/lit/ - Literature

>>18043673
its not about whether mathematicians will ever come across this obscure type of propostion within their work.
As i understand it, its that you cant create a system of mathematics which is both complete and consistent, which was a big question around the start of the 20th century when people the foundations of mathematics where a bit shakey. So to answer your question the specifics of this encoding of a mathematically analagous statement of contradiction is probably not something mathematicians care about but the results are very important.

If it wasnt already obvious im not a mathematician, never finished that book and could be very off point

>incompleteness only arises because of the unprovability
IIRC, it doesn't only arise in that case, what's special about it is that you can prove that it forces you into a system that is either inconsistent or incomplete, and that situation will happen in every system of sufficient complexity. It's like a proof by contradiction, who knows how many other sentences are not provable, the point is that this one is not, unless you allow inconsistency

>>18043673
>When will mathematicians ever care about those types of propositions?
a physicist or engineer wouldn't worry about such things, but this is the exact kind of shit mathfags sperg out about.
math isn't about flexing the human mind's capacity for symbolic manipulation, one old bloke said something like
>God can turn water into wine, heal the lame, and cure the blind, but God cannot make a triangle whose interior angles do not add up to 180
through understanding mathematics, one is able to pull the curtains back beyond man and beyond god, to touch deeply the most sublime faculties of existence. To know, then, that we are fundamentally unable to attain a mathematical understanding that is entirely whole and cohesive, is the cosmic equivalent of knowing we're doomed to 'chase the dragon' forever

>>18043673
Because indirect self reference is also possible.

The idea is something like:
1. neurons are logic circuits
2. they make a bunch of self-referential propositions
3. some self-referential propositions are true but unprovable
4. consciousness is a true but unprovable self-referential proposition

Conscioussnes magically emerges from any complex enough self-referring system, but we prove it in the system becase Gödel.

>>18043673

From what I understand, the self referential statements, were used to prove that consistent systems were necessarily incomplete. That is there were statements that were true within a given mathematical system that could not be arrived at by that system. Further, my understanding is that if mathematicians could prove that consistent systems are only incomplete with respect to a particular class of constructed self-referential statements, this wouldn't be an issue. As they could say there are no true meaningful statements that cannot be proved within a consistent system. However, I don't believe such a thing has been proven. Also not a mathematician.

>>18044003
>>18044070
these are correct
this means that there are possibly truths in any given logical system which can never be proven. godel's incompleteness threorems proof are not about showing incompleteness arising only as a result of some specific self-referential proposition as much as they are showing that incompleteness is endemic to all systems. which is obviously a thing that mathematicians (and really any logician) would care about.

>>18044060
>t. Jordan Peterson

>>18044106

>>18044121
>Proof is impossible without an axiom (as Godel proved)
nigga what

>>18044037
>God cannot make a triangle whose interior angles do not add up to 180
With God all things are possible. Matthew 19:26

>>18044060
>but we prove it in the system becase Gödel
*but we CAN'T prove it in the (language of the) system
>>18044106
>>18044121
I don't see how this relates to my post.

>>18044281
You don't need a god for that. It is possible make a triangle whose interior angles do not add up to 180.

>>18044314
But God is in us all. Everything you do, you do with God.

>>18044325
creepy

>>18043673
>When will mathematicians ever care about those types of propositions?
Hofstadter uses those verbal gimmicks because it's easier to demonstrate to the layman the concept of self-reference. This verbal sentences just happen to share the one same property with the math.

In 1900 mathematician David Hilbert proposed an agenda for the future of mathematics at the International Congress of Mathematicians in Paris. Of the 23 problems Hilbert listed, the 2nd was to prove whether the axioms of arithmetic are consistent.
This is a vital question for the whole of mathematics because arithmetic is central to the entire edifice. Godel basically proved that any axiomatic system of arithmetic can't be both complete and consistent, which means there are statements that are true but unprovable through the application of its rules. So there is no way to have a perfect mathematical system that can give you all the truths and cover all the bases.

This was shocking because many took it to mean that if arithmetic is incomplete, then all of mathematics must be as well. The self-reference comes in with the fact that the system references its own axioms. A different set of axioms can prove statements which are unprovable in the first, and vice versa, but there is no single universal frame of reference or axiomatic superset that contains them all. Conjoining the axioms of one system with the other might cause them to logically contradict. This isn't good because mathematicians fantasize about math being a beautifully logically coherent and seamless enterprise without knicks and bumps.

>>18044060
Yeah Terrence Deacon uses cognitive neuroscience and semiotics to come to the same conclusion. Penrose came to the same conclusion as well. Why are regular philosophers of mind falling so behind? Scientists are doing their job better than them.
>>18044070
This is the best response in the thread, thanks. There are many true meaningful statements independent of ZFC. This still leaves a possibility for a system with every meaningful proposition proveable, but like you said there doesn't seem to be evidence in favor of it. It hasn't been proven to be impossible, either.

There are also unprovable (within the system) propositions that one actually cares about. But the proof needs to work for an arbitrary consistent system (that is strong enough). So, how could you come up with an actually interesting statement in any such system? That's clearly very hard if not impossible.

What does completeness mean? It just says that for any proposition that can be formulated in the system it is either provable within the system or its negation is provable within the system. To disprove completeness one only needs to find ONE proposition that can be formulated but not proven or disproven in the system. The question whether this proposition is interesting does not matter for the proof.

Gödel chose this type of sentences because the proof works with them and that is enough. There are basically the easiest examples that one could come up with.

Also, most mathematicians don't really care about this. They may think it's somehow neat but it doesn't impede their daily work. The only ones that might care about it are people that spezialize in set theory or logic both of which are regarded as quite obscure on a non-elementary level by most mathematicians

>>18043673

In any field of human endeavor, amoing people who care, it is a very natural thing to try to do it as well as you can. Whitehead and Russell had the broader idea of doing math from the ground up, completely, in a certain sense. Godel showed that this particular approach couldn't work, that's all.

>>18043673
>GEEG
what did he mean by this?