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Here is a definition:. Now given this definition, we can reformulate the General Principle of Induction more strictly as:. We may sketch the proof strategy as follows. That is, assume:. Recall that Fact 7 is:. Thus, if the antecedent can be established, the proof is done. However, this claim can be established straightforwardly from things we know to be true and, in particular, from facts contained in the antecedent of the Principle we are trying to prove, which we assumed as part of our conditional proof. The reader is encouraged to complete the proof as an exercise. For those who would like to check their work, we give the complete Proof of the General Principle of Induction here:.

Proof of the General Principle of Induction.

Multisets and a new framework for arithmetic - Data Structures Math Foundations 187

Frege uses the Principle of Mathematical Induction to prove that every natural number has a successor that's a natural number. We may formulate the theorem as follows:. Frege then considers the concept member of the predecessor-series ending with n , i. That is, Frege proves that every natural number has a successor by proving the following Lemma on Successorsby induction:. Frege can establish Theorem 5 by proving the Lemma on Successors and by showing that the successor of a natural number is itself a natural number. For example, the third member of this sequence is true because there are 3 natural numbers 0, 1, and 2 that are less than or equal to 2; so the number 2 precedes the number of numbers less than or equal to 2.

So, given this intuitive understanding of the Lemma on Successors, Frege has a good strategy for proving that every number has a successor.


Now to prove the Lemma on Successors by induction, we need to reconfigure this Lemma to a form which can be used as the consequent of the Principle of Mathematical Induction; i. The result is therefore something that has been proved and that we therefore know to be true:. Gg I , Theorem :. Given this proof of the Lemma on Successors, Theorem 5 is not far away. We still have to show that such successor cardinals are natural numbers.

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Theorem 5 now follows from the Lemma on Successors and the fact that successors of natural numbers are natural numbers. Before we turn to the last section of this entry, it is worth mentioning the mathematical significance of this theorem. It is an immediate consequence Theorem 5 and the fact that Predecessor is a functional relation that every number has a unique successor.

The Foundations of Arithmetic | Northwestern University Press

These definitions constitute the foundations of arithmetic. We discuss the thinking behind this attitude, and other things, in what follows. If Frege could show that the basic laws of number theory are derivable from analytic truths of logic, then he could argue that we need only appeal to the faculty of understanding as opposed to some faculty of intuition to explain our knowledge of the truths of arithmetic.

The Kantian model here is that of geometry; Kant thought that our intuitions of figures and constructions played an essential role in the demonstrations of geometrical theorems. This in turn required that he show that the latter are derivable using only rules of inference, axioms, and definitions that are purely analytic principles of logic. Here is what Frege says:. Since it is only in the context of a proposition that words have any meaning, our problem becomes this: To define the sense of a proposition in which a number word occurs.

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Arithmetic thus becomes simply a development of logic, and every proposition of arithmetic a law of logic, albeit a derivative one. In the present book, this shall be confirmed, by the derivation of the simplest laws of Numbers by logical means alone.

The foundations of arithmetic : a logico-mathematical enquiry into the concept of number

By what means are we justified in recognizing numbers as objects? Even if this problem is not solved to the degree I thought it was when I wrote this volume, still I do not doubt that the way to the solution has been found. Given the above discussion, it should be clear that Frege at some point in Gg endorsed existence claims, either directly in his formalism or in his metalanguage, for the following entities:.

Although Frege attempted to reduce the latter two kinds of entities truth-values and numbers to extensions, the fact is that the existence of concepts and extensions are derivable from his Rule of Substitution and Basic Law V, respectively. In light of these existence claims, a Kantian might well suggest not only that explicit existence claims are synthetic rather than analytic i.

If the Kantian is right, then some other faculty such as intuition might still be needed to account for our knowledge of the existence claims of arithmetic. Boolos suggests a defense for Frege with respect to this particular aspect of his logic, namely, to reinterpret by paraphrasing the second-order quantifiers so as to avoid commitment to concepts.

See Boolos for the details. The question of how we obtain knowledge of such principles is still an open question in philosophy. In contemporary philosophy, this question is still poignant, since many philosophers do accept that properties and relations of various sorts exist.

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  • Moreover, he thought that an appeal to extensions would answer one of the questions that motivated his work:. Now it is unclear why Frege thought that he could answer the question posed here by saying that we apprehend numbers as the extensions of concepts. Some philosophers do argue that certain consistent principles having the same logical form as Basic Law V are analytic, and that such principles justify reference to the entities described in the left-side condition by grounding such reference in the truth of the right-side condition.

    To hold that Basic Law V is analytic, it seems that one must hold that the right-side condition implies the corresponding left-side condition as a matter of meaning. Suppose the right hand condition implies the left-side condition as a matter of meaning. That is, suppose that R implies L as a matter of meaning:.

    Now note that L itself can be analyzed, from a logical point of view. But if R implies L as a matter of meaning, and L implies D as a matter of meaning, then R implies D as a matter of meaning. In other words, the suggestion that Va i. The moral to be drawn here is that, even if Basic Law V were consistent, it is not exactly clear how its right side analytically implies the existence of extensions. In the end, we may need some other way of justifying our knowledge of principles like Basic Law V, that imply the existence of abstract objects — the justification discussed so far seems to contain a gap.

    We might agree that there must be logical objects of some sort if logic is to have a subject matter, but if Frege is to achieve his goal of showing that our knowledge of arithmetic is free of intuition, then at some point he has to address the question of how we can know that numbers exist. A reason must be given as to why the claim:.

    Concerning this definition, Frege says:. In Gl , Frege solves the problem by giving his explicit definition of numbers in terms of extensions. Even if Frege somehow could have successfully restricted the quantifiers of Gg to avoid the Julius Caesar problem, he would no longer have been able to apply his system by extending it to include names of ordinary non-logical objects. That means his logical system could not be used for the analysis of ordinary language. But it was just the analysis of ordinary language that led Frege to his insight that a statement of number is an assertion about a concept.

    The second question arises because the Julius Caesar problem applies to Hume's Principle; without a solution to that problem, Frege can't claim to have precisely specified which objects the numbers are, so as to delineate them within the domain of all logical and non-logical objects?

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    These contextual definitions combine two jobs which modern logicians now typically accomplish with separate principles. The latter should specify identity conditions for logical objects in terms of their most salient characteristic, one which distinguishes them from other objects. By way of example, consider modern set theory. For example, consider the well-known Subset or Separation Axiom:.

    Note that the second principle offers identity conditions in terms of the most salient features of sets, namely, the fact that they, unlike other objects, have members. This leads us naturally to a very general principle of identity for any objects whatever:. The question of existence is thus laid bare. In some classic essays and , Boolos appears to recommend this very procedure of using separate existence and identity principles. It openly faces the epistemological questions head-on: Do we know that Numbers is true, and if so, how?

    This is where philosophers need to concentrate their energies. Those separate existence claims should be the focus of attention. I was motivated to write the present entry after reading an early draft of an essay by William Demopoulos. The draft was eventually published as Demopoulos and Clark Demopoulos kindly allowed me to quote certain passages from that early draft in the footnotes to the present entry.

    I am also indebted to Roberto Torretti, who carefully read this piece and identified numerous infelicities; to Franz Fritsche, who noticed a quantifier transposition error in Fact 2 about the strong ancestral; to Seyed N.