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.

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.

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.

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

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 (...) there to hear it, then the answer to the question is evidently "no." But if noise means that, if there were (counterfactually) someone there, then he would hear it, then just as obviously, the answer becomes "yes.". (shrink)

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.

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 (...) shown that a particular sub-system of Peano Arithmetic is powerful enough to express assertions about syntax, provability, consistency, and models, while being too weak to allow the standard proofs of the theorems to go through. An alternative proof is available for the First Incompleteness Theorem, but is of such a different nature that the import of the theorem changes. And there are no alternative proofs for (certainly) the Completeness and (apparently) the Second Incompleteness Theorems. It is therefore perfectly rational for someone to be skeptical about Godel’s results. (shrink)

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. (...) In particular, F proves some (but not all) of the Peano Axioms; that is, F proves the axioms of a sub-theory - call it FPA - of second-order Peano-Arithmetic. This short technical note will demonstrate that the converse also holds, in the following sense. F has the same language as second-order Peano Arithmetic except that, in addition, it has a two-place predicate symbol “Μ”. Then it is possible to provide a definition, indeed a reasonable definition, for “Μ” such that FPA proves all the axioms of F. So F and FPA effectively have the same proof-theoretic strength. In particular FPA, which lacks the Successor Axiom stating that every natural number has a successor, is able to prove the Euclidean Algorithm, Unique Prime Factorization, and Quadratic Reciprocity, indeed (again) a sizable amount of arithmetic. (shrink)

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 (...) Multiplication, but also Lagrange’s Four-Square Theorem. Adding one more axiom, the one-oneness of succession, one can prove many more theorems, such as Quadratic Reciprocity and Fermat’s Little Theorem. By looking at arithmetic in this general setting, one receives a deeper understanding of the underlying structures. (shrink)

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.

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.

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.

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 (...) is a serious one, because in the end, one can lose sight of what one should be doing in the first place. Perhaps it is taking matters too literally, but it seems that there is something to be said for taking the term to represent X. Doing so and accepting the term to have some kind of significance, it is then natural to focus on the question of what a foundation of arithmetic should be; and, if one exists, what one is. Whatever the case, that is what shall be done in this paper. (shrink)

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?

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 (...) evidently one-to-one, and the image of phi is contained in S. Indeed, it is properly contained in S, because I myself can be an object of my thoughts and so belong to S, but I myself am not a mere thought. Thus S is infinite. (shrink)