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  1. Andrew Boucher, Proving Bertrand's Postulate.
    Bertand's Postulate is proved in Peano Arithmetic minus the Successor Axiom.
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  2. Andrew Boucher, Against Angles and the Fregean-Cantorian Theory of Number.
    How-many numbers, such as 2 and 1000, relate or are capable of expressing the size of a group or set. Both Cantor and Frege analyzed how-many number in terms of one-to-one correspondence between two sets. That is to say, one arrived at numbers by either abstracting from the concept of correspondence, in the case of Cantor, or by using it to provide an out-and-out definition, in the case of Frege.
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  3. Andrew Boucher, A A.... G.
    These notes are meant to continue from the paper on Consistency, in proving number-theoretic theorems from the second-order arithmetical system called FFFF. Its ultimate target is Quadratic Reciprocity, although it introduces and proves some facts about the least common multiple at the start.
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  4. Andrew Boucher, A Note On the Berry Paradox.
    For those who have understood the solution to the Liarʼs Paradox and the Paradoxes of Predication, presented in A Comprehensive Solution to the Paradoxes and The Solution to the Liarʼs Paradox1, it will come as no surprise how the Berry Paradox should be solved. Nonetheless, the solution will be presented here in a short note, for completenessʼ sake.
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  5. Andrew Boucher, Arithmetic Without the Successor Axiom.
    Second-order Peano Arithmetic minus the Successor Axiom is developed from first principles through Quadratic Reciprocity and a proof of self-consistency. This paper combines 4 other papers of the author in a self-contained exposition.
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  6. Andrew Boucher, Equivalence of F with a Sub-Theory of Peano Arithmetic.
    In a short, technical note, the system of arithmetic, F, introduced in Systems for a Foundation of Arithmetic and "True" Arithmetic Can Prove Its Own Consistency and Proving Quadratic Reciprocity, is demonstrated to be equivalent to a sub-theory of Peano Arithmetic; the sub-theory is missing, most notably, the Successor Axiom.
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  7. Andrew Boucher, Logical Philosophy.
    Note to the reader: To avoid confusion and possible misinterpretations of the author's intentions, whenever a paragraph contains a definition or explication of how the author means the meaning of a word, asterisks have been placed after the paragraph number and before the word or words in question. The reader is warned that some words may be meant idiosyncratically.
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  8. Andrew Boucher, Proving Quadratic Reciprocity.
    The system of arithmetic considered in Consistency, which is essentially second-order Peano Arithmetic without the Successor Axiom, is used to prove more theorems of arithmetic, up to Quadratic Reciprocity.
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  9. Andrew Boucher, Sub-Theory of Peano Arithmetic.
    The system called F is essentially a sub-theory of Frege Arithmetic without the ad infinitum assumption that there is always a next number. In a series of papers (Systems for a Foundation of Arithmetic, True” Arithmetic Can Prove Its Own Consistency and Proving Quadratic Reciprocity) it was shown that F proves a large number of basic arithmetic truths, such as the Euclidean Algorithm, Unique Prime Factorization (i.e. the Fundamental Law of Arithmetic), and Quadratic Reciprocity, indeed a sizable amount of arithmetic. (...)
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  10. Andrew Boucher, "True" Arithmetic Can Prove its Own Consistency.
    Using an axiomatization of second-order arithmetic (essentially second-order Peano Arithmetic without the Successor Axiom), arithmetic's basic operations are defined and its fundamental laws, up to unique prime factorization, are proven. Two manners of expressing a system's consistency are presented - the "Godel" consistency, where a wff is represented by a natural number, and the "real" consistency, where a wff is represented as a second-order sequence, which is a stronger notion. It is shown that the system can prove at least its (...)
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  11. Andrew Boucher, Who Needs (to Assume) Hume's Principle?
    Neo-logicism uses definitions and Hume's Principle to derive arithmetic in second-order logic. This paper investigates how much arithmetic can be derived using definitions alone, without any additional principle such as Hume's.
