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Different sets of axioms open the doors to different mathematical universes? I'm given to understand that's the general concensus on what Gödel's incompletenes

Different sets of axioms open the doors to different mathematical universes? I'm given to understand that's the general concensus on what Gödel's incompleteness theorems mean, practically speaking.

datameta 3y ago
Can these universes be mutual subsets? I'm grappling with the incompleteness theorems as they pertain to statements that are not contained in the set of possibilities which they describe, as in Gödel Escher Bach when discussing ideas foundational to Hofstadter's strange loop concept. The self-descriptive nature of information or lack thereof, essentially the existence of recursion without exclusion of the seed itself is stretching my mind. I think I need to start G.E.D. from the beginning (i.e. re-read the 40 pages I made it through).
- mtreis86 3y ago
  Keep reading it, page 1 really isn't the start. Each thread starts in a different place and we are introduced to some of them in the middle of their story.
- nyssos 3y ago
  If you want to understand the incompleteness theorems you need an introductory model theory textbook. GEB doesn't really have anything to do with it.
  - markisus 3y ago
    I recall that GEB gives an informal proof that systems that can formalize arithmetic are incomplete. But a standard textbook would certainly give a terser and more rigorous proof.
a_wild_dandan 3y ago
Gödel tells us that every formal system contains contradictions and unprovable statements (save for very simple systems). I’ve never heard a summary remotely akin to yours, so I’m unsure if it’s equivalent.
- greiskul 3y ago
  > contains contradictions It definitely does not say that. Only that we cannot make a proof of consistency of them. It is possible, and believed, that the Zermelo–Fraenkel axioms we use for set theory for example are consistent. But we cannot prove that because of Godel.
  - markisus 3y ago
    I think the statement may be corrected by replacing the “and” with “or”. Does it work to say that “Every sufficiently powerful formal system is incomplete or has a contradiction.”?
- sgt101 3y ago
  I was introduced to incompleteness in the context of Turing Machines and the halting problem in a CS course. I think that if you are introduced to it as part of a foundations of mathematics course then the focus is on what it implies about mathematical reasoning. As ever if you can find the right bit of Wikipedia then there's a fascinating explanation clearly written by someone who really knows about it that I don't understand but hints at worlds of thought that I will spend time exploring when I don't have to get this fucking demo to work or deal with my boss. So maybe never... https://en.wikipedia.org/wiki/Foundations_of_mathematics#Philosophical_consequences_of_G%C3%B6del's_completeness_theorem https://en.wikipedia.org/wiki/Foundations_of_mathematics#Phi...
cjohnson318 3y ago
Not really. Goedel's incompleteness theorem says that no consistent set of axioms can prove all true statements. Moreover, once you take a few unprovable, but true, things and add them to you axioms, there will still be things that are true that you cannot prove. This means that you're not going to find two mutually exclusive sets of axioms that, between the two of them, can prove any and all true things. This suggests to me that there's not a Library of Babel of mathematical universes, it's more or less one big universe, with a lot of black holes.
- cjohnson318 3y ago
  Not sure why this was downvoted. If I said something inaccurate or misunderstood Goedel's incompleteness theorem, then I'd like to know. It's something that's fascinated me for decades.

Different sets of axioms open the doors to different mathematical universes? I'm given to understand that's the general concensus on what Gödel's incompleteness theorems mean, practically speaking.

datameta 3y ago

Can these universes be mutual subsets? I'm grappling with the incompleteness theorems as they pertain to statements that are not contained in the set of possibilities which they describe, as in Gödel Escher Bach when discussing ideas foundational to Hofstadter's strange loop concept. The self-descriptive nature of information or lack thereof, essentially the existence of recursion without exclusion of the seed itself is stretching my mind. I think I need to start G.E.D. from the beginning (i.e. re-read the 40 pages I made it through).

mtreis86 3y ago
Keep reading it, page 1 really isn't the start. Each thread starts in a different place and we are introduced to some of them in the middle of their story.
nyssos 3y ago
If you want to understand the incompleteness theorems you need an introductory model theory textbook. GEB doesn't really have anything to do with it.
- markisus 3y ago
  I recall that GEB gives an informal proof that systems that can formalize arithmetic are incomplete. But a standard textbook would certainly give a terser and more rigorous proof.

a_wild_dandan 3y ago

Gödel tells us that every formal system contains contradictions and unprovable statements (save for very simple systems). I’ve never heard a summary remotely akin to yours, so I’m unsure if it’s equivalent.

greiskul 3y ago
> contains contradictions It definitely does not say that. Only that we cannot make a proof of consistency of them. It is possible, and believed, that the Zermelo–Fraenkel axioms we use for set theory for example are consistent. But we cannot prove that because of Godel.
- markisus 3y ago
  I think the statement may be corrected by replacing the “and” with “or”. Does it work to say that “Every sufficiently powerful formal system is incomplete or has a contradiction.”?
sgt101 3y ago
I was introduced to incompleteness in the context of Turing Machines and the halting problem in a CS course. I think that if you are introduced to it as part of a foundations of mathematics course then the focus is on what it implies about mathematical reasoning. As ever if you can find the right bit of Wikipedia then there's a fascinating explanation clearly written by someone who really knows about it that I don't understand but hints at worlds of thought that I will spend time exploring when I don't have to get this fucking demo to work or deal with my boss. So maybe never... https://en.wikipedia.org/wiki/Foundations_of_mathematics#Philosophical_consequences_of_G%C3%B6del's_completeness_theorem https://en.wikipedia.org/wiki/Foundations_of_mathematics#Phi...

cjohnson318 3y ago

Not really. Goedel's incompleteness theorem says that no consistent set of axioms can prove all true statements. Moreover, once you take a few unprovable, but true, things and add them to you axioms, there will still be things that are true that you cannot prove. This means that you're not going to find two mutually exclusive sets of axioms that, between the two of them, can prove any and all true things. This suggests to me that there's not a Library of Babel of mathematical universes, it's more or less one big universe, with a lot of black holes.

cjohnson318 3y ago
Not sure why this was downvoted. If I said something inaccurate or misunderstood Goedel's incompleteness theorem, then I'd like to know. It's something that's fascinated me for decades.