0.00 undecidability of Hlting Problem

0.01 definition of 0'

0.1 RCA0 and ACA0

0.2 (minimal) models of reverse maths

1 Ramsey's theorem (e.g. any pair formed by elements in H is colored by the same colour)

2 RT_n^n can be reduced to RT_2^n二染色

• 3 proof of Ramsey's theorem (Ramsey used induction.for simplicity, we only go from RT^1_2 to ^2_2)

6 Another proof.可能如4跟0是染是紅的，跟1染是藍(lán)的（總有位置沒人占，因此對(duì)數(shù)n，最多問n次，最后每個(gè)數(shù)都會(huì)出現(xiàn)）

7 analysis of two proofs

7.5 left turns mean red

8 ACA0 implies RTnk

9 RT32 implies ACA0

10 停機(jī)問題限制到前段（前a個(gè)階段，如2，就看第0個(gè)程序（輸入是0）和第1個(gè)程序（輸入是1））(不可能永遠(yuǎn)變化，因?yàn)樽兂赏C(jī)了就不會(huì)再變化，如a=100最多變100次)

11 如果e要停機(jī)一定在b之前停

12 RT22 and WKL0 (Liu, etc.)