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. His major works published with the goal of doing this are these:

• in 1879, Begriffsschrift, defining his “axiomatic-deductive” predicate calculus for the ultimate purpose of proving the basic truths of arithmetic "by means of pure logic."
• in 1884, Die Grundlagen der Arithmetik, using his predicate calculus to present an axiomatic theory of arithmetic.
• in 1893, the first volume of Die Grundgesetze der Arithmetik, containing an intuitive collection of axioms and formal proofs of number theory.

As we have seen, Boole’s developed his algebra of logic as a means by which deduction becomes calculation. Frege's predicate calculus in the Begriffsschrift 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

Driven by “an over-ruling passion to demonstrate his position conclusively” and not “content with the usual informal mathematical standard of rigour,” Frege’s exposition in Grundgesetze is characterized by a great degree by precision and explicitness.

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.

Central to all of this work was a distinction that Frege was developing, but only finally published in 1892, namely that every concept, mathematical or otherwise, had two important, entirely distinct aspects:

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

This distinction of Frege's 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. In other words, a concept partitions $V$ (a universe of discourse) into two classes: 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, $V$ -- what we have seen as Dedekind’s Gedankenwelt
2. an unrestricted principle of Comprehension as a basic law of thought: Given any well-defined property (formalized as an open, unquantified sentence $Φ(x)$) there exists the set $S = \{ x: Φ(x)\}$.

Just one of assumptions (1) and (2) suffices for “naïve” set theory:

• deriving the Universal Set from the principle of Comprehension: replace $Φ(x)$ by a truism, such as the property $x = x$.
• deriving Comprehension from the Universal Set: 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!

To these two, one adds Dedekind's principle of Extensionality (Dedekind 1888, 345).

Frege’s Grundgesetze made an implicit appeal to such an unrestricted Comprehension principle according to which every predicate (concept/property) defines a set. This is in contrast to the use of restricted predicates found in Cantor's early theory of sets.

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Primary sources

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

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