🗑️ 🧵 Untitled Thread
Anonymous at Sat, 7 Sep 2024 12:29:00 UTC No. 16365763
Gödel's incompleteness theorems are two theorems of mathematical logic that are concerned with the limits of provability in formal axiomatic theories.
https://en.wikipedia.org/wiki/G%C3%
Kurt Friedrich Gödel (April 28, 1906 – January 14, 1978) was a logician, mathematician, and philosopher. Considered along with Aristotle and Gottlob Frege to be one of the most significant logicians in history, Gödel profoundly influenced scientific and philosophical thinking in the 20th century (at a time when Bertrand Russell, Alfred North Whitehead, and David Hilbert were using logic and set theory to investigate the foundations of mathematics), building on earlier work by Frege, Richard Dedekind, and Georg Cantor.
https://en.wikipedia.org/wiki/Kurt_
Science is limited. Spirituality is not. Somebody had to tell you that. Science cannot be your God.
Anonymous at Sat, 7 Sep 2024 12:35:19 UTC No. 16365765
I'm yet to see spirituality achieve anything of import.
Anonymous at Sat, 7 Sep 2024 12:38:52 UTC No. 16365769
>>16365765
Yeah. You never look into it, so you're completely ignorant about it, showing your ignorance to everybody else.
Anonymous at Sat, 7 Sep 2024 12:44:21 UTC No. 16365777
>>16365763
Godel also proved that literally everything that is true in a formal system can be proven. So there's that as well.
Anonymous at Sat, 7 Sep 2024 12:50:41 UTC No. 16365786
>>16365763
If I write
>this statement is false
Godelfags assume 'this' refers to the above statement, but there's no reason to make that assumption. If you create a scheme that explicitly forbids a statement referring to itself then there's no issue.
For some reason OP mistakes this reddit tier ambiguity for spirituality.
Anonymous at Sat, 7 Sep 2024 12:58:02 UTC No. 16365793
>>16365786
I thought the point of the incompleteness theorems is that you can construct such a self-referential statement certain systems e.g. arithmatic. I assume if you were to get rid of it's ability to do that then you would also have to get rid of important parts of maths. Counting or something
Anonymous at Sat, 7 Sep 2024 14:37:18 UTC No. 16365914
Math doesn't have a solid foundation in first-order logic. You define a radius by assuming the existence of a circle and assume a circle by using a radius.
https://www.youtube.com/watch?v=mhI
Anonymous at Sat, 7 Sep 2024 14:43:52 UTC No. 16365922
>>16365793
I haven't studied this area in depth but I don't see why self reference is necessary for counting. The statement 'n+1' refers to n, +, and 1, 'n+1' doesn't need to refer to itself.
Anonymous at Sat, 7 Sep 2024 15:03:25 UTC No. 16365960
>>16365922
But the point of the incompleteness theorem is that it proves this. The quotes in the OP about Gödel being one of the most important logicians ever do indicate to me that it makes a lot of sense that you and I wouldn't see that immediately and it was presumably really difficult to prove.
Anonymous at Sat, 7 Sep 2024 15:11:49 UTC No. 16365971
>>16365763
>Science cannot be your God.
Aren't most IFLSers atheists? So they've already negated God and aren't looking for Him? I don't understand how the missionary overtones of your last reddit paragraph are supposed to work with its purported audience.
Anonymous at Sat, 7 Sep 2024 15:12:31 UTC No. 16365972
>>16365763
It just says that some true statements can't be proven under any set of axioms.
Not a big deal, these fringe unprovable statements are either self referential or show a need for stronger/weaker axioms, or are unprovable by construction.
Anonymous at Sat, 7 Sep 2024 15:21:20 UTC No. 16366011
>>16365972
So it's essentially a don't walk off a cliff sign for autists?
Anonymous at Sat, 7 Sep 2024 15:32:27 UTC No. 16366026
>>16365972
>It just says that some true statements can't be proven under any set of axioms.
Wrong. That's not what it says, and the statement you made is false. Every statement, including every true statement, can be proven in some axiomatic system.
>>16365922
>>16365960
The point of Godel's theorem is that any system that allows you to talk about basic arithmetic (i.e. adding and multiplying natural numbers) will necessarily include statements about natural numbers which are true in the standard model of natural numbers yet unprovable.
