[Math Challenge]: Prove me wrong.

AlexPontik · 2021-10-01 02:36:30

Hi all,

Relax and be patient while reading, as below I am claiming that one of your famous mathematicians is wrong, that I am right, and on top of all of this bragging, that all of you can prove that to yourselves after you read what is written below, and...I am no famous mathematician...I am no one really...but...none of you are someone either...or do you think you are?

We start with some examples, regarding the argument I am making here.

Examples (the formal argument is provided under the challenge section)
for a child to walk, a child balances its body on its feet, and for the child to learn to walk, the child has to be able to prove that to itself, within its system, meaning once the child learns how to balance as described earlier, the child thinks, "I think I get how to walk",
or otherwise said,
"you have to balance your body on your feet to walk" is an axiom in order to be able to walk, and even children can prove this to themselves within their system.

"Most birds can fly" is an axiom in the reality I observe, that birds seem to be able to prove to themselves.
"If you are a bird, and you really sense that you can fly from your nature, then you can learn how to really fly", is an axiom that I observe birds are able to prove to themselves within their own system.

But, if we look at humans again, since one may say that animals are off limits for what we are discussing...
being able to make sense of the world around you by just observing and talking about it, is an axiom humans follow in order to have fun in reality
as when humans are talking nonsense, they haven't thought yet, how reality...really makes sense,
and other humans can prove that to them, so that...

then they can prove what makes sense to themselves, within their own system, after thinking about what was said alone, without bothering other humans with nonsense, before they prove it to themselves...by thinking about it.

Challenge <-- Prove to me that I am wrong (examples are provided above, if you don't understand below)
Why say that the following phrase is nonsense?
“The consistency of axioms cannot be proved within their own system.”
Because:
A system which has axioms for itself, in order for the system to call them axioms for itself, the system has to have a consistent behavior around those axioms and so when it behaves inconsistently with regard to those axioms, the inconsistency between those axioms and the system’s behavior the system can prove to itself.
If what is written above is false, then when a system behaves inconsistently with regard to some axioms it has for itself, that inconsistency it cannot prove to itself, and it keeps behaving inconsistently with regard to those axioms…but…
if the system keeps behaving inconsistently with regard to some axioms and cannot prove to itself that it does so with regard to those axioms, then it doesn’t seem to me it can consistently keep regarding them as axioms for the system, and then something else replaces them, and that something else is what the system calls axioms for itself.

Or if you want it explained in any other way...let's experiment with physics:
AXIOM: In any experiment conducted in reality, nothing can happen as a result, some-thing can happen as a result, or...something else can happen as a result.
This is an axiom that seems consistent and complete to me, and I dare say...logical.
Isn't it? If it doesn't seem to you, here's why I think that:

Physics as a science, progresses as follows:
1.There is a current theory, at any given time.
2.A candidate theory, which is more exact regarding what really is happening, appears from research as a proposed new theory.
3. Experiments have to be conducted to verify the new theory.
4. When experiments are conducted, they can have the following results.
5. Nothing happens, the experiments fail to show any results, which has happened in the past.
6. Something happens, the experiments had the expected results, which has hap-pened in the past, and science keeps following its path.
7. Something else happens...which was the case with some previous experiments...or else we wouldn't be looking for a new theory, as then all experiments would point only to something, and nothing else...but up to now, this isn't the case, and the future still happens next, and not before next happens.
8. What seems to be happening, is that before people actually make things in their lives that do something...they make things that don't do something exactly...and they find that early at best, or late at worst...but the complete story they all know from the beginning, pretty consistently, it seems to me...as it could be the case with the argument I am making here and below.

Or if you are still unconvinced, and you want it explained using casual language:
After someone says that "the consistency of axioms cannot be proved within their own system", can someone prove to oneself that in order for one to relax, one simply starts relaxing and waits, or is this inconsistent with someone's logic, and then someone cannot relax? Why?
If you want to naturally relax, you simply start doing that(relaxing) and you wait... and you can notice yourself after doing that again and again, that this is an axiom to relax, that you can consistently prove to yourself.
If you want to understand the logic of relaxing, the logic of relaxing is to have a simple common word which describes the starting point of naturally relaxing...if you want to use more words to describe that starting point of naturally relaxing, then this is less relaxing...than simply relaxing.
If you don't want to understand the nature of relaxing, don't worry too much about it, just relax and do something else.

