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And the series continues!
Last week was about the history behind one of the greatest discoveries of all times: Gödel's theorems
Today, I'm going to explain those theorems along with some misconceptions about them
Are you ready? Let's go
https://twitter.com/josejorgexl/status/1350870594565320710
Last week was about the history behind one of the greatest discoveries of all times: Gödel's theorems
Today, I'm going to explain those theorems along with some misconceptions about them
Are you ready? Let's go
https://twitter.com/josejorgexl/status/1350870594565320710
But...
. First things first. Let's talk about some important concepts.
Last week we saw what an axiomatic theory is. Well, there are two properties that you'd like to have if you were an axiomatic theory
Consistency and Completeness
. First things first. Let's talk about some important concepts.Last week we saw what an axiomatic theory is. Well, there are two properties that you'd like to have if you were an axiomatic theory
Consistency and Completeness
ConsistencyA consistent theory is one in which a proposition can be either true or false but not both
In other words, a theory without contradictions
Inconsistent theories are useless because you can prove anything from them... Yeah, *anything*
There's a funny story of Bertrand Russell proving that if 2+2=5 then he was the Pope
There's a funny story of Bertrand Russell proving that if 2+2=5 then he was the Pope
CompletenessA complete theory is one in which all the *true* propositions are provable inside the theory
The doctoral thesis of Gödel was the demonstration of the completeness of the first-order logic (with 23 yo
)
Now we can continue with the history
So, mathematicians were trying to build math on top of other ground different from geometry
They picked number theory (arithmetic, natural numbers) as the new foundations
The main reason: it was axiomatized some years before
So, mathematicians were trying to build math on top of other ground different from geometry
They picked number theory (arithmetic, natural numbers) as the new foundations
The main reason: it was axiomatized some years before
To give you an idea of the magnitude of Gödel discovery I'm going to mention some of the mathematicians trying to rebuild math:
-David Hilbert
-Bertrand Russel
-Ackerman
-John Von Neuman
-David Hilbert
-Bertrand Russel
-Ackerman
-John Von Neuman
They were trying to prove that the number theory was both Consistent and Complete. That way Math would be safe 
The entire Math would be contradictions free and everything could be proven
The entire Math would be contradictions free and everything could be proven
It seemed to be a matter of time before the proof arrived
Actually, some sub-theories of the arithmetic was proven to be both consistent and complete
Gödel himself was working on that but he realized this ( coming)
Actually, some sub-theories of the arithmetic was proven to be both consistent and complete
Gödel himself was working on that but he realized this (
coming)
Theorem 1: About incompletenessFor any axiomatic theory *that includes a certain part of arithmetic*, if it is consistent then, it is incomplete
This means that all theories that include the number theory, contain true propositions that we'll never be able to prove inside that theory
All the work of some of the greatest mathematicians of all times was in vain
Jon Von Neuman never worked again in Logic
All the work of some of the greatest mathematicians of all times was in vain
Jon Von Neuman never worked again in Logic
But for those who have some hope in their hearts 
I remind you that there are two theorems...
I remind you that there are two theorems...
Theorem 2: About consistencyFor any consistent theory *that contains a certain part of arithmetic* the consistency of the theory is not provable
Precisely one of those true but not provable propositions is the consistency of the theory itself!!!!
So, 0 out of 2
No consistency, and no completeness
Math can't be built that way... We have to live with that. There are true propositions out there we'll never prove
No consistency, and no completeness
Math can't be built that way... We have to live with that. There are true propositions out there we'll never prove
End of story
Now, let's talk about some misconceptions generated from the theorems
First I'd like you to note that both theorems say "with a certain amount of arithmetic". We will be talking about that amount next week. For now, just suppose a theory containing the arithmetic
Now, let's talk about some misconceptions generated from the theorems
First I'd like you to note that both theorems say "with a certain amount of arithmetic". We will be talking about that amount next week. For now, just suppose a theory containing the arithmetic
Misconception number one
Gödel said: for any sufficiently complex theory if it is consistent, then it is incomplete
There is this idea of anything more complex than number theory has the conditions to apply Gödel's incompleteness theorem
Real numbers is a complete theory
Gödel said: for any sufficiently complex theory if it is consistent, then it is incomplete
There is this idea of anything more complex than number theory has the conditions to apply Gödel's incompleteness theoremReal numbers is a complete theory
And real numbers are at least as complex as natural numbers
It is not about complexity... It is about how natural numbers are defined. That definition has the "poison"
It is not about complexity... It is about how natural numbers are defined. That definition has the "poison"
Misconception number two
The truth is unreachable for scientists
Ok, some true propositions can't be proven in some theories. But maybe there are other alternative theories. Furthermore, experiments and observations are other methods to discover the truth about our universe
The truth is unreachable for scientists
Ok, some true propositions can't be proven in some theories. But maybe there are other alternative theories. Furthermore, experiments and observations are other methods to discover the truth about our universe
Misconception number three
There is no philosophic system that can explain the universe
The explanation of the universe doesn't have to do with natural numbers necessarily. And Gödel's theorems don't apply when there is no arithmetic in the theory
There is no philosophic system that can explain the universe
The explanation of the universe doesn't have to do with natural numbers necessarily. And Gödel's theorems don't apply when there is no arithmetic in the theory
Of course, there are lots more misconceptions about Gödel's results
But I'll stop here
Next week I'll be talking about the demonstrations of the theorems
Hope you have enjoyed
Feel free to reply, @-me, retweet, or whatever
Stay tuned and don't miss the end of the series!
But I'll stop here
Next week I'll be talking about the demonstrations of the theorems
Hope you have enjoyed
Feel free to reply, @-me, retweet, or whatever
Stay tuned and don't miss the end of the series!
