
725What 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 the same individual sometimes, if not usually or always, means the same word in …Read more

712In 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 neologicism was begun by Crispin Wright. Neologicism holds that Fr…Read more

584It 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. Howe…Read more

65I 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 tablerocks exist, fairies do not. Argument can appear longwinded 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…Read more

492The 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 secondorder 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 its own …Read more

342Consider a onegood 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 of the good, and barter these promi…Read more

15In "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 evidently onetoone, and the image of phi is contained in S…Read more

683These notes are meant to continue from the paper on Consistency, in proving numbertheoretic theorems from the secondorder 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.

21On the one hand, firstorder 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 firstorder 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?

18As 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 philosophicological 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 is a serious one, because in the e…Read more

112A 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.

524General 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 onetoone. 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 A…Read more

717A 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.

715I 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 wrongheaded in several major ways, both in its methodology and its content. Herein is a brief explanation why.

638In 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 ʻshould,ʼ although …Read more

402A new secondorder 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

376The system called F is essentially a subtheory 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…Read more

637For 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.

439These notes are meant to continue from the paper on Consistency, in proving numbertheoretic theorems from the secondorder 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.

37Howmany numbers, such as 2 and 1000, relate or are capable of expressing the size of a group or set. Both Cantor and Frege analyzed howmany number in terms of onetoone 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 outandout definition, in the case of Frege.

14Note 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.

21The system of arithmetic considered in Consistency, which is essentially secondorder Peano Arithmetic without the Successor Axiom, is used to prove more theorems of arithmetic, up to Quadratic Reciprocity.

49Neologicism uses definitions and Hume's Principle to derive arithmetic in secondorder logic. This paper investigates how much arithmetic can be derived using definitions alone, without any additional principle such as Hume's.

562Using an axiomatization of secondorder arithmetic (essentially secondorder 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 secondorder sequence, which is a stronger notion. It is shown t…Read more

24In 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 subtheory of Peano Arithmetic; the subtheory is missing, most notably, the Successor Axiom

36Secondorder Peano Arithmetic minus the Successor Axiom is developed from first principles through Quadratic Reciprocity and a proof of selfconsistency. This paper combines 4 other papers of the author in a selfcontained exposition

575Bertand's Postulate is proved in Peano Arithmetic minus the Successor Axiom.
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