Reverse mathematics | Wikipedia audio article

This is an audio version of the Wikipedia Article: https://en.wikipedia.org/wiki/Reverse_mathematics 00:00:32 1 General principles 00:01:04 1.1 Use of second-order arithmetic 00:01:37 1.2 Use of higher-order arithmetic 00:02:09 2 The big five subsystems of second-order arithmetic 00:02:41 2.1 The base system RCAsub0/sub 00:03:14 2.2 Weak Kőnig's lemma WKLsub0/sub 00:03:46 2.3 Arithmetical comprehension ACAsub0/sub 00:04:18 2.4 Arithmetical transfinite recursion ATRsub0/sub 00:04:51 2.5 Πspansup1/supsub1/sub 00:05:23 3 Additional systems 00:05:55 4 ω-models and β-models 00:06:28 5 See also Listening is a more natural way of learning, when compared to reading. Written language only began at around 3200 BC, but spoken language has existed long ago. Learning by listening is a great way to: - increases imagination and understanding - improves your listening skills - improves your own spoken accent - learn while on the move - reduce eye strain Now learn the vast amount of general knowledge available on Wikipedia through audio (audio article). You could even learn subconsciously by playing the audio while you are sleeping! If you are planning to listen a lot, you could try using a bone conduction headphone, or a standard speaker instead of an earphone. Listen on Google Assistant through Extra Audio: https://assistant.google.com/services/invoke/uid/0000001a130b3f91 Other Wikipedia audio articles at: https://www.youtube.com/results?search_query=wikipedia+tts Upload your own Wikipedia articles through: https://github.com/nodef/wikipedia-tts Speaking Rate: 0.957401342365991 Voice name: en-AU-Wavenet-D "I cannot teach anybody anything, I can only make them think." - Socrates SUMMARY ======= Reverse mathematics is a program in mathematical logic that seeks to determine which axioms are required to prove theorems of mathematics. Its defining method can briefly be described as "going backwards from the theorems to the axioms", in contrast to the ordinary mathematical practice of deriving theorems from axioms. It can be conceptualized as sculpting out necessary conditions from sufficient ones. The reverse mathematics program was foreshadowed by results in set theory such as the classical theorem that the axiom of choice and Zorn's lemma are equivalent over ZF set theory. The goal of reverse mathematics, however, is to study possible axioms of ordinary theorems of mathematics rather than possible axioms for set theory. Reverse mathematics is usually carried out using subsystems of second-order arithmetic, where many of its definitions and methods are inspired by previous work in constructive analysis and proof theory. The use of second-order arithmetic also allows many techniques from recursion theory to be employed; many results in reverse mathematics have corresponding results in computable analysis. Recently, higher-order reverse mathematics has been introduced, in which the focus is on subsystems of higher-order arithmetic, and the associated richer language. The program was founded by Harvey Friedman (1975, 1976) and brought forward by Steve Simpson. A standard reference for the subject is (Simpson 2009), while an introduction for non-specialists is (Stillwell 2018). An introduction to higher-order reverse mathematics, and also the founding paper, is (Kohlenbach (2005)).