*
arXiv
*
pre-print

But then by algebraising the rest of

arXiv:1508.05840v1
fatcat:xsyr345hgzf4natorz4ds6hy6q
*Henkin's**proof*, we get that rich algebras are representable. ... Rich theories occuring in*Henkin's**completeness**proof*eliminate existential quantifiers in existential formulas via individual constants more commonly referred to as witnesses. ... Proving the strictness of the last inclusion can be easily distilled from the*proof*of [43, Theorem 36] . The rest of the alleged statements can also be found in [43] . ...##
###
From Hilbert's program to a logic tool box

2008
*
Annals of Mathematics and Artificial Intelligence
*

At the moment of

doi:10.1007/s10472-009-9115-z
fatcat:5czgb6guqzb3hdq46spqz24wdy
*completion*we were in the fourth week of our teaching strike which lasted 10 weeks (October 2007-January 2008), and which was a resounding success. ... Act VI: 100*years**later*-Fixing Frege If only G. Frege had not been so intimidated by B. Russel's letter. C. Wright, P. Geach and H. Hodes suggested, and G. ... Our students, however, should rather follow the advise of the Rabbinic Sages, who admonish us not to study Kabbala (Jewish Mysticism) before the mature age of*forty**years*and before serious exposure to ...##
###
Book Review: Wittgenstein's lectures on the foundations of mathematics, Cambridge,

1978
*
Bulletin of the American Mathematical Society
*

(which is 'useless' if we know that

doi:10.1090/s0002-9904-1978-14413-9
fatcat:krecdpjskrab5ps6xinv7cfbd4
*later*we shall go much farther); cf. [4]. involved.) ... Sure, by the*completeness*theorem, F A has a*proof*using only the rules of ordinary predicate calculus, too; but this fact, which is certainly fundamental if the assumptions of t.f. are granted, has turned ... scientifically, if not artistically-speaking, such*proofs*, for example, of the 4-color conjecture, involve incomparably more progress than, say, the use of large cardinals in (ii) above. ...##
###
On consistency and existence in mathematics
[article]

2020
*
arXiv
*
pre-print

The question is related historically to the formulation,

arXiv:2007.10167v1
fatcat:72llgzqmenhivmhgorpdn5xohq
*proof*, and reception of Gödel's*Completeness*Theorem. ...*completeness**proof*. ... For instance in the case of Gödel's original*proof*, b will be a natural number -say 17 -and the case of*Henkin's*(1949)*proof*b will be one of countably many witness constants -say c 17 . ...##
###
On the finitizability problem in algebraic logic; recent results
[article]

2013
*
arXiv
*
pre-print

The ω extra dimensions play the role of added constants or witnesses in

arXiv:1302.1368v1
fatcat:bguykr2afvfajda4qskyr265ge
*Henkin's*classical*proof*of the*completeness*theorem. ... Furthermore, A can be chosen to be the minimal*completion*of B. More on that*later*on, theorems 2.29, 2.32, 4.27) . ...##
###
"Mathematics is the Logic of the Infinite": Zermelo's Project of Infinitary Logic

2021
*
Studies in Logic, Grammar and Rhetoric
*

Set
theory had been developing for about

doi:10.2478/slgr-2021-0042
fatcat:nnowhogl3bhklm234xa7dion7u
*forty**years*as a mathematical theory, and only*later*did investigations into models of this theory begin. 674 “Mathematics is the Logic of the Inﬁnite”: Zermelo ... The algebraic inspiration for*Henkin’s*work may be traced as follows: 1. ...##
###
The Origin of the Group in Logic and the Methodology of Science

2018
*
Journal of Humanistic Mathematics
*

Well-known for his new

doi:10.5642/jhummath.201801.19
fatcat:yklp4r74yrafpocseoca3qluwq
*proof*of the*completeness*theorem for first-order logic and the theory of types (1950), and for his work in algebraic logic, Henkin was a principal investigator in all the grants ... For this section, see also*Henkin's*"In Memoriam" [3] . A recent volume celebrating*Henkin's*work is [15] . The Victor Lenzen Papers are preserved at the Bancroft Library at UC Berkeley. ...##
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The Literary Impact of Vestiges of Creation

1956
*
Modern Language Quarterly: A Journal of Literary History
*

One reason why he did not

doi:10.1215/00182702-17-3-213
fatcat:uewefvijwjhzbmwd66d3j5qi44
*complete*his chosen task, why in this whole direction he did comparatively little after a brilliant start, was that he devoted much of the energy of his last*years*to works of ... See also pages 120 ff. of Leo*Henkin’s*valuable Darwinism and the English Novel (New York, 1940). ...##
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Page 60 of Mathematical Reviews Vol. 25, Issue Index
[page]

*
Mathematical Reviews
*

ness

*proof*:*forty**years**later*. 93c:03010 Mo, Shao Kui Establishment of a logical functional calculus. ... (English summary) 93m:03017 Hand, Michael Meaning, truth-conditions, and substitutional quantification. 93k:03007 Leblanc, Hugues (with Roeper, Peter; Thau, Michael; Weaver, George)*Henkin’s**complete*- ...##
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The good, the bad and the ugly

2007
*
Synthese
*

In like manner, an analogous abstraction on well-orderings (or relations generally) is inconsistent, falling to the Burali-

doi:10.1007/s11229-007-9263-9
fatcat:d77yi56kt5ae3jsxfj5z4v27d4
*Forti*paradox. ... If the theories are first-order, or secondorder with Henkin semantics, then deductive consistency does suffice for satisfiability, in light of Gödel's and*Henkin's**completeness*theorems. ... Any attempt to generate a Russell or Burali-*Forti*paradox on the defined abstracts fails. ...
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