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@yuuki26

yuuki@yuuki26

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737日(2021/11/25より)
ツイート数
884(1.1件/日)

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2023年12月01日(金)3 tweetssource

2023年11月30日(木)5 tweetssource

2023年11月28日(火)1 tweetsource

2023年11月26日(日)2 tweetssource

2023年11月25日(土)8 tweetssource

2023年11月23日(木)10 tweetssource

11月23日

@yuuki26

yuuki@yuuki26

- hobbies: shogi/programming/math/english
- stan of: nothing
- fav game: pokemon go
- comments:
never drink or smoke / never get my driver's license / never fall in love or get married

posted at 03:31:24

   

11月23日

@yuuki26

yuuki@yuuki26

- what's the acc for?: lifestreaming
- mainly tweet: my thoughts
- twitter stance: anything is OK
- what kind of ppl do u want to connect with?: ppl who share my hobbies
- free space:
5th grade~ unschooler / lives alone in late grandma's house / receives $700 a month from parents

posted at 03:31:23

   

2023年11月21日(火)6 tweetssource

11月21日

@yuuki26

yuuki@yuuki26

¹ this indicates "direct implications or relative consistency implications", tho.
(e.g., huge < supercompact)

in order of both size and strength:
• inaccessible < measurable < huge < rank-into-rank < 0=1

posted at 06:20:22

   

11月21日

@yuuki26

yuuki@yuuki26

in the large picture,
• the size of a number
is proportional to
• how many numbers it can prove to be consistent.

the largest number proves the consistency of all numbers, including itself.
by Gödel's theorem, it is a "contradiction".

posted at 04:04:38

   

11月21日

@yuuki26

yuuki@yuuki26

let us write Fin + Inf as ZFC.

similarly, if we add "a large cardinal exists", then we can prove the consistency of ZFC.
• ZFC + LC ⊢ Con(ZFC)

posted at 04:02:04

   

11月21日

@yuuki26

yuuki@yuuki26

by Gödel's second incompleteness theorem, the axioms "finite numbers exist" can't prove their own consistency.
• Fin ⊬ Con(Fin)

now if we add "ℵ₀ exists", then we can prove the consistency of finite numbers.
• Fin + Inf ⊢ Con(Fin)

this shows Fin + Inf > Fin.

posted at 03:59:23

   

2023年11月20日(月)2 tweetssource

11月20日

@yuuki26

yuuki@yuuki26

actually, large cardinals refer to large ♾ with specific properties.

(the successor of an inaccessible cardinal is not inaccessible by definition.)

posted at 02:37:09

   

2023年11月18日(土)2 tweetssource

11月18日

@yuuki26

yuuki@yuuki26

infinities are written
• ℵ₀, ℵ₁, ℵ₂, …, ℵ_ω, …
in order from smallest to largest.

for the number of reals ℶ₁,
• ℶ₁ = ℵ_?
is not provable from the standard axioms of mathematics.¹

the ℶ₁ = ℵ₁ conjecture is called the
• continuum hypothesis (CH).

posted at 21:50:24

   

2023年11月15日(水)1 tweetsource

2023年11月12日(日)8 tweetssource

11月12日

@yuuki26

yuuki@yuuki26

¹ already known to the Jains of India (400 BC), tho.
but they made mistakes such as the number of points on a line |ℝ| and a plane |ℝ²| are not equal. (both are 2^ℵ₀.)

posted at 23:17:26

   

11月12日

@yuuki26

yuuki@yuuki26

actually, the number of natural, integer, and rational numbers are all the same.

that infinite number is called ℵ₀ (aleph zero) or ℶ₀ (beth zero).

Cantor (1874) proved that the number of reals is larger than that.¹
that is called 𝔠 = 2^ℵ₀ = ℶ₁ (beth one).

posted at 23:15:52

   

11月12日

@yuuki26

yuuki@yuuki26

the idea that having the same number is the same as having a 1-1 correspondence is called
• Hume's principle.¹

¹ neither Hume nor Galileo thought it applied to ♾, tho (unlike Cantor).

posted at 21:40:16

   

11月12日

@yuuki26

yuuki@yuuki26

♾ is such a strange thing.
the whole and part can be the same size.¹
cf.
• Galileo's paradox.

¹ contrary to Euclid's 5th Common Notion (300 BC).
although some say that some CNs are by, e.g., Theon of Alexandria (4c).

posted at 21:16:42

   

11月12日

@yuuki26

yuuki@yuuki26

mathematically, the answer is A✔
they're the same.
(even tho evens are part of integers.)

this is bc one integer corresponds to one even.
1 ↦ 2
2 ↦ 4
3 ↦ 6
4 ↦ 8
5 ↦ 10

n ↦ 2n

posted at 13:53:58

   

11月12日

@yuuki26

yuuki@yuuki26

let me explain from scratch.

Q: which is greater,
• the number of all integers
• the number of all even numbers

both are infinite.
there's many ways to think abt it.

posted at 09:42:34

   

2023年11月11日(土)3 tweetssource

11月11日

@yuuki26

yuuki@yuuki26

mathematicians consider infinity a number.

this is bc in the 19c, a man named Cantor found that there are "multiple infinities".
i.e., infinity 1, infinity 2, infinity 3, ….

just like 1, 2, 3, ….
can also do arithmetic with ♾.

posted at 08:13:31

   

11月11日

@yuuki26

yuuki@yuuki26

The largest number is a "contradiction".
let me explain this.

u may be reminded of a googolplex or graham's number.
the problem is: graham's number+1 is larger.

they're still finite.

posted at 07:38:31

   

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