Book contents
- Frontmatter
- Contents
- Introduction
- Historical remarks on Suslin's problem
- The continuum hypothesis, the generic-multiverse of sets, and the Ω conjecture
- ω-models of finite set theory
- Tennenbaum's theorem for models of arithmetic
- Hierarchies of subsystems of weak arithmetic
- Diophantine correct open induction
- Tennenbaum's theorem and recursive reducts
- History of constructivism in the 20th century
- A very short history of ultrafinitism
- Sue Toledo's notes of her conversations with Gödel in 1972–5
- Stanley Tennenbaum's Socrates
- Tennenbaum's proof of the irrationality of √2
Introduction
Published online by Cambridge University Press: 07 October 2011
- Frontmatter
- Contents
- Introduction
- Historical remarks on Suslin's problem
- The continuum hypothesis, the generic-multiverse of sets, and the Ω conjecture
- ω-models of finite set theory
- Tennenbaum's theorem for models of arithmetic
- Hierarchies of subsystems of weak arithmetic
- Diophantine correct open induction
- Tennenbaum's theorem and recursive reducts
- History of constructivism in the 20th century
- A very short history of ultrafinitism
- Sue Toledo's notes of her conversations with Gödel in 1972–5
- Stanley Tennenbaum's Socrates
- Tennenbaum's proof of the irrationality of √2
Summary
§1. Introduction. It is a unique feature of the field of mathematical logic, that almost any technical result from its various subfields: set theory, models of arithmetic, intuitionism and ultrafinitism, to name just a few of these, touches upon deep foundational and philosophical issues. What is the nature of the infinite? What is the significance of set-theoretic independence, and can it ever be eliminated? Is the continuum hypothesis a meaningful question? What is the real reason behind the existence of non-standard models of arithmetic, and do these models reflect our numerical intuitions? Do our numerical intuitions extend beyond the finite at all? Is classical logic the right foundation for contemporary mathematics, or should our mathematics be built on constructive systems? Proofs must be correct, but they must also be explanatory. How does the aesthetic of simplicity play a role in these two ideals of proof, and is there ever a “simplest” proof of a given theorem?
The papers collected here engage each of these questions through the veil of particular technical results. For example, the new proof of the irrationality of the square root of two, given by Stanley Tennenbaum in the 1960s and included here, brings into relief questions about the role simplicity plays in our grasp of mathematical proofs. In 1900 Hilbert asked a question which was not given at the Paris conference but which has been recently found in his notes for the list: find a criterion of simplicity in mathematics. The Tennenbaum proof is a particularly striking example of the phenomenon Hilbert contemplated in his 24th Problem.
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- Information
- Set Theory, Arithmetic, and Foundations of MathematicsTheorems, Philosophies, pp. ix - xivPublisher: Cambridge University PressPrint publication year: 2011