Frege's theorem
Encyclopedia
Frege's theorem states that the Peano axioms of arithmetic
can be derived in second-order logic
from Hume's principle
. It was first proven, informally, by Gottlob Frege
in his Die Grundlagen der Arithmetik (Foundations of Arithmetic), published in 1884, and proven more formally in his Grundgesetze der Arithmetik (Basic Laws of Arithmetic), published in two volumes, in 1893 and 1903. The theorem was re-discovered by Crispin Wright
in the early 1980s and has since been the focus of significant work. It is at the core of the philosophy of mathematics
known as neo-logicism.
Arithmetic
Arithmetic or arithmetics is the oldest and most elementary branch of mathematics, used by almost everyone, for tasks ranging from simple day-to-day counting to advanced science and business calculations. It involves the study of quantity, especially as the result of combining numbers...
can be derived in second-order logic
Second-order logic
In logic and mathematics second-order logic is an extension of first-order logic, which itself is an extension of propositional logic. Second-order logic is in turn extended by higher-order logic and type theory....
from Hume's principle
Hume's principle
Hume's Principle or HP—the terms were coined by George Boolos—says that the number of Fs is equal to the number of Gs if and only if there is a one-to-one correspondence between the Fs and the Gs. HP can be stated formally in systems of second-order logic...
. It was first proven, informally, by Gottlob Frege
Gottlob Frege
Friedrich Ludwig Gottlob Frege was a German mathematician, logician and philosopher. He is considered to be one of the founders of modern logic, and made major contributions to the foundations of mathematics. He is generally considered to be the father of analytic philosophy, for his writings on...
in his Die Grundlagen der Arithmetik (Foundations of Arithmetic), published in 1884, and proven more formally in his Grundgesetze der Arithmetik (Basic Laws of Arithmetic), published in two volumes, in 1893 and 1903. The theorem was re-discovered by Crispin Wright
Crispin Wright
Crispin Wright is a British philosopher, who has written on neo-Fregean philosophy of mathematics, Wittgenstein's later philosophy, and on issues related to truth, realism, cognitivism, skepticism, knowledge, and objectivity....
in the early 1980s and has since been the focus of significant work. It is at the core of the philosophy of mathematics
Philosophy of mathematics
The philosophy of mathematics is the branch of philosophy that studies the philosophical assumptions, foundations, and implications of mathematics. The aim of the philosophy of mathematics is to provide an account of the nature and methodology of mathematics and to understand the place of...
known as neo-logicism.