CURED - 2017-08-22

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Martin Sleziak

04:41

@amWhy You wrote a comment like this under the above post:

> All those downvotes, can we get more delete votes here, please? It is SPAM.

In fact, whenever somebody flags as spam or rude/abusive, automatically also a downvote (-1) is added. So the negative score might have been consequence of flags.

It seems that the post has been deleted relatively quickly - metasmoke says: "deleted 10 minutes after being reported".

Martin Sleziak

04:54

Also I think it's better not to edit spam posts, so that others can see clearly that it is spam. However, I've seen this recommendation usually with posts which edited out the offending link - I guess your edit was ok (although probably not really necessary).

7 hours later…

user84215

11:56

Why is the following question going to be deleted?:

user84215

-4

Q: Undecidable sentence in Godel's incompleteness theorem

Please assume that this is a question of the one who knows little mathematics (a bit higher than high school mathematics) As far as I know, in the proof of the Godel's incompleteness theorem the undecidable sentence is $g: \text{I am not provable}$. So there exists a true sentence in every arith...

user21820

14:58

@amWhy I'm not sure what to do with it. Sure the asker shouldn't be putting 4 questions into 1 without giving much effort, but I don't really know what the question is asking so I can't judge what is reasonable effort to expect.

@amWhy Post deleted so message unstarred.

@SimplyBeautifulArt Deleted by asker, so do you want to unstar or keep for now?

user21820

15:14

@MathematicsAminPhysics You'd have to ask the people who voted to delete it. I didn't vote to delete so I can't give a reliable explanation of their thinking to you.

Since you only have a bit more than high-school mathematical background, I'd say that you need to spend a significant amount of time and effort to learn logic properly before you can understand the import of the incompleteness theorems. However, there is one thing I can tell you; many results hold only because of the incompleteness theorem. Carl Mummert has already given you a good example, namely that no finite collection of axioms can substitute for the induction schema in PA.

Basically PA can prove the consistency of any finite collection of its axioms, and hence that finite collection cannot suffice to generate all theorems of PA otherwise it would violate the incompleteness theorem. So the Godel sentence is not at all a useless sentence of arithmetic.

Also, the whole point of incompleteness is that you can never has a complete consistent implementable axiomatization of the natural numbers, because any such theory cannot prove its Godel sentence. The exact symbolic content of the sentence itself is not so important, but its mere existence has far-reaching consequences.

Finally, "there exists a true sentence in every arithmetic theory (formal axiomatic system containing basic arithmetic) that cannot be proved" is false. Truth is only defined with respect to a world/structure. See the following posts:

A: How could a statement be true without proof?

Your confusing stems from the way many articles about Godel's incompleteness theorems are extremely imprecise. Here is a proper definition. $\def\nn{\mathbb{N}}$ We say that a sentence $φ$ over a language $L$ is true in an $L$-structure $M$ iff $M \vDash φ$. For convenience, when $L$ is ...

A: Does a $\Pi_2^0$ sentence becomes equivalent to a $\Pi_1^0$ sentence after it has been proven?

The problem is with your notion of logic and truth. Before you even can talk about a sentence, you need to specify your language, and before you can talk about whether it is provable or disprovable (or neither), you need to specify the formal system (which must use the same language or larger), a...

@MathematicsAminPhysics: If you wish to clarify any point about what I've just written about the incompleteness theorems, you are welcome to ask in the logic chat-room. But please stick to mathematical logic, thank you.

Paramanand Singh

15:36

Hello @user21820. It was nice to hear about Godel incompleteness theorem and the wonderful argument about its utility even in this forum.

user21820

@ParamanandSingh: Haha I just thought it might be worth stating brief points here to address the question about which he asked why it is closed and getting deleted. =)

Since of course I can't answer a closed question.

→ 4 messages moved to This is the Realm of Simply Beautiful Art

Simply Beautiful Art

16:13

@user21820 Sure

user21820

16:25

This post is ready for deletion.

Simply Beautiful Art

@user21820 @amWhy PSQ that is also not a well-defined problem

amWhy

17:18

@MartinSleziak Thank you;

@MartinSleziak Thank you.

@user21820 You don't need to justify any reluctance to act. But, if you read the image of the assigned problems, you'll surely see 4 questions. That usually counts as "too broad." Any way, the post is closed,

Simply Beautiful Art

@amWhy Open for deletion

amWhy

Already have voted :P Only one more delete vote needed. (Also added my vote to close the link you've given before the last.

Simply Beautiful Art

17:36

@amWhy :P

amWhy

@SimplyBeautifulArt Open for deletion: math.stackexchange.com/questions/2401206/…

Oops, @SimplyBeautifulArt I meant the question asked, not the answer.

4 hours later…

amWhy

21:55

@SimplyBeautifulArt, @user21820, @Did, @ParamanandSingh, @anyone interested: This question is surely a typo'd question, whether in source, or by asker. It cannot be answered, currently. I think the best route is to close as unclear.

amWhy

22:28

Scratch my last request (immediately above this), as it seems there was, or must be, a typo or error in the source of the question.

amWhy

23:22

Close votes? math.stackexchange.com/questions/2402766/…, @Did, @SimplyBeautifulArt, @user21820, @ParamanandSingh ?

amWhy

23:37

Problem statement(s): Three sentences, three questions, absolutely nothing else: math.stackexchange.com/questions/2402923/…

Simply Beautiful Art

Mhm

amWhy