When Did Set Theory Become the Dominant Framework in Mathematics?

[aha!]

Dear members,

it has recently been put forth to me the question as to when and by which means did it become clear to the majority of logicians and mathematicians that set theory prevailed over Russell's theory of types (as in Principia)?

(By set theory I mean as in the axioms of Zermelo-Fraenkel's or ZFC, NBG etc.)

Not being an expert, I wasn't exactly secure as to how to answer this question. Both Whitehead-Russell's and Zermelo-Fraenkel's axioms are victims to Gödel's second Incompleteness theorem, with the difference that the consistency of ZFC can be proved by assuming the existence of an inaccessible cardinal.

What are your views on this?

 

[matt grime]

IN what sense was the question put to you? Examined essay or just general inquiry? I doubt there is a date, but I an explain the reasons why it is preferred, or at least for my money, the main reason: ZF(C) etc are now in a form (axiomatic) that agrees with other styles of mathematics and is of minimal complexity.

 

[tacman]

Dear ,

Thank you for bringing up this interesting question. The prevailing of set theory over Russell's theory of types can be traced back to the early 20th century, specifically around the 1920s. This was a time when the foundations of mathematics were being reexamined and many mathematicians were looking for a more rigorous and consistent approach to mathematics.

One of the key figures in this movement was David Hilbert, who proposed his famous program to formalize all of mathematics using a consistent set of axioms. Hilbert's program was heavily influenced by the work of Georg Cantor, who introduced the concept of sets and set theory in the late 19th century.

In 1908, Ernst Zermelo published his axioms for set theory, which became known as Zermelo-Fraenkel set theory (ZFC). This set of axioms was seen as a more rigorous and consistent approach to set theory compared to Russell's theory of types, which had been introduced in 1903.

However, it wasn't until the 1920s that ZFC gained widespread acceptance and became the dominant set theory in mathematics. This was due in part to the work of mathematicians such as John von Neumann, who helped to develop the modern concept of a set as a collection of objects satisfying certain properties.

Furthermore, the discovery of Gödel's incompleteness theorems in 1931 had a major impact on the acceptance of ZFC. While it showed that ZFC (and any other consistent formal system) cannot prove its own consistency, it also demonstrated that ZFC was a powerful and flexible enough system to capture most of mathematics.

In conclusion, while there were debates and discussions about the foundations of mathematics in the early 20th century, it was around the 1920s that ZFC emerged as the prevailing set theory. Its acceptance was solidified by its ability to capture most of mathematics and withstand the challenges posed by Gödel's incompleteness theorems.

I hope this helps to answer your question. Thank you for bringing up this interesting topic for discussion.