The statement that Godel comes up with can be INTERPRETED as self-referential weakly speaking, but in a strict sense it's NOT self-referential, because the statement talks only about numbers and the statement itself is not a number, it would be a category error to say that the statement talks about itself. However, there is some interpretation of those numbers. Given a fixed encoding scheme, some numbers are encodings for statements, and in this sense Godels statement talks about numbers which encode statements and one of those numbers that it talks about is the number encoding the statement itself.
This is nothing surprising. I can easily come up with much simpler such "self referential" statements.
For example, I can have an encoding of mathematical expressions by numbers such that
1 encodes the expression "1". Then 1 expression can be viewed as self referential.
Anonymous at Sat, 7 Sep 2024 15:34:11 UTC No. 16366030
>>16365914
>Math doesn't have a solid foundation in first-order logic
Wrong.
> You define a radius by assuming the existence of a circle and assume a circle by using a radius.
You don't define a circle using a radius. You define a circle using the notion of a distance. A circle is simply the set of points equidistant from some fixed point in the euclidean plane. No notion of radius employed in the definition.
Anonymous at Sat, 7 Sep 2024 15:37:52 UTC No. 16366033
>>16365765
Spirituality is the pastime of the brain damaged and the uneducated plebs.
Anonymous at Sat, 7 Sep 2024 15:39:10 UTC No. 16366034
>>16366026
>category theory
I don't hate random people who are sexually dysmorphic because of a social or genetic brain defect, I just have no interest in worshipping them.
Anonymous at Sat, 7 Sep 2024 15:40:18 UTC No. 16366036
>>16366034
Category theory has nothing to do with anything I said. What the fuck
Anonymous at Sat, 7 Sep 2024 15:45:27 UTC No. 16366041
>>16366026
Thank you
Anonymous at Sat, 7 Sep 2024 15:45:28 UTC No. 16366042
>>16366036
I don't hate you or your specialization just because you don't pass.
Anonymous at Sat, 7 Sep 2024 15:57:35 UTC No. 16366056
To add on the idea of "self-referentials", When encoding statements with natural numbers, Godel only talks about provability not it's truth value. Let's analyze the cases:
Statement K : The statement K (encoded by some number) isn't provable in this system
If K is proveable, then it is a true statement by default in a sound and logic, but that would mean that K is a false statement even if it's something that can be proven true. Contradiction. So let's try the other possibiliy, K is not proveable so K is a false statement (in our sound system, for example i can't prove ever that 3=2 under peano axioms, so its false) AND it matches the description of statement K.
Anonymous at Sat, 7 Sep 2024 16:04:41 UTC No. 16366061
>>16366056
>Statement K : The statement K (encoded by some number) isn't provable in this system
>If K is proveable, then it is a true statement by default in a sound and logic, but that would mean that K is a false statement even if it's something that can be proven true. Contradiction.
Did Godel add anything useful to Cantor's ∞^{m,w} argument?
Anonymous at Sat, 7 Sep 2024 16:22:35 UTC No. 16366083
>>16366061
The only thing related to the two of them (other than the fact that system with arithmetic over numbers are incomplete), is that Godel proved that the Continuum Hypothesis ("Any subset of the real numbers is either finite, or countably infinite, or has the cardinality of the real numbers") cannot be proven or disproven using ZFC (the axioms we usually use)
Anonymous at Sat, 7 Sep 2024 16:41:58 UTC No. 16366113
>>16366083
But how is Godel's
>k (encoded by N) not provable
meaningfully different than Cantor's
>mm...
>mw...
>mmw...
>mww...
>mmmw...
Anonymous at Sat, 7 Sep 2024 16:59:53 UTC No. 16366134
>>16366113
If you mean the diagonalization argument, the technique is different but the idea of encoding a number to prove that there's a contradiction somewhere is the same.
Anonymous at Sat, 7 Sep 2024 17:03:32 UTC No. 16366136
>>16366134
Yes, thank you, I'm curious how the technique is different iff it's more layered.
Anonymous at Sat, 7 Sep 2024 17:57:07 UTC No. 16366190
>>16366136
You have a set of symbols associated with a range of numbers (like < is 0 > is 1 = is 2, "for every" = 3, etc...)
then at each position u use the nth prime number and raise it to the number associated to teh symbol at the position of the statements.
since we use prime numbers we can ensure that two different statements aren't represented by the same number. For example let's say 0 is "not", 1 is =, 2 is "the successor of" and 3 is 0.