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  12. Andrew Boucher, A Comprehensive Solution to the Paradoxes.
    A solution to the paradoxes has two sides: the philosophical and the technical. The paradoxes are, first and foremost, a philosophical problem. A philosophical solution must pinpoint the exact step where the reasoning that leads to contradiction is fallacious, and then explain why it is so.
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  13. Andrew Boucher, A Natural First-Order System of Arithmetic Which Proves Its Own Consistency.
    Herein is presented a natural first-order arithmetic system which can prove its own consistency, both in the weaker Godelian sense using traditional Godel numbering and, more importantly, in a more robust and direct sense; yet it is strong enough to prove many arithmetic theorems, including the Euclidean Algorithm, Quadratic Reciprocity, and Bertrand’s Postulate.
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  14. Andrew Boucher, A Philosophical Introduction to the Foundations of Elementary Arithmetic by V1.03 Last Updated: 1 Jan 2001 Created: 1 Sept 2000 Please Send Your Comments to Abo. [REVIEW]
    As it is currently used, "foundations of arithmetic" can be a misleading expression. It is not always, as the name might indicate, being used as a plural term meaning X = {x : x is a foundation of arithmetic}. Instead it has come to stand for a philosophico-logical domain of knowledge, concerned with axiom systems, structures, and analyses of arithmetic concepts. It is a bit as if "rock" had come to mean "geology." The conflation of subject matter and its study (...)
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  15. Andrew Boucher, A Theory of Meaning.
    What an individual means by a word sometimes, if not always, is dependent on the individual, on what he believes, and on his memories; and so on what kind of life he has lived and what kind of experiences he has had, the manner in which he learned the word, and so forth. For instance, someone who lives in a hot climate will surely mean the word ʻcoldʼ in a different way than someone who comes from a cold one. Indeed (...)
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  16. Andrew Boucher, Consistency and Existence by V1.00 Last Updated: 1 Oct 2000 Please Send Your Comments to Abo.
    On the one hand, first-order theories are able to assert the existence of objects. For instance, ZF set theory asserts the existence of objects called the power set, while Peano Arithmetic asserts the existence of zero. On the other hand, a first-order theory may or not be consistent: it is if and only if no contradiction is a theorem. Let us ask, What is the connection between consistency and existence?
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  17. Andrew Boucher, Created: 9 June 2003 12 November 2003 Version 1.1 Www.Andrewboucher.Com/Papers/Quadratic_reciprocity.Pdf.
    These notes are meant to continue from the paper on Consistency, in proving number-theoretic theorems from the second-order arithmetical system called FFFF. Its ultimate target is Quadratic Reciprocity, although it introduces and proves some facts about the least common multiple at the start.
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  18. Andrew Boucher, Comments on Naming and Necessity.
    I recently had the occasion to reread Naming and Necessity by Saul Kripke. NaN struck me this time, as it always has, as breathtakingly clear and lucid. It also struck me this time, as it always has, as wrong-headed in several major ways, both in its methodology and its content. Herein is a brief explanation why.
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  19. Andrew Boucher, Depression in a One-Good Barter Economy.
    Consider a one-good economy where money is not used and only barter holds. As is traditional, the unique good can be exchanged for labor, which itself is used to produce the good; and there are capitalists, who own the means of production, who contract for the labor and keep whatever of the good is left from production after paying the workers. The only unusual feature of the economy is that the various economic agents can also make promises of future delivery (...)
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  20. Andrew Boucher, Dedekind's Proof by V2.0 Last Updated: 10 Dec 2001 Created: 1 Sept 2000 Please Send Your Comments to Abo.
    In "The Nature and Meaning of Numbers," Dedekind produces an original, quite remarkable proof for the holy grail in the foundations of elementary arithmetic, that there are an infinite number of things. It goes like this. [p, 64 in the Dover edition.] Consider the set S of things which can be objects of my thought. Define the function phi(s), which maps an element s of S to the thought that s can be an object of my thought. Then phi is (...)
