Hacker News Board Hacker News, browsed like a classic message board
Live data from Hacker News

Kurt Gödel and the romance of logic

prospectmagazine.co.uk

Post reply on HN
Print view View original

31–40 of 52 posts

Re: Kurt Gödel and the romance of logic

#31

Post by mietek »

antidesitter wrote: post #23
whatshisface wrote: post #19

Earlier quoted context omitted.

How would you be able to take every true statement as an axiom? Without proving anything I don't see how you could identify any statements as true.

The point is that the set of all true statements about arithmetic [1] exists “out there” (and thus as a “theory” in a very general sense) even if we can’t identify it. [1] https://en.wikipedia.org/wiki/True_arithmetic

You are invoking a classical notion of existence.

This is not acceptable to a constructivist, for whom “a statement is true if we have a proof of it, and false if we can show that the assumption that there is a proof for the statement leads to a contradiction”[1]. The parent poster, whatshisface, may be a constructivist.

[1] Troelstra A., D. van Dalen (1988) “Constructivism in mathematics: an introduction”

Re: Kurt Gödel and the romance of logic

#32

Post by mietek »

ginnungagap wrote: post #27
whatshisface wrote: post #19

Earlier quoted context omitted.

How would you be able to take every true statement as an axiom? Without proving anything I don't see how you could identify any statements as true.

Whenever you have a structure M of some language L you can take the so called complete theory of the structure, denoted Th(M), which is just the set of all L-sentences true in M. In particular if L is the language of PA and M are the standard natural number with the standard operations Th(M) is a theory called true arithmetic. This theory is complete (that's because Th(M) is always complete) and clearly enough to tal…

You seem to be forgetting that constructive truth means provability.

Also, it doesn’t seem right to use the informal “take” when the object in question is not computable.

Re: Kurt Gödel and the romance of logic

#33

Post by ginnungagap »

mietek wrote: post #32
ginnungagap wrote: post #27

Earlier quoted context omitted.

Whenever you have a structure M of some language L you can take the so called complete theory of the structure, denoted Th(M), which is just the set of all L-sentences true in M. In particular if L is the language of PA and M are the standard natural number with the standard operations Th(M) is a theory called true arithmetic. This theory is complete (that's because Th(M) is always complete) and clearly enough to tal…

You seem to be forgetting that constructive truth means provability. Also, it doesn’t seem right to use the informal “take” when the object in question is not computable.

I specified I'm working in standard first order logic. And I'm not sure what do you mean with the informal take.

Also intuitionist logic is not my area, but isn't it divided in inference rules for provability and (Heyting or Kripke) semantic for model theory and truth just like FOL?

Re: Kurt Gödel and the romance of logic

#34

Post by infinity0 »

skh wrote: post #12
infinity0 wrote: post #5

> rescued the idea that there are truths that humans can never prove This is a gross misinterpretation of Gödel's actual theorem that helps perpetuate irrational superstitious attitudes against science, mathematics, and logic. What Gödel showed was that proofs are relative to some underlying axiomatic model and that for any particular axiomatic model there are always truths unprovable by it. That doesn't mean "there…

You are using the word “truth” but it is more correct to use provable/non-provable. What Godel showed is that -limiting the discussion to the natural numbers for simplicity - there are statements that are true in the standard model of the natural numbers that are not provable in the first order Peano Axiomatic system for the natural numbers. What this means is that such a statement will be false in some non-standard…

> You are using the word “truth” but it is more correct to use provable/non-provable.

I used both the words "truth" and "provable" in the correct and appropriate ways. Both are distinct concepts that form an important part of the theory.

Re: Kurt Gödel and the romance of logic

#35

Post by DyslexicAtheist »

> The theoretical physicist and mathematician Roger Penrose, for example, has argued that Gödel’s theorem shows that “Strong AI” is false: our minds cannot be computers, and that by extension the intelligence of computers will never fully replicate them.

Note that there are camps that have been disputing this (e.g. McCullough’s Objection):

http://www.deepideas.net/godels-incompleteness-theorem-and-i...

and

https://www.iep.utm.edu/lp-argue/#H3

sadly, I'm not even close to figuring out if the Roger-Penrose argument is valid. Nada, not even a gut-feeling!