We can illustrate the statement "0 is not equal to 1" with 0 not = successor of 0 -> 2^3 * 3^0 * 5^1 * 7^2 * 11^3 = 2608760. Now, Godel cleverly makes up a statement that talks about a specific number, and when you encode that statement, you get the number that it was talking about, it's basically solving the equation symbol for number y is x then y = p_1^m_1...p_k^x...p_n^m_n
where x=y. Then you can add as many useless symbols as you want (since it doesn't affect the consistency of the system assuming you have all the basic symbols), until you solve the equation. And now you have a number that encodes a statement about itself. This encoding is only valid is the system has arithmetic and numbers though.
Anonymous at Sat, 7 Sep 2024 17:59:46 UTC No. 16366194
>>16366190
>y is x then y = p_1^m_1...p_k^x...p_n^m_n
How is that different from Cantor is what I'm asking?
Anonymous at Sat, 7 Sep 2024 18:13:12 UTC No. 16366211
>>16365972
>true statements
>can't be proven
Then how do you know it's true? I swear to God, mathfags are schizophrenic. On the one hand they'll tell you how important precision in statements is. Then on the other hand use retarded language games to claim knowledge where no knowledge exists.
Anonymous at Sat, 7 Sep 2024 18:15:19 UTC No. 16366214
>>16366026
>are true in the standard model of natural numbers
> yet unprovable
Then how do you know they're true?
Anonymous at Sat, 7 Sep 2024 18:16:53 UTC No. 16366215
>>16366211
>your shoes are white
>if i piss on your shoes theyll be offwhite
Why attach youself to something retarded?
Anonymous at Sat, 7 Sep 2024 18:32:45 UTC No. 16366249
>>16366214
You know they're true because you prove them to be true, assuming the consistency of the system that you're considering.
For example, assuming PA is consistent, then Con(PA) (= an arithmetic statement expressing the consistency of PA) is true yet unprovable in PA.
Anonymous at Sat, 7 Sep 2024 18:43:00 UTC No. 16366266
>>16365763
His theorem sounds cool to me but I wish I actually understood what the fuck it means. Anyone QRD?
Anonymous at Sat, 7 Sep 2024 18:46:55 UTC No. 16366273
>>16366266
If you understand how Cantor's diagonal theorem works, then unfortunately no poster in this thread can differentiate Godel's theorem from the implicit model of Godel's theorem.
Anonymous at Sat, 7 Sep 2024 18:49:51 UTC No. 16366279
>>16366273
I understand Cantor's theorem because I watched a Vsauce video about it, but the rest of your post went over my head so to speak
Anonymous at Sat, 7 Sep 2024 18:52:41 UTC No. 16366290
The incompleteness theorem was a rather simple application of logic. Gödel's much greater achievement was the Gödel metric which exposed the insufficiency of Einsteins field equations.
Anonymous at Sat, 7 Sep 2024 18:56:37 UTC No. 16366302
>>16366279
>I masturbated to a man fucking a woman therefore I watch hetero porn
Anonymous at Sat, 7 Sep 2024 19:02:32 UTC No. 16366317
>>16365777
and if something is not true wouldn't the absence of a proof that it is true prove that it is not true?
Anonymous at Sat, 7 Sep 2024 19:18:42 UTC No. 16366349
>>16365763
>Science cannot be your God
Neither can something that doesn't exist
Anonymous at Sat, 7 Sep 2024 19:20:18 UTC No. 16366351
>>16365763
Math =/= Science
Anonymous at Sun, 8 Sep 2024 00:51:41 UTC No. 16367027
>>16365763
>modern mathematics is inconsistent
wow amazing revelation.
Raphael at Sun, 8 Sep 2024 01:32:26 UTC No. 16367197
>>16365786
Your high Iq post made me laugh anon
Raphael at Sun, 8 Sep 2024 01:36:50 UTC No. 16367215
>>16365914
First order theory is category error
Midwit spotted
Raphael at Sun, 8 Sep 2024 01:38:50 UTC No. 16367219
>>16365914
I knew it was a fallacy but my vci is low so I wouldn’t know the correct term
>t. 100 FSIQ anti memer
Raphael at Sun, 8 Sep 2024 01:42:03 UTC No. 16367230
>>16365763
Philosophical math is bullshit
Logical operators are based on math category error nigga
Anonymous at Sun, 8 Sep 2024 03:19:00 UTC No. 16367359
>>16366249
>It's unprovable
>You prove it's true
Schizo.
