Kleene's recursion theorem | Wikipedia audio article
Channel: wikipedia tts
Date: 2018-12-31

This is an audio version of the Wikipedia Article:
https://en.wikipedia.org/wiki/Kleene%27s_recursion_theorem


00:00:43 1 Notation
00:02:31 2 Rogers's fixed-point theorem
00:03:37 2.1 Proof of the fixed-point theorem
00:09:20 2.2 Fixed-point free functions
00:10:21 3 Kleene's second recursion theorem
00:12:25 3.1 Comparison to Rogers's theorem
00:13:05 3.2 Application to quines
00:14:31 3.3 Application to elimination of recursion
00:17:53 3.4 Reflexive programming
00:18:33 4 The first recursion theorem
00:20:39 4.1 Example
00:20:46 4.2 Proof sketch for the first recursion theorem
00:24:16 4.3 Comparison to the second recursion theorem
00:24:42 5 Generalized theorem
00:26:23 6 See also
00:27:30 7 References
00:28:49 8 External links



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: 0.9385419917650819
Voice name: en-US-Wavenet-C


"I cannot teach anybody anything, I can only make them think."
- Socrates


SUMMARY
=======
In computability theory,  Kleene's recursion theorems are a pair of fundamental results about the application of computable functions to their own descriptions.    The theorems were first proved by Stephen Kleene in 1938 and appear in his 1952 book Introduction to Metamathematics. A related theorem which constructs fixed points of a computable function is known as Rogers's theorem and is due to Hartley Rogers, Jr. (1967).
The recursion theorems can be applied to construct fixed points of certain operations on computable functions, to generate quines, and to construct functions defined via recursive definitions.