🧵 Math's incompleteness
Anonymous at Fri, 8 Mar 2024 02:46:55 UTC No. 16062189
So we know that math is
1. Inconsistent
2. Incomplete
3. Indecidable
Based on the brilliant works of Turing and Godel.
However, is it possible to create different formal systems based on different axioms to circumvent some of the holes our current mathematical system has? These different formal systems also fulfill the same 3 constraints, but perhaps they have different holes in them in certain places and can be used to prove problems in other places our current system can't?
Anonymous at Fri, 8 Mar 2024 02:59:57 UTC No. 16062200
We don't know math is inconsistent. We just don't (can't) know if it is consistent if it is consistent.
Anonymous at Fri, 8 Mar 2024 03:07:23 UTC No. 16062208
>>16062189
you misunderstand what incompleteness and unsolvability actually mean. if your symbolic system is powerful enough to encode first order logic and arithmetic then it is incomplete if it is consistent, i.e. there are statements in your symbolic systems which are consistent with the axioms but not derivable from the axioms in a finite number of steps
Anonymous at Fri, 8 Mar 2024 03:35:31 UTC No. 16062241
>>16062208
>different formal systems
I take OP as suggesting to have multiple systems to cover for each other, like how an atlas of multiple charts can be used to map the Earth even while no single chart can.
In which case >>16062189 there's been some work in this direction: assuming that your theories are expressed in classical first-order logic (following Godel), the basic idea is to convert them into coherent theories via Morleyization, and then use the Morita equivalence to pass between them. This isn't my field so I can't say much more about this, but you may find the following to be a useful starting point:
https://www.oliviacaramello.com/Uni
Anonymous at Fri, 8 Mar 2024 03:44:53 UTC No. 16062250
>>16062189
This reminds me of the primal-dual approach in convex analysis. I am certainly not a logician, but I can tell you that from my experience with analysis on convex duals that there's always a trade-off. Nobody gets free lunch. If you're getting more power in one direction it's coming at the cost of something else.
Anonymous at Fri, 8 Mar 2024 03:48:45 UTC No. 16062256
>>16062200
we know its inconsistent, thats why division by 0 isnt allowed.
Anonymous at Fri, 8 Mar 2024 04:10:01 UTC No. 16062277
>>16062189
>However, is it possible to create different formal systems based on different axioms to circumvent some of the holes our current mathematical system has
Yes you can complete a formal system based on any set of axioms you want
Anonymous at Fri, 8 Mar 2024 04:12:13 UTC No. 16062278
>>16062189
just shift the symbols and don't think too much about why
Anonymous at Fri, 8 Mar 2024 04:50:16 UTC No. 16062324
>>16062256
How is that inconsistent? It's consistently not allowed.
Anonymous at Fri, 8 Mar 2024 04:52:00 UTC No. 16062326
>>16062237
>Math doesn't have a firm foundation in first-order logic.
Some math does. It just depends on how many extensions deep you are in what directions.
Anonymous at Fri, 8 Mar 2024 07:07:10 UTC No. 16062450
>>16062324
if its allowed then any number can equal another
if its not allowed then you’re missing an operation for a number.
its either contradictory or inconsistent
Anonymous at Fri, 8 Mar 2024 07:18:24 UTC No. 16062461
>>16062189
Gödel's third theorem: Threads on /sci/ about the incompleteness theorems are always made by the worst retards displaying the most pop soiish misunderstanding of said theorems.
Anonymous at Fri, 8 Mar 2024 07:48:24 UTC No. 16062497
>>16062237
buy an ad
Anonymous at Fri, 8 Mar 2024 07:49:32 UTC No. 16062498
>>16062461
yeah, it's not even sad at his point, i'm numb to it
Anonymous at Fri, 8 Mar 2024 08:34:41 UTC No. 16062529
>>16062450
>if its not allowed then you’re missing an operation for a number.
Nothing says all operations need to have the same numbers of solutions, or solutions at all. sqrt(1000) has 2 solutions for example. arcsin(1000) has no solution.
Anonymous at Fri, 8 Mar 2024 08:46:43 UTC No. 16062537
>>16062189
You don't understand Godel's theorems
now shut the fuck up forever until you've actually fucking read them.