Reverse mathematics | Wikipedia audio article
Channel: wikipedia tts
Date: 2019-10-07

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 &#969;-models and &#946;-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)).