Philosophy Now Forum

Godel's 2nd theorem ends in paradox

http://gamahucherpress.yellowgum.com/bo ... GODEL5.pdf
Godel's 2nd theorem



Godel's 2nd theorem is about

"If an axiomatic system can be proven to be consistent and complete from
within itself, then it is inconsistent.”


But we have a paradox

Gödel is using a mathematical system
his theorem says a system cant be proven consistent


THUS A PARADOX

Godel must prove that a system cannot be proven to be consistent based upon the premise that the logic he uses must be consistent . If the logic he uses is not consistent then he cannot make a proof that is consistent. So he must assume that his logic is consistent so he can make a proof of the impossibility of proving a system to be consistent. But if his proof is true then he has proved that the logic he
uses to make the proof must be consistent, but his proof proves that
this cannot be done
THUS A PARADOX

Copy paste from another thread

I am going to argue this from my own paradigm and say that: not all Mathematics end in contradiction. Only those founded upon set theory.
And so keep in mind that when you say "Mathematics" and when I say "Mathematics" we mean different things. When I say "Mathematics" I actually mean "Computation".

Which is why my mathematical foundations is Lambda calculus expressed as Type theory. https://en.wikipedia.org/wiki/Type_theory
Any Turing-equivalent logic is sufficient so if you are a set-theorist this should make perfect sense to you:

λ-calculus ⇔ λ-calculus ⇔ λ-calculus ⇔ λ-calculus ⇔ λ-calculus ⊇ Type theory ⊇ Mathematics

Equilibrium!!!

https://en.wikipedia.org/wiki/Turing_completeness

That is not entirely true - there is no way to eliminate all contradictions, but some contradictions are far more trivial than others.
Using paraconsistent logic we are able to contain local inconsistencies, which maximizes global consistency. How counter-intuitive is that?

This way that I am assured (by the Curry-Howard isomorphism) that the global system remains functional, even though we may have localized inconsistencies.

The problem with mathematics is systemic and strategic. We are chasing after a consistent system, but we forget that the consistent system is going to be used by inconsistent humans. This is pure idealism and it needs to die. Humans are humans. You aren't going to teach us to think in something as unnatural, unintuitive and rigid as set theory! And when it comes to deal with complexity (where we have to perform millions of calculations) surely you don't expect humans to NOT make any errors in grammar, in semantics, in vocabulary? That's just wishful thinking!
And if ONE contradiction is all it takes for the entire system to come crashing down then that is just setting us up for failure!

Which is fundamentally why I evangelise for paraconsistent logics in the context of computation! That way you can deal with contradictions on case-by-case basis without spending the rest of your life chasing the pipe-dream of global consistency.

Some inconsistency is just fine. In fact - some localized/minor inconsistency/entropy is absolutely necessary for the entire system to remain in tact.
In managing trivial inconsistencies as they arise, we maximize global consistency in the system.
I demonstrate this very thing in this thread: viewtopic.php?f=26&t=26202

I am not going to take credit for this. That is Chomsky's work. I am simply making it tangible and accessible to the average philosopher/mathematician who has no computational background.

https://en.wikipedia.org/wiki/Chomsky_hierarchy

In the formal languages of computer science and linguistics, the Chomsky hierarchy is a containment hierarchy of classes of formal grammars.

I remove the axiom on which all of mathematics supposedly rests and the system does not explode.
Metaphorically speaking: I have removed the foundation and the skyscraper remains standing.
I guess it must be hanging from the sky or something?

Plato would be so proud

Another copy-paste

Every classical logician understands the three laws of thought.

Identity: (A == A) is True
LEM: (A OR not A) is True
LNC: (A AND not A) is False

P1: The laws of Classical logic are the universal laws of thought.
P2: IF we violate the law of identity THEN LNC will be violated also.

Proof that premise P2 is false follows: https://repl.it/repls/StrangeLiquidPolyhedron

Conclusion1. It is not always necessary to adhere to the law of Identity.
Conclusion 2. We can use Para-consistent logic to contradict Classical logic.
Conclusion 3. The classical laws of logic are not universal.