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  21. Andrew Boucher, General Arithmetic.
    General Arithmetic is the theory consisting of induction on a successor function. Normal arithmetic, say in the system called Peano Arithmetic, makes certain additional demands on the successor function. First, that it be total. Secondly, that it be one-to-one. And thirdly, that there be a first element which is not in its image. General Arithmetic abandons all of these further assumptions, yet is still able to prove many meaningful arithmetic truths, such as, most basically, Commutativity and Associativity of Addition and (...)
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  22. Andrew Boucher, Introduction.
    The Successor Axiom asserts that every number has a successor, or in other words, that the number series goes on and on ad infinitum. The present work investigates a particular subsystem of Frege Arithmetic, called F, which turns out to be equivalent to second-order Peano Arithmetic minus the Successor Axiom, and shows how this system can develop arithmetic up through Gauss' Quadratic Reciprocity Law. It then goes on to represent questions of provability in F, and shows that F can prove (...)
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  23. Andrew Boucher, On Descriptive Ethics.
    In its descriptive sense ethical language allows one to make assertions, which like other assertions may be true or not. “One should not torture,” descriptively, makes an assertion about torture - that it is an act that one should not do. While the peculiar force of ethical language comes from its overloading of different types of uses - descriptive, imperative, and emotive -, our concern here will be with the descriptive. Many of our assertions will focus on the English word (...)
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  24. Andrew Boucher, Systems for a Foundation of Arithmetic.
    A new second-order axiomatization of arithmetic, with Frege's definition of successor replaced, is presented and compared to other systems in the field of Frege Arithmetic. The key in proving the Peano Axioms turns out to be a proposition about infinity, which a reduced subset of the axioms proves.
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  25. Andrew Boucher, The Existence of Numbers (Or: What is the Status of Arithmetic?) By V2.00 Created: 11 Oct 2001 Modified: 3 June 2002 Please Send Your Comments to Abo. [REVIEW]
    I begin with a personal confession. Philosophical discussions of existence have always bored me. When they occur, my eyes glaze over and my attention falters. Basically ontological questions often seem best decided by banging on the table--rocks exist, fairies do not. Argument can appear long-winded and miss the point. Sometimes a quick distinction resolves any apparent difficulty. Does a falling tree in an earless forest make noise, ie does the noise exist? Well, if noise means that an ear must be (...)
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  26. Andrew Boucher, The Solution to the Liar's Paradox.
    A solution to the Liar must do two things. First, it should say exactly which step in the Liar reasoning - the reasoning which leads to a contradiction - is invalid. Secondly, it should explains why this step is invalid.
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  27. Andrew Boucher, Three Theorems of Godel.
    It might seem that three of Godel’s results - the Completeness and the First and Second Incompleteness Theorems - assume so little that they are reasonably indisputable. A version of the Completeness Theorem, for instance, can be proven in RCA0, which is the weakest system studied extensively in Simpson’s encyclopaedic Subsystems of Second Order Arithmetic. And it often seems that the minimum requirements for a system just to express the Incompleteness Theorems are sufficient to prove them. However, it will be (...)
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  28. Andrew Boucher, Who Needs (to Assume) Hume's Principle? July 2006.
    In the Foundations of Arithmetic, Frege famously developed a theory which today goes by the name of logicism - that it is possible to prove the truths of arithmetic using only logical principles and definitions. Logicism fell out of favor for various reasons, most spectacular of which was that the system, which Frege thought would definitively prove his thesis, turned out to be inconsistent. In the early 1980s a movement called neo-logicism was begun by Crispin Wright. Neo-logicism holds that Frege (...)
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  29. Andrew Boucher (1997). Parallel Machines. Minds and Machines 7 (4):543-551.
    Because it is time-dependent, parallel computation is fundamentally different from sequential computation. Parallel programs are non-deterministic and are not effective procedures. Given the brain operates in parallel, this casts doubt on AI's attempt to make sequential computers intelligent.
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