Re: Kurt Gödel and the romance of logic

#36

Post by DyslexicAtheist »

craigr1972 wrote: post #2

" He announced that he had studied the US constitution in detail, and—no doubt, forensically examining its propositions one at a time and perhaps testing it against thought experiments against wild possible futures in which the president was allowed to get out of control—he had discovered how the US could legally be turned into a dictatorship."

sadly this article skips a lot of detail on what happened during and before the hearing and how Einstein tried to coach him. The New Yorker had a much better summary on this. Here in all its hilarity:

----

from https://www.newyorker.com/magazine/2005/02/28/time-bandits-2

So naïve and otherworldly was the great logician that Einstein felt obliged to help look after the practical aspects of his life. One much retailed story concerns Gödel’s decision after the war to become an American citizen. The character witnesses at his hearing were to be Einstein and Oskar Morgenstern, one of the founders of game theory. Gödel took the matter of citizenship with great solemnity, preparing for the exam by making a close study of the United States Constitution. On the eve of the hearing, he called Morgenstern in an agitated state, saying he had found an “inconsistency” in the Constitution, one that could allow a dictatorship to arise. Morgenstern was amused, but he realized that Gödel was serious and urged him not to mention it to the judge, fearing that it would jeopardize Gödel’s citizenship bid. On the short drive to Trenton the next day, with Morgenstern serving as chauffeur, Einstein tried to distract Gödel with jokes. When they arrived at the courthouse, the judge was impressed by Gödel’s eminent witnesses, and he invited the trio into his chambers. After some small talk, he said to Gödel, “Up to now you have held German citizenship.”

No, Gödel corrected, Austrian.

“In any case, it was under an evil dictatorship,” the judge continued. “Fortunately that’s not possible in America.”

“On the contrary, I can prove it is possible!” Gödel exclaimed, and he began describing the constitutional loophole he had descried. But the judge told the examinee that “he needn’t go into that,” and Einstein and Morgenstern succeeded in quieting him down. A few months later, Gödel took his oath of citizenship.

----

Another one: https://www.quickanddirtytips.com/education/science/when-g-d...

---

EDIT: something I also missed in this piece was that Gödel developed rheumatic fevers as a child and started reading medical books with the age of 8 to learn more about the condition. He concluded that he had a weak heart :D

Gödel is really worth studying closer and this article just leaves out a lot.

Re: Kurt Gödel and the romance of logic

#38

Post by skh »

infinity0 wrote: post #34
skh wrote: post #12

Earlier quoted context omitted.

You are using the word “truth” but it is more correct to use provable/non-provable. What Godel showed is that -limiting the discussion to the natural numbers for simplicity - there are statements that are true in the standard model of the natural numbers that are not provable in the first order Peano Axiomatic system for the natural numbers. What this means is that such a statement will be false in some non-standard…

> You are using the word “truth” but it is more correct to use provable/non-provable. I used both the words "truth" and "provable" in the correct and appropriate ways. Both are distinct concepts that form an important part of the theory.

I’m not used to seeing “axiomatic model”. When I read that I thought you meant axiomatic system and not model. Sorry.

Re: Kurt Gödel and the romance of logic

#39

Post by skh »

mietek wrote: post #32
ginnungagap wrote: post #27

Earlier quoted context omitted.

Whenever you have a structure M of some language L you can take the so called complete theory of the structure, denoted Th(M), which is just the set of all L-sentences true in M. In particular if L is the language of PA and M are the standard natural number with the standard operations Th(M) is a theory called true arithmetic. This theory is complete (that's because Th(M) is always complete) and clearly enough to tal…

You seem to be forgetting that constructive truth means provability. Also, it doesn’t seem right to use the informal “take” when the object in question is not computable.

Informally speaking we can take the set of all real numbers. Assuming we are using standard mathematics that most working mathematicians use then this includes non-computable numbers. Sets don’t have to be computable to be used unless you are someone who works with non-standard models of set theory.

Re: Kurt Gödel and the romance of logic

#40

Post by chewz »

neokantian wrote: post #25

The biggest qualm that I have with the example for the incompleteness theorem, is the use of two-valued, Boolean/Aristotelian logic, in which "not true" automatically becomes "false". If you allow the use of a three-valued logic, for example, with (true, undetermined, false), the Gödel statement, “I am not provable" amounts to saying "My provability is false or undetermined". The same problem occurs in Russell's para…

You don't grasp. Gödel's incompleteness theorems is not dependent on chosen logic or any set of axioms. It stands in 3-, 4- and N-value logic as well.
Post reply on HN