Does Gödel's Incompleteness Theorems means 2+2=5?

[francis20520]

I am a layman trying to understand above theorems. This could be a stupid question. 

Does these theorems imply that we actually cannot prove that 2+2 = 4???

Is this one of the implications of these theorems???

[studiot]

32 minutes ago, francis20520 said:

I am a layman trying to understand above theorems. This could be a stupid question. 

Does these theorems imply that we actually cannot prove that 2+2 = 4???

Is this one of the implications of these theorems???

Hello Francis and welcome.

The short answer is no Godel is not about statements we can prove, like yours, but statements we will miss out because we cannot derive them from our axioms.

There is a really good introductory book by Raymond Smullyan about this called

Forever Undecided

Which is fun to read. I think a pdf may be available.

Meanwhile here, what do you know about axiomatic structures in Mathematics?

 

Edited June 27, 2020 by studiot

[joigus]

No. Who said that? 2+2=4 can be obtained immediately from definitions.

2+2 = 4 is the same as (1+1)+(1+1) = (1+1+1)+1 which is obviously true following the axioms.

The scope of Gödel's theorem is (presumably) about divisibility, number theory, primes. Things like that. Things that are (or may be) out of reach of finite (algorithmic) proofs from the axioms.

x-posted with Studiot. +1

10 minutes ago, studiot said:

The short answer is now Godel is not about statements we can prove, like yours, but statements we will miss out because we cannot derive them from our axioms.

Sorry for overlapping with your answer, @studiot.

Edited June 27, 2020 by joigus
x-post / added adverb "presumably" to account for swansont's (quite correct) comment below

[swansont]

Some true statements can’t be proven, but AFAIK the incompleteness theorem doesn’t say which statements.

[francis20520]

Thanks for your quick responses. 

I don't much about  axiomatic structures in Mathematics. Only things I know is what I read on Wikipedia.

Just out of curiosity, is it PROVED in mathematics (like proving the Pythagoras theorem) that 2 + 2 = 4??

I.e. 0 + 0 = 0. 0 + 1 = 1, 1 + 1 = 2. Are these Axioms in mathematics???

Or is there a proof??? What is it called?

[joigus]

17 minutes ago, francis20520 said:

Thanks for your quick responses. 

I don't much about  axiomatic structures in Mathematics. Only things I know is what I read on Wikipedia.

Just out of curiosity, is it PROVED in mathematics (like proving the Pythagoras theorem) that 2 + 2 = 4??

I.e. 0 + 0 = 0. 0 + 1 = 1, 1 + 1 = 2. Are these Axioms in mathematics???

Or is there a proof??? What is it called?

Good question. +1

1+1=2 is a definition. The axioms only require the existence of 0 and 1.

Were it not for the definitions (symbols, substituters) 2, 3, 4, etc., we would have to write 7 as,

1+1+1+1+1+1+1

If the axioms (associative law for sum) didn't allow for the proof that 

(1+1)+1=1+(1+1), etc.

we would have to distinguish between "these 2 kinds of three": One for the left sum and another for the right.

This actually happens in more general algebraic systems, like groups, octonions, etc.

I hope that helps.

Edited June 27, 2020 by joigus
minor addition

[studiot]

34 minutes ago, francis20520 said:

Thanks for your quick responses. 

I don't much about  axiomatic structures in Mathematics. Only things I know is what I read on Wikipedia.

Just out of curiosity, is it PROVED in mathematics (like proving the Pythagoras theorem) that 2 + 2 = 4??

I.e. 0 + 0 = 0. 0 + 1 = 1, 1 + 1 = 2. Are these Axioms in mathematics???

Or is there a proof??? What is it called?

You seem to have some idea about axioms and seem to understand that axioms are what you start with (accept as true without question).

Using these axioms you can then develop theorems (= very important results) (such as Pythagoras) and lemmas (less important results)

What is not often said is that the axioms must be about something.

These 'somethings' are given in definitions, which usually outnumber the axioms themselves.

