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

[joigus]

26 minutes ago, ahmet said:

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 

Bourbaki, really? The OP has a simple enough question about Gödel's theorem and simple relations between real numbers and your suggestion is a treatise that proposes to redo mathematics from scratch by a group of (brilliant) mathematicians that proposed to change the whole structure of maths under a pseudonym?

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

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???

This is the line that you misunderstood, split into independent lines:

0+0=0

0+1=1

1+1=2

You understood:

6 hours ago, ahmet said:

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

Incorrect interpretation of OP's question. Honest mistake, so far. But then,

You bring up division by zero, which is irrelevant, as it was not implied by the OP.

Then you bring up my level of knowledge. Not that I care. I don't.

Then you bring up Bourbaki. I happen to know Bourbaki and I coincide with @studiot that it's nothing to do with the OP, nor does it have any bearing on the question, nor is it advisable in order to answer it, among other things, on account of what the OP said, very clearly,

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

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

All of this when the question had already been answered to the satisfaction of the OP, as I understand. I wonder, what's next? Algebraic topology? Apparently there's no limit to how far off-topic you're willing to go in order to bring more confusion to the discussion or not to recognize that you misunderstood the initial question.

Now, I suggest you ask this yourself: Am I really helping here?

[ahmet]

18 minutes ago, joigus said:

 

This is the line that you misunderstood, split into independent lines:

0+0=0

0+1=1

1+1=2

 

then how will you reach the title's implication: 2+2=5 with your this map??

23 minutes ago, joigus said:

The OP has a simple enough question about Gödel's theorem and simple relations between real numbers

I think this is your own supposition. check also simply wikipedia here with a part of this original title here: https://en.wikipedia.org/wiki/G%C3%B6del%27s_incompleteness_theorems#:~:text=G%C3%B6del's%20incompleteness%20theorems%20are%20two,capable%20of%20modelling%20basic%20arithmetic.&text=The%20second%20incompleteness%20theorem%2C%20an,cannot%20demonstrate%20its%20own%20consistency.

25 minutes ago, joigus said:

All of this when the question had already been answered to the satisfaction of the OP, as I understand.

this is your (own) understanding.  

28 minutes ago, joigus said:

Algebraic topology?

if the question was relevant,then no problem for me.

I recommend that you check the ordered/given references and try to correlate them well. but of course, this requires mathematical sight. I marked as OP's own reference in paranthesis.All are relevant each other and in the conformity.

 

[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 

[2]  https://en.wikipedia.org/wiki/Peano_axioms#Addition  (this is the reference that OP provided. please check carefully the axioms here such as "Equivalent axiomatizations"  check also please the note What you see here,in the OP's this link/reference)

[3] https://en.wikipedia.org/wiki/G%C3%B6del%27s_incompleteness_theorems#:~:text=G%C3%B6del's%20incompleteness%20theorems%20are%20two,capable%20of%20modelling%20basic%20arithmetic.&text=The%20second%20incompleteness%20theorem%2C%20an,cannot%20demonstrate%20its%20own%20consistency. ( a piece of OP's title )

 

general comment: this is a mathematics forum. 

 

 

[joigus]

1 minute ago, ahmet said:

then how will you reach the title's implication: 2+2=5 with your this map??

I already told you about this. I'm certainly not going to repeat just because you don't care enough to read.

[ahmet]

59 minutes ago, joigus said:

Now, I suggest you ask this yourself: Am I really helping here?

I am sure, I have tried  ..

[joigus]

 

9 minutes ago, ahmet said:

I recommend that you check the ordered/given references and try to correlate them well.

And I really recommend that you read the OP:

Quote

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

By francis20520, Saturday at 04:40 PM in Mathematics

And then:

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

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

(My emphasis)

Now what? Do you realize you haven't understood the OP already? Don't worry, I'm waiting for you to catch up.

IOW: Just because we cannot prove everything (following Gödel's theorem), does it follow that we cannot prove that 2+2=4? Should we accept that 2+2=5? Or anything else?

