Gödel’s 1st theorem is meaningless

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bas

Jun 2010
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Gödel’s 1st theorem is meaningless

http://gamahucherpress.yellowgum.com/wp-content/uploads/GODEL5.pdf

Gödel’s 1st theorem

a) “Any effectively generated theory capable of expressing elementary arithmetic cannot be both consistent and complete. In particular, for any consistent, effectively generated formal theory that proves certain basic arithmetic truths, there is an arithmetical statement that is true,[1] but not provable in the theory (Kleene 1967, p. 250)


note
"... there is an arithmetical statement that is true..."

In other words, there are true mathematical statements which can't be proven.
But the fact is Gödel can't tell us what makes a mathematical statement true thus his theorem is meaningless.
 
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topsquark

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Oh God, not again.

-Dan
 

skeeter

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The "truth" is out there ...

 
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But the fact is Gödel can't tell us what makes a mathematical statement true thus his theorem is meaningless.
Gödel has a precise definition of what makes a statement true and, if alive, could surely tell you (he already told us... :rolleyes: ) what it is.

I trust that concludes this thread.
 
Oct 2009
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364
Gödel’s 1st theorem is meaningless

http://gamahucherpress.yellowgum.com/wp-content/uploads/GODEL5.pdf

Gödel’s 1st theorem

a) “Any effectively generated theory capable of expressing elementary arithmetic cannot be both consistent and complete. In particular, for any consistent, effectively generated formal theory that proves certain basic arithmetic truths, there is an arithmetical statement that is true,[1] but not provable in the theory (Kleene 1967, p. 250)


note
"... there is an arithmetical statement that is true..."

In other words, there are true mathematical statements which can't be proven.
But the fact is Gödel can't tell us what makes a mathematical statement true thus his theorem is meaningless.
OK, I sincerely hope you are not a crackpot, since I'm going to explain this for you. And I will be very angry if you just turn out to be dismissing my explanations because you are just another crackpot troll.

Suppose we are together in a room. I have a bag with a certain contents (the contents might be empty). Furthermore, you and I know that whatever I say is the complete truth.

So I tell you the following:
1) There is money in that bag, dollar bills (an integer number of dollar bills)
2) There is at least 10€ in that bag
3) There is an even amount of dollars in this bag
4) There is less than 21€ in that bag

These are axioms that I am giving you and I am ensuring you that they are 100% correct, this is called soundness of the axiom system: every axiom is a true statement.

Now you can make certain derivations. You can for example think a while and tell me that you know for sure that the amount of dollar bills is not a prime number. This is something you know for sure from my axioms. If the axioms are true, then this statement MUST be true. This is provability: from axioms we derive true statements. Soundness of the axiom system yields truth of the axioms and hence truth of the proved statements.

However, consider the following statements:
A) There is 10€ in the bag
B) There is 12€ in the bag
C) There is 14€ in the bag
D) There is 16€ in the bag
E) There is 18€ in the bag
F) There is 20€ in the bag

One of these statements must be true. However, ALL of these statements are also not provable. This is incompleteness: I have not given you enough axioms in order to describe the system completely. It is a weakness of the axiom system, of the information given to you.

So you see that provable => true, by soundness. But true =/=> provable.

You can however take (A) as an axiom, and deduce new statements. Taking any of A-F as an axiom means you can derive very different statements. The fun thing is that statement (A) (and the others) are consistent: you can imagine a world where statement (A) is true, it doesn't yield contradictions with the known axioms. Statement
H) There is 44$ in the bag
would yield an inconsistent system.
 
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Denis

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Oct 2011
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This guy's 1st post was Jun.15 2010; title:
"Godels incompleteness theorems are invalid ie illegitimate"

So he's worked over 8 years on it...give him "A" for effort!!
 
Oct 2009
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This guy's 1st post was Jun.15 2010; title:
"Godels incompleteness theorems are invalid ie illegitimate"

So he's worked over 8 years on it...give him "A" for effort!!
Oh damnit, another crackpot...

At least I hope my post makes things clearer for others.
 
Jun 2014
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For the record, I appreciated that explanation even if the OP turns out to be a troll.
 
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