In your example (Pythagoras) in Euclidian Geometry there are 5 axioms and 23 definitions.

 

The axioms you need for ordinary arithmetic are known as Peano's axioms

https://mathworld.wolfram.com/PeanosAxioms.html

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

 

Edited June 27, 2020 by studiot

[francis20520]

Just to confirm,  here (https://en.wikipedia.org/wiki/Peano_axioms#Addition ) is (I think) where they show that 2 + 3 = 5, right???

 

 

Edited June 27, 2020 by francis20520

[studiot]

12 hours ago, francis20520 said:

Just to confirm,  here (https://en.wikipedia.org/wiki/Peano_axioms#Addition ) is (I think) where they show that 2 + 3 = 5, right???

 

 

An apology for posting the Wikipedia link, I was in rather a hurry last night.

The article was really too complicated, perhaps the result of a bit of showing off by the authors.
In that respect the Mathworld article is better.

Very often the axioms that are finally chosen (Note Wikipedia offers modern vesions and says that they are not Peano's originals) are a bit obscure and it is the theorems or lemmas we deduce from them that are what we actually want to use.
In fact usually these theorems are properties we have observed and found useful so the axioms are constructed later to produce these theorems as deductiuons.

The point is that in Mathematics to prove something means to show that it is consistent with the axioms.
Since we have already shown that the theorems are consistent when we 'deduced them', this may simply mean showing that the something is consistent with the theorem(s).

Anyway here are a couple of extracts from two books, one very famous, one very modern, that may help.
They tell us about the something (numbers) that lead to the rules of arithmetic (the theorems we want)
Both books are highly recommended.

Unknown Quantity a real and Imagined History of Algebra

John Derbyshire

 

What is Mathematics

Courant and Robbins

 

[ahmet]

On 6/27/2020 at 7:17 PM, francis20520 said:

Thanks for your quick responses. 

I don't much about  axiomatic structures in Mathematics. Only things I know is what I read on Wikipedia.

Just out of curiosity, is it PROVED in mathematics (like proving the Pythagoras theorem) that 2 + 2 = 4??

I.e. 0 + 0 = 0. 0 + 1 = 1, 1 + 1 = 2. Are these Axioms in mathematics???

Or is there a proof??? What is it called?

this is wrong as I know. because of some rules in algebra (general algebra)

in fact you tried to write

0+0= 0(1+1)= 0 .2 if we would simplify both parts then we would find that (0=2)

or equivalently

0*0=0 

0*0=0+0

0*0=0(1+1) 

simplify 0 at both parts. then 0=2 which is wrong. you cannot do this , because of the rules in algebra. 

(that mentioned rule/property is (presumably) this one: for all elemnts in R,Q,Z that are not equal to zero (other than zero), the multiplication of two elements  will never be zero)  

that mathematically said 

∀ x,y ϵ R or Q or Z and x≠0 ,y≠0 --> x.y≠0

 

Edited June 29, 2020 by ahmet

[studiot]

18 minutes ago, ahmet said:

0+0= 0(1+1)= 0 .2 if we would simplify both parts then we would find that (0=2)

When you finish the job, (ie write out the simplification) how is your conclusion reached?

[ahmet]

24 minutes ago, studiot said:

When you finish the job, (ie write out the simplification) how is your conclusion reached?

 

theorem 1)

when <H,+,.> is a circle. a,b ϵ H-{0H} , if a.b=0H   ,then a,b are called zero division of H circle. if tere is no such elements ,then  H is called as it has had no this property.

theorem 2:) H is a circle, and ∀ x,y,z ϵ H , for ∃       x≠ 0_H  ,if this; , x.y=x.z <=> y=z   condition is satisfied then ,we can conclude that simplification exists in H.

theorem 3) if H is a circle, to be able to apply simplification , theorem 1 is sufficient and required  (<=>).

 

thus, you cannot write the simplification in that way because of theorem 1 ,theorem 2 and theorem 3 (check please theorem 2).