Anonymous at Sun, 8 Sep 2024 03:52:41 UTC No. 16367382
Geometry is self evidently true. Unless you believe shapes dont exist, in which idk you can just kys. Any useful math can be derived from relations between geometry that are self evident.
All models are wrong, some models are useful.
Anonymous at Sun, 8 Sep 2024 05:47:07 UTC No. 16367484
>>16367382
Actually there's weaker assumptions that make geometry arise:
Definition of objects using sets (points, pairs, etc.)
A distance metric (euclidean, L2 norm, projected euclidean distance)
A concept of betweeness and congruence (mainly for lines and angles)
and continuity.
You can define TONS of geometries in non-euclidean spaces, extreme examples would be like space curvature, projected elliptic curves being able to define additions using points in space, etc.
Anonymous at Sun, 8 Sep 2024 06:06:24 UTC No. 16367520
>>16366317
It might be that we just haven't found the proof yet, there is no way to differentiate until there is a definite proof of falsehood or truthfulness.
Anonymous at Sun, 8 Sep 2024 09:28:39 UTC No. 16367731
>>16365793
If i recall correctly, one can construct arithmetic and counting without needing the incompleteness theorem.
Atm the name has eluded me, but it might be related to the work of Stanisław Leśniewski or Boolos and Frege.
Maybe smb else remembers
Anonymous at Sun, 8 Sep 2024 10:14:40 UTC No. 16367786
>>16367359
I should have clarified, by unprovable means unprovable in the system you're considering. For example, the consistency of arithmetic system PA is unprovable in PA but provable in ZFC.
Anonymous at Sun, 8 Sep 2024 10:18:14 UTC No. 16367794
>>16366302
accurate summary
Anonymous at Sun, 8 Sep 2024 20:11:36 UTC No. 16368615
OP is a faggot
OP is in Gödel's cult
OP is a cult member
>I will accept everything the cult leader says on faith
this is a /lit/ or /his/ thread about a charismatic leader's cult
Anonymous at Sun, 8 Sep 2024 20:12:50 UTC No. 16368618
spiritual topics can be discussed on these boards:
x
lit
his
pol
Anonymous at Sun, 8 Sep 2024 20:23:10 UTC No. 16368633
The anti-math garbage Godel spewed turned into a German assault on English language mathematics. For example, some claim that the four color conjecture has been proven, but it's a clown show:
> Appel and Haken's announcement was widely reported by the news media around the world
> However, the unavoidability part of the proof was verified in over 400 pages of microfiche, which had to be checked by hand with the assistance of Haken's daughter Dorothea Blostein.[20]
source: https://en.wikipedia.org/wiki/Four_
The so-called "four color theorem" appears to be an operation that isn't related to mathematics at all. In fact, it's unclear why the press would interfere in mathematics this way unless there were a good reason to do so. For example: a foreign intelligence operation, possibly of German or Russian origin.
This is the work of the sore losers of World War II.
Anonymous at Sun, 8 Sep 2024 20:27:43 UTC No. 16368640
It isn't just a Russian and German problem either, apparently this American Landon Clay decided to try to inject money into math research, but it's a farce of legalistic intrigue, not anything substantive
https://en.wikipedia.org/wiki/Clay_
https://www.claymath.org/wp-content
you can see that these people have created a legal problem for themselves, it's more entertainment
it makes me think that anyone working on such a problem should ignore the CMI with extreme prejudice unless they want to hand you the password to a bank account worth $1,000,000
Anonymous at Sun, 8 Sep 2024 21:42:04 UTC No. 16368757
>>16368633
>>16368640
You have no idea what you're talking about. I hope those are just LLM generated posts.
Anonymous at Sun, 8 Sep 2024 21:54:23 UTC No. 16368783
>>16368757
this is a demoralization reply
it's just an insult with zero contribution
Anonymous at Sun, 8 Sep 2024 22:37:44 UTC No. 16368833
>>16365763
is that Logic?