That's called "reading between the lines." If you can't read between the lines, the OP looks like a contradiction. Which it isn't. But you must take some time to really read carefully the OP and really want to help. Just trying to appear cleverer than everybody else just because you can quote Bourbaki, or link to it, doesn't really help.

14 minutes ago, ahmet said:

general comment: this is a mathematics forum. 

Oh, really? I hadn't noticed. I thought we were talking about the history of pudding (sigh).

[ahmet]

3 minutes ago, joigus said:

 

 

Oh, really? I hadn't noticed. I thought we were talking about the history of pudding (sigh).

hahahahhahhaa        hahahhaa 

ok. I really spent enough effort to help the OP. 

sorry, but I won't reply anymore (at least until some contexts more be provided by OP)

but already beacuse of laughing ...  I suppose    I can't by by now.

  

 

[studiot]

@ahmet

Quote

 

@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. )

3 hours 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)

 

It seems I nearly lost a post when I went to dinner.

Luckily the input editor here didn't loose it.

 

I don't have Bourbaki, but here is an excerpt from Lang

How does it help ?

Can I also ask by circle do you mean ring? (I am particularly intested in the answer to thi since I wonder if there is a language difficulty)

 

 

 

[ahmet]

36 minutes ago, studiot said:

Can I also ask by circle do you mean ring?

yes. but..it has been a bit late here and i had intented not to write until OP gives more contexts. 

... 

have a good night. 

[studiot]

2 minutes ago, ahmet said:

yes. but..it has been a bit late here and i had intented not to write until OP gives more contexts. 

... 

have a good night. 

Thank you for that, it only just occurred to me and certainly shows the need to be careful with language. +1

[francis20520]

I like this ask this question regarding implications of Gödel's Incompleteness Theorems.

That is, from what I have read one of the implications of Gödel is that either "Maths is inconsistent" or  that "we will NEVER know everything".

What does "maths is inconsistent" mean?? Can you give an example where maths is inconsistent?? Have they discovered such an inconsistency??

And does this mean that for example, we can never discover the smallest particle or the smallest time segment or whether the universe is finite or infinite?? What does it mean????

[studiot]

11 minutes ago, francis20520 said:

I like this ask this question regarding implications of Gödel's Incompleteness Theorems.

That is, from what I have read one of the implications of Gödel is that either "Maths is inconsistent" or  that "we will NEVER know everything".

What does "maths is inconsistent" mean?? Can you give an example where maths is inconsistent?? Have they discovered such an inconsistency??

And does this mean that for example, we can never discover the smallest particle or the smallest time segment or whether the universe is finite or infinite?? What does it mean????

Good that you are still with us after all that squabbling.

 

Please explain where you got those ideas post a link or reference.

Because they are seditious.

The whole point of Maths is that it is self consistent.

But that does not mean that any part of it is complete ie tells everything.

That is the point of Godel's incompleteness theorems   - we will never know everything unless everything is limited. But we do not know and I do not believe that 'everything is limited'.

 

If you like ? a bit of background about the Erlangen program and David Hilbert might help place Godel in context.

[ahmet]

26 minutes ago, francis20520 said:

I like this ask this question regarding implications of Gödel's Incompleteness Theorems.

That is, from what I have read one of the implications of Gödel is that either "Maths is inconsistent" or  that "we will NEVER know everything".

What does "maths is inconsistent" mean?? Can you give an example where maths is inconsistent?? Have they discovered such an inconsistency??

And does this mean that for example, we can never discover the smallest particle or the smallest time segment or whether the universe is finite or infinite?? What does it mean????

hereby, within this comment, I recommend the moderation to move this thread to another forum. (maybe,speculations or one of other forums might be more suitable)

 

Edited June 30, 2020 by ahmet

[francis20520]

23 hours ago, studiot said:

Good that you are still with us after all that squabbling.

 

Please explain where you got those ideas post a link or reference.

Because they are seditious.

The whole point of Maths is that it is self consistent.

