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Papers by Henry Africk

The College Mathematics Journal, 1985

This text is intended for a brief introductory course in plane geometry. It covers the topics fro... more This text is intended for a brief introductory course in plane geometry. It covers the topics from elementary geometry that are most likely to be required for more advanced mathematics courses. The only prerequisite is a semester of algebra. The emphasis is on applying basic geometric principles to the numerical solution of problems. For this purpose the number of theorems and definitions is kept small. Proofs are short and intuitive, mostly in the style of those found in a typical trigonometry or precalculus text. There is little attempt to teach theorem proving or formal methods of reasoning. However the topics are ordered so that they may be taught deductively. The problems are arranged in pairs so that just the odd-numbered or just the even-numbered can be assigned. For assistance, the student may refer to a large number of completely worked-out examples. Most problems are presented in diagram form so that the difficulty of translating words into pictures is avoided. Many proble...

This membership list may not be used for commercial purposes, for bulk mailing, or to prepare mai... more This membership list may not be used for commercial purposes, for bulk mailing, or to prepare mailing lists, without written permission from the Association. For information regarding use of this list, contact the Secretary-Treasurer of the Association. Corrections to this list should ...

This membership list may not be used for commercial purposes, for bulk mailing, or to prepare mai... more This membership list may not be used for commercial purposes, for bulk mailing, or to prepare mailing lists, without written permission from the Association. For information regarding use of this list, contact the Secretary-Treasurer of the Association. Corrections to this list should ...

Journal of Symbolic Logic, Mar 1, 1974

The Journal of Symbolic Logic, 1974

In [1] we proved the following interpolation theorem for first-order (finitary) logic: Theorem (S... more In [1] we proved the following interpolation theorem for first-order (finitary) logic: Theorem (Scott). Let A and B be sentences. There is an -sentence C such that A → C and C → B iff whenever and are -isomorphic structures and satisfies A then satisfies B. We show here that the Theorem holds for A and B in L ω1, ω only if we permit the interpolant C to be in L (2 ω )+ , ω, where (2 ω )+ is the successor of 2 ω . Our language contains the usual logical symbols and the relation symbols R i, is i ∈ J. If I ⊆ J define an I-sentence to be a sentence containing only relations with subscripts in I. If define an -sentence to be a boolean combination of I-sentences with , i.e., a sentence consisting of I-sentences, for various I's in , joined together by ∨, ∧ and ⌝; e.g., ∀xR 1(x) ∨ ∀xR 2(x)is a{{1}, {2}}-sentence but ∀x(R 1(x) ∨ R 2(x)) is not. If is a structure let , be the structure obtained by restricting to just the relations with subscripts in I. We say that is -isomorphic to if i...

The Two-Year College Mathematics Journal, 1981

The Journal of Symbolic Logic, 1966

as the last three days of a Summer School in Mathematical Logic. One hour invited adresses were g... more as the last three days of a Summer School in Mathematical Logic. One hour invited adresses were given by Prof. J.

Notre Dame Journal of Formal Logic, 1992

The College Mathematics Journal, 1985

The College Mathematics Journal, 1985

This text is intended for a brief introductory course in plane geometry. It covers the topics fro... more This text is intended for a brief introductory course in plane geometry. It covers the topics from elementary geometry that are most likely to be required for more advanced mathematics courses. The only prerequisite is a semester of algebra. The emphasis is on applying basic geometric principles to the numerical solution of problems. For this purpose the number of theorems and definitions is kept small. Proofs are short and intuitive, mostly in the style of those found in a typical trigonometry or precalculus text. There is little attempt to teach theorem proving or formal methods of reasoning. However the topics are ordered so that they may be taught deductively. The problems are arranged in pairs so that just the odd-numbered or just the even-numbered can be assigned. For assistance, the student may refer to a large number of completely worked-out examples. Most problems are presented in diagram form so that the difficulty of translating words into pictures is avoided. Many proble...

This membership list may not be used for commercial purposes, for bulk mailing, or to prepare mai... more This membership list may not be used for commercial purposes, for bulk mailing, or to prepare mailing lists, without written permission from the Association. For information regarding use of this list, contact the Secretary-Treasurer of the Association. Corrections to this list should ...

This membership list may not be used for commercial purposes, for bulk mailing, or to prepare mai... more This membership list may not be used for commercial purposes, for bulk mailing, or to prepare mailing lists, without written permission from the Association. For information regarding use of this list, contact the Secretary-Treasurer of the Association. Corrections to this list should ...

Journal of Symbolic Logic, Mar 1, 1974

The Journal of Symbolic Logic, 1974

In [1] we proved the following interpolation theorem for first-order (finitary) logic: Theorem (S... more In [1] we proved the following interpolation theorem for first-order (finitary) logic: Theorem (Scott). Let A and B be sentences. There is an -sentence C such that A → C and C → B iff whenever and are -isomorphic structures and satisfies A then satisfies B. We show here that the Theorem holds for A and B in L ω1, ω only if we permit the interpolant C to be in L (2 ω )+ , ω, where (2 ω )+ is the successor of 2 ω . Our language contains the usual logical symbols and the relation symbols R i, is i ∈ J. If I ⊆ J define an I-sentence to be a sentence containing only relations with subscripts in I. If define an -sentence to be a boolean combination of I-sentences with , i.e., a sentence consisting of I-sentences, for various I's in , joined together by ∨, ∧ and ⌝; e.g., ∀xR 1(x) ∨ ∀xR 2(x)is a{{1}, {2}}-sentence but ∀x(R 1(x) ∨ R 2(x)) is not. If is a structure let , be the structure obtained by restricting to just the relations with subscripts in I. We say that is -isomorphic to if i...

The Two-Year College Mathematics Journal, 1981

The Journal of Symbolic Logic, 1966

as the last three days of a Summer School in Mathematical Logic. One hour invited adresses were g... more as the last three days of a Summer School in Mathematical Logic. One hour invited adresses were given by Prof. J.

Notre Dame Journal of Formal Logic, 1992

The College Mathematics Journal, 1985