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www.logicmatters.net | ||
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dvt.name
Gödel's incompleteness theorems have been hailed as "the greatest mathematical discoveries of the 20th century" - indeed, the theorems apply not only to mathematics, but all formal systems and have deep implications for science, logic, computer science, philosophy, and so on. In this post, I'll give a simple but rigorous sketch of Gödel's First Incompleteness ... |
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xorshammer.com
There are a number of applications of logic to ordinary mathematics, with the most coming from (I believe) model theory. One of the easiest and most striking that I know is called Ax's Theorem. Ax's Theorem: For all polynomial functions $latex f\colon \mathbb{C}^n\to \mathbb{C}^n$, if $latex f$ is injective, then $latex f$ is surjective. Very... |
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www.umsu.de
[AI summary] A blog post explores the philosophical conflict between mathematical pluralism and logical consistency using Peano Arithmetic, followed by a comment section debating the implications for semantic meaning and privileged models. |
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math.andrej.com
[AI summary] The discussion revolves around the interpretation of inductive types and their elements in type theory, particularly focusing on the concept of 'reachability' through constructors. The key points include: 1) The idea that inductive types may have elements that are not reachable by constructor application, which is a non-absolute property. 2) The distinction between internal and external perspectives in models, where the 'standard' natural numbers (N) in a model may differ from the meta-theoretical N. 3) The role of exotic models in highlighting the non-absoluteness of inductive types and their elements. 4) The importance of recognizing that inductive types are not freely generated, as seen in identity types, and the challenges of defining genera... |
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