But that does not mean that any part of it is complete ie tells everything.

That is the point of Godel's incompleteness theorems   - we will never know everything unless everything is limited. But we do not know and I do not believe that 'everything is limited'.

 

If you like ? a bit of background about the Erlangen program and David Hilbert might help place Godel in context.

Actually I was told this in a Youtube comment discussion.

I kind of get it that "math is self consistent". For example 2 + 2 should always give 4 under ALL circumstances. So, even in a parallel universe in the multiverse 2 + 2 will still be 4. So that is consistency from the point of view of a layman, right?

But does Godel's theorems have implications in physics where we search for "knowledge"???

I.e. Does these 2 theorems put a limit to knowledge we can obtain about the universe through physics???

[Strange]

1 hour ago, francis20520 said:

But does Godel's theorems have implications in physics where we search for "knowledge"???

I.e. Does these 2 theorems put a limit to knowledge we can obtain about the universe through physics???

No. They are purely about the "completeness" of formal systems.

In other words, can a formal system (e.g. mathematics) prove that anything that can be written down using that formal system is either true of false.

And the answer to that is no. You can write something using mathematics that you cannot use the same mathematics to prove or disprove. You can extend your formal system to make it more complete and allow you to prove that statement. But then there will be other statements that this extended system cannot prove. And so ad infinitum.

Physics uses mathematics but it is not limited by that mathematics in the same way. (And some mathematicians complain that physicists are a bit "ad hoc" and don't really stick to absolutely formal derivations.)

[studiot]

1 hour ago, francis20520 said:

Actually I was told this in a Youtube comment discussion.

I kind of get it that "math is self consistent". For example 2 + 2 should always give 4 under ALL circumstances. So, even in a parallel universe in the multiverse 2 + 2 will still be 4. So that is consistency from the point of view of a layman, right?

But does Godel's theorems have implications in physics where we search for "knowledge"???

I.e. Does these 2 theorems put a limit to knowledge we can obtain about the universe through physics???

Yes Strange's description of 'formal systems' is a good one.

And whilst much of mathematics is embodied in formal systems some important parts are not.

For instance the technique of iterative refinement is widely used in Science and Engineering.

That is

Have a guess at the answer or value you seek.
Substitute in your guesstimate and see how close to the correct outcome you are.
Refine your first guess and test again.
Repeat until your guess matches observation to the required degree of accuracy.

This technique appears in statistics as Bayes Theorem.

Remember that in Physics well conducted observation always trumps any 'proof' or theoretical determination of a result.

[francis20520]

1 hour ago, Strange said:

No. They are purely about the "completeness" of formal systems.

In other words, can a formal system (e.g. mathematics) prove that anything that can be written down using that formal system is either true of false.

And the answer to that is no. You can write something using mathematics that you cannot use the same mathematics to prove or disprove. You can extend your formal system to make it more complete and allow you to prove that statement. But then there will be other statements that this extended system cannot prove. And so ad infinitum.

Physics uses mathematics but it is not limited by that mathematics in the same way. (And some mathematicians complain that physicists are a bit "ad hoc" and don't really stick to absolutely formal derivations.)

This is mindbogglingly complex stuff.

I am assuming that there are SOME statements that CAN be proved. For example we can "prove" that 2 + 3 = 5 is true.

But Godel "proved" that there are certain statements that cannot be proven to be correct or not??

I suppose that is what Godel's theorems show and where his genius lies.

So, is there any statement yet discovered in mathematics that satisfies Godel's proof??? Have they found a statement that cannot be shown to be either true or false?? 

 

[Strange]

34 minutes ago, francis20520 said:

I am assuming that there are SOME statements that CAN be proved.

Pretty much everything in mathematics can be proved. There are a few well-known problems that have not been proved yet (some have substantial prizes associated with them).

It took over 300 years before someone proved Fermat's Last Theorem. That is partly because it had to build on a huge amount of mathematics that was developed in the meantime. Some very simple looking problems can be very hard to solve. Some quite complex sounding problems, that people have struggled with for decades, might turn out to have a really simple proof. (Mathematics is one of the few files where outsiders can, and do, make breakthroughs.)

