User talk:Whayes43

Frege’s theory of arithmetic

Frege, by virtue of his work creating predicate logic, is one of the founders of modern (mathematical) logic. His view was that mathematics is reducible to logic.

In 1892, Frege published his now famous discussion showing that concepts (mathematical and otherwise) have two important aspects that are distinct from one another:

1. Sinn: a “meaning” or “sense” or “connotation”
2. Bedeutung: an “extension” or “reference” or “denotation”

This distinction is the basis of what Gödel (very much later) characterized as the dichotomic conception:

Any well-defined concept (property or predicate) $P(x)$ establishes a dichotomy of all things into those that are $P$s and those that are non-$P$s, i.e., a concept partitions $V$ (a universe of discourse) into the class $\{ x : P(x) \}$ and its absolute complement, the class $\{ x : \neg P(x) \}$.

This standpoint is based on two key assumptions:

1. the existence of a universal set, $V$ -- Dedekind’s Gedankenwelt -- and
2. the principle of comprehension as a basic law of thought: Given a well-defined property (an open, unquantified sentence $Φ(x)$), there exists the set $S = \{ x: Φ(x)\}$.

To them, one adds Dedekind's principle of extensionality (Dedekind 1888, 345). Just one of assumptions (1) and (2) suffices for “naïve” set theory. The path from the principle of comprehension to assumption (1) is well known:

simply replace $Φ(x)$ by a truism, such as the property $x = x$.

The converse path is also true:

establish the existence of an all-encompassing domain $V$ given as a set; in order to establish the principle of comprehension, the key idea is that, since $V$ is assumed to be a set, any part of it should again be a set. Hence, since a well-defined concept $P(x)$ defines a subset of $V$, the set $\{ x: P(x) \}$ exists!

As we have seen, Boole’s developed his algebra of logic as a means by which deduction becomes calculation. In 1879, Frege published his “axiomatic-deductive” presentation of the predicate calculus. In doing so, he stood Boole’s purpose on its head:

• Frege’s goal: “to establish ... that arithmetic could be reduced to logic” and, thus, to create a logic by means of which calculation becomes deduction
• Frege’s program: to develop arithmetic as an axiomatic system such that all the axioms were truths of logic

Frege gave the following reason for developing his logic as he did:

Because we cannot enumerate all of the boundless number of laws that can be established, we can obtain completeness only by a search for those which, potentially, imply all the others.

Frege identified as the kernel of his system the axioms (laws) of his logic that potentially imply all the other laws. His statement above implies that he thought his system to be complete, though he did not provide either a precise definition of completeness or a proof that his system was actually complete.

In 1879, Frege published Begriffsschrift, in which he defined his predicate calculus and stated his ultimate aim of proving the basic truths of arithmetic "by means of pure logic."

In 1884, he published Die Grundlagen der Arithmetik to achieve the aim of the Begriffsschrift by presenting an axiomatic theory of arithmetic.

In 1893, he published the first volume of Die Grundgesetze der Arithmetik, in which he axiomatized arithmetic with an intuitive collection of axioms, and proofs of number theory results which he had only sketched earlier he now gave formally.

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• Frege, G. (1884). Die Grundlagen der Arithmetik, K¨obner, Breslau.

• Frege, G. (1892) Uber Sinn und Bedeuting.

O’Connor and Robertson (2002)

Frege (1892)

Ferreiros pp. 18-19

Gillies pp. 74-75

Frege (1879) p. 136 cited in Gillies p. 71

O’Connor and Robertson (2002)