Edited June 29, 2020 by ahmet

[Ghideon]

On 6/27/2020 at 6:17 PM, francis20520 said:

I.e. 0 + 0 = 0. 0 + 1 = 1, 1 + 1 = 2. Are these Axioms in mathematics???

@ahmet I interpret the dot "." as end of a sentence. Reformatting the above using separate lines for each sentence:

0+0=0

0+1=1

1+1=2

 That seems to be ok examples to illustrate the question asked? I do not see 0=2. Maybe I miss something.

Edited June 29, 2020 by Ghideon
spelling

[ahmet]

2 minutes ago, Ghideon said:

@ahmet I interpret the dots "." as end of a sentence. Reformatting the above using separate lines for each sentence:

0+0=0

0+1=1

1+1=2

 That seems to be ok examples to illustrate the question asked? I do not see 0=2. Maybe I miss something.

to be honest, I did not understand the equation as it stands correctly so,I wrote my supposition/prediction. sorry if I failing.

and how would we reach 0=1 in that way?

[studiot]

21 minutes ago, ahmet said:

 

theorem 1)

when <H,+,.> is a circle. a,b ϵ H-{0H} , if a.b=0H   ,then a,b are called zero division of H circle. if tere is no such elements ,then  H is called as it has had no this property.

theorem 2:) H is a circle, and ∀ x,y,z ϵ H , for ∃       x≠ 0_H  ,if this; , x.y=x.z <=> y=z   condition is satisfied then ,we can conclude that simplification exists in H.

theorem 3) if H is a circle, to be able to apply simplification , theorem 1 is sufficient and required  (<=>).

 

thus, you cannot write the simplification in that way because of theorem 1 ,theorem 2 and theorem 3 (check please theorem 2).

All I see is your statement 0+0 = 0*2,  which is true since anything times 0 is still 0.

What I don't see is a formal continuation of your line of reasoning to the end.

[ahmet]

ok.it is easy.

0+0+0+0= 0.(0+0+0+0+0)

0(1+1+1+1)=0.(1+1+1+1+1)

simplify 0s.

the n 4=5 ,you can similarly obtain 1=0 etc.

the core reason is effective here.

50 minutes ago, studiot said:

All I see is your statement 0+0 = 0*2,  which is true since anything times 0 is still 0.

What I don't see is a formal continuation of your line of reasoning to the end.

have you understood the theorems ? I meant that you would not be able to write the simplification in that way. 

On 6/27/2020 at 5:40 PM, francis20520 said:

I am a layman trying to understand above theorems. This could be a stupid question. 

Does these theorems imply that we actually cannot prove that 2+2 = 4???

Is this one of the implications of these theorems???

meanwhile,apart from our conversation, while I do not know specifically Gödel's that mentioned teorem, as I know, incompleteness is different subject (i.e. potentially irrelevant)

one of our issues is from Algebra (general algebra) and the other one is presumably from functional analysis. 

of course,they are of intersections  but specifically these two issues seems to me irrelevant. 

Edited June 29, 2020 by ahmet

[ahmet]

@studiot ,I did not check the book , but again I think that you would find the relevant theorems or further considerations in this resource and the continuation (series of ) the resource:

 

https://www.springer.com/gp/book/9783540642435

Edited June 29, 2020 by ahmet

[joigus]

4 hours ago, Ghideon said:

@ahmet I interpret the dot "." as end of a sentence. Reformatting the above using separate lines for each sentence:

0+0=0

0+1=1

1+1=2

 That seems to be ok examples to illustrate the question asked? I do not see 0=2. Maybe I miss something.

Totally agree. +1. Nobody understood division by zero here, except you, @ahmet.

Plus the question,

On 6/27/2020 at 4:40 PM, francis20520 said:

Does [Gödel's theorem] imply that we actually cannot prove that 2+2 = 4???

Is this one of the implications of these theorems???

has been satisfactorily answered, I think. Unless the OP has any further question.