48 minutes ago, francis20520 said:

So, is there any statement yet discovered in mathematics that satisfies Godel's proof??? Have they found a statement that cannot be shown to be either true or false?? 

There are lots of unknowns in mathematics (e.g the continuum hypothesis; that there is no infinity between the (infinite) set of integers and the (infinitely larger) set of reals) that have not yet been proved either way.

And there are some problems for which it can be proved that there's no solution.

But I don't think you can ever say that a particular problem is unprovable because of Gödel Incompleteness. Unless it is a problem specifically constructed to be unprovable for that reason (which is how Gödel proved the theorem; by constructing a mathematical statement that could not be proved in the rules of the system).

[studiot]

1 hour ago, francis20520 said:

This is mindbogglingly complex stuff.

I am assuming that there are SOME statements that CAN be proved. For example we can "prove" that 2 + 3 = 5 is true.

But Godel "proved" that there are certain statements that cannot be proven to be correct or not??

I suppose that is what Godel's theorems show and where his genius lies.

So, is there any statement yet discovered in mathematics that satisfies Godel's proof??? Have they found a statement that cannot be shown to be either true or false?? 

 

Every schoolboy studying applied maths or physics learns the 'principle of static indeterminancy'.

A formal system is the principle of static equilibrium applied to a structure.

This says that the sum of the applied vertical reactions, the applied horizontal reactions and the applied moments must each be separately zero.

This provides three equations for a structure.

Some structures are statically determinate in that those three equations are sufficient to fully determine the reactions.
For example a simply supported beam.

For other structures those equations are insufficient for the determination, although the reactions must exist.
Such structures are called statically indeterminate.
For example a beam, built in at both ends.

Edited July 1, 2020 by studiot

[Strange]

4 minutes ago, studiot said:

Every schoolboy studying applied maths or physics learns the 'principle of static indeterminancy'.

Is it bad that I have never heard of that before? Can I blame my teachers?

[Eise]

10 hours ago, studiot said:

Every schoolboy studying applied maths or physics learns the 'principle of static indeterminancy'.

Why am I remembered of Hilbert, saying:

Quote

Every boy in the streets of Göttingen understands more about four-dimensional geometry than Einstein.

(But added: "Yet, in spite of that, Einstein did the work, and not the mathematicians.")

[studiot]

59 minutes ago, Eise said:

Why am I remembered of Hilbert, saying:

(But added: "Yet, in spite of that, Einstein did the work, and not the mathematicians.")

I don't understant the connection, but Godel did discover a 'timeloop' solution to Einstein's equations.

remembered 

reminded

Edited July 2, 2020 by studiot

[Eise]

26 minutes ago, studiot said:

I don't understant the connection...

Just the schoolboy...

26 minutes ago, studiot said:

remembered 

reminded

Thanks. I try to remember that...

Edited July 2, 2020 by Eise

[joigus]

11 hours ago, Strange said:

Is it bad that I have never heard of that before? Can I blame my teachers?

Count me in. In my case it may have been attention deficit. 🤦‍♂️

[studiot]

11 hours ago, Strange said:

Is it bad that I have never heard of that before? Can I blame my teachers?

 

20 minutes ago, joigus said:

Count me in. In my case it may have been attention deficit

It is common practice (especially in schools) to state, somewhere near the beginning, "We will only consider statically determinate frames/beams/structures in this text."
Sometimes a simple example of an indeterminate structure (such as a propped cantilever or a doubly cross braced frame) is shown as an example but not discussed further.

 

 

[Strange]

1 hour ago, studiot said:

It is common practice (especially in schools) to state, somewhere near the beginning, "We will only consider statically determinate frames/beams/structures in this text."
Sometimes a simple example of an indeterminate structure (such as a propped cantilever or a doubly cross braced frame) is shown as an example but not discussed further.

I must have been away that day.