To me, end of story.

Edit: Unless you have any further comments on how Gödel's theorem could imply that we can prove 2+2=5, in which case, I unrest my case.

Edit 2: It seems the OP made a mistake in the title. They meant, I think, ¿Can't we prove (from Gödel's theorem) that 2+2=4?

Edited June 29, 2020 by joigus
Addition

[ahmet]

2 minutes ago, joigus said:

Totally agree. +1. Nobody understood division by zero here, except you, @ahmet.

 

I think that you were NOT a mathematician. 

[joigus]

3 minutes ago, ahmet said:

I think that you were NOT a mathematician. 

I'm well educated enough to withhold my opinion of what you (or any other member of this forum) are or are not.

I will always concentrate on the arguments and document them properly wherever necessary.

I suggest you do the same.

[ahmet]

1 minute ago, joigus said:

I'm well educated enough to withhold my opinion of what you (or any other member of this forum) are or are not

then,I can say that being well educated will not bring you a guarantee to know everything   

anyway,as I see that your comments are going to be off topic,maybe I had better go until seeing a relevant comment. 

[joigus]

16 minutes ago, ahmet said:

then,I can say that being well educated will not bring you a guarantee to know everything   

Have you seen my profile?:

Quote

I was born, then I started learning. I'm still learning.

Who says I say I know everything? It's you who seems to think that people are saying things they're not really saying. Read whatever people say and then say whatever you have to say. Enough said.

Edited June 29, 2020 by joigus
minor correction

[ahmet]

1 hour ago, joigus said:

Totally agree. +1. Nobody understood division by zero here, except you, @ahmet.

 

check please this resource [1] ,in fact it is same with the above. 

 

[1]     N. BOURBAKI Elements of Mathematics Algebra I Chapters 1 - 3 ISBN 2-7056-5675-8 (Hermann) ISBN 0-201-00639-1 (Addison-Wesley) Library of Congress catalog card number LC  72- 5558 American Mathematical Society (MOS) Subject Classification Scheme (1970) : 15-A03, 15-A69, 15-A75, 15-A78 Printed in Great Britain page: 96-99 

Edited June 29, 2020 by ahmet

[studiot]

1 hour ago, ahmet said:

@studiot ,I did not check the book , but again I think that you would find the relevant theorems or further considerations in this resource and the continuation (series of ) the resource:

Thank you for offering this however I think Bourbaki is way ouside the OP comfort zone.

I was trying to interpret your response in the light of simpler mathematics.

Of course there, division by zero is forbidden.

On 6/27/2020 at 3:40 PM, francis20520 said:

I am a layman trying to understand above theorems.

 

However

5 hours ago, ahmet said:

to be honest, I did not understand the equation as it stands correctly so,I wrote my supposition/prediction. sorry if I failing.

 

I think your response to Ghideon is the most appropriate since I also agree with Ghideon, on the understanding that the OP has a full stop in place of a comma

I admit to being confused by your introduction of fancy mathematical notation, most especially the H notation, as used for Octonions.

As far as I am aware the octonion ring is not a division ring with the necessary structure to include division by zero.
The only possibly suitable one I can think of is a 'wheel' which has three defined operations, not two.

Anyway as far as I know the OP is satisfied with the information.

[ahmet]

@studiot check please once again my previous post. (with the stated/given page infromation please, because it seems  somebody who claims that he was well educated but not aware of 0 divisors of a circle. )

17 minutes ago, studiot said:

Thank you for offering this however I think Bourbaki is way ouside the OP comfort zone.

 

I disagree to this idea. because our keywords seems suitable: (* incompleteness)

however,

17 minutes ago, studiot said:

I was trying to interpret your response in the light of simpler mathematics.

 

I have commonly experienced in mathematics, something (which seems even very  simple ),can cause big discussions. 

so, I do not recommend thinking like this : "this is so much simple,I can easily resolve it" ,

Nah   

Edited June 29, 2020 by ahmet