Links regarding the challenge. Check on section "incompleteness theorem" number 2 (links are provided so that you understand what we are discussing
https://en.wikipedia.org/wiki/Kurt_G%C3%B6del

Kind regards, and thank you for your patience,
no one.

p.s.
"Ok, but WHY ARE YOU SENDING THIS TO US?", someone may ask.
Use the links below, follow the money, and...be patient:

https://www.facebook.com/media/set/?set=a.250991363520389&type=3
https://www.youtube.com/watch?v=hNFYMORvM_o
https://www.youtube.com/watch?v=KFrv57zoPq0

GrandStrategos · 2021-10-06 09:57:19

Simple answer: Systems exist in mathematical logic, and they cannot be applied to life in general, because there is no set of formulas that explain life.

Let me start by first noting that this is not necessarily a math challenge, only requiring an understanding of what mathematical logic is, and its applications. Real life is not one of these applications. In order to even start with these claims, you would need to show first how you can apply it, specifically how you can apply first-order logic to explain the totality of natural phenomena. I remember seeing a post where you did make an attempt to map it emotions, but the list as far as I can see is incomplete and inconsistent. That can be left for another time.

The statements you use are easily disproven by reading the theory in full:

"Assume F is a consistent formalized system which contains elementary arithmetic. Then
F ⊬ Cons(F)."

The key here is “formalized.” When we refer to axioms and the like, we refer to formal systems. Formal systems consist of a few things:

1) A finite set of symbols, which we call an alphabet, from which we form finite strings of symbols, which are formulas.
2) A grammar, from which we get rules on how to best create formulas from simpler ones.
3) A set of axioms, consisting of well-formed formulas. AXIOMS DO NOT HAVE TO BE TRUE. They just have to be taken to be true.
4) A set of inference rules. The more appropriate term here is theorems.

The key problem here seems to be a misunderstanding of what a formal system actually is. The child and the birds are NOT formal systems, because there is no set of axioms that fully defines them. Children cannot be defined by a set of axioms, because we cannot fully define the child. Physically, yes, but you cannot prove their mental state or their unconscious desires. “But the child can speak!” Yes, but you will receive an answer in the formal system of English, or whatever language they speak, because languages are formal systems of their own. We need a definitive set of axioms from the source. The same goes for the birds; why does one bird go in one direction to find food, while another chooses a completely new one? There is no axiom that describes what direction it goes in and whether it finds food. You can argue that they use their eyesight, but how do it’s eyes work, and why do they interpret it in the way they do? Do all members of the set “bird” recognize the same items as food, and if so, why not? Arguing about habitat or evolution will not work, because birds are part of different system. Of course, we can use the classical philosophy definition, since we are definitely leaning outside of mathematics at this point, but axioms there must be undeniably true. This is certainly false, because children and birds do not all share the exact same sequences of genes as others in their set. If something is not a system, then we cannot use these examples to refute the second theorem of incompleteness.

The third example can follow a similar line of logic. Humans do not formulate sentences before speaking by using their own “system” because humans cannot be defined by a set of axioms, and are thus not a system. Rather, we use the formal system of the English language, specifically using the formal language defined by English, specifically its syntax (how it is spelled) and its semantics (what it means). If human were individual formal systems, then everybody would have their own language, since systems cannot learn from other systems.
There is another thing that makes your first 3 examples non sensical, and that is the fact that all 3 examples show living beings learning, not proving. Implying that a child has proven how to walk, or a bird to fly, or a human to speak logically all imply that they had it since birth, which is false. And you can’t prove learning, since humans, animals, and other living beings interpret knowledge in different ways.

You can prove a proof, but you can’t knowingly observe something proving a proof unless you knew it’s axioms. And as stated before, a bird has no axioms to even form a proof about itself. The child cannot prove himself, mainly because he does not understand the axioms of himself, because that requires knowledge of what he is. Children do not even begin to question their identity as a person. Similarly, the child and the human in the two scenarios cannot prove themselves because that would mean that there is a set of axioms concerning themselves and their behavior. But there isn’t, because we cannot formulate human behavior, or any sentient behavior, and write it down as a short, well-made list of axioms. And if we can’t formulate sentient behavior, we can’t formulate a system for sentient life, because a part of sentient life is sentient behavior. Life is not bounded by a set of mathematical rules that can always be proven to be true.

Physics is a formal system. However, what you describe is not physics. What you describe is the scientific method. Replace “physics” with another science and the implied point still stands. Physics is not the research papers, but the actual meat and norms; the formulas, the variables, and the rules that we believe it follows. Physics has axioms such as Newton’s Laws of Motion. A mistake here is that you are equating consistency to consistency in results. The system of physics on paper is everything we know about motion. When we discover a new part of physics, the system is not inconsistent, we just discover something that may or may not change our understanding of the system of physics.

If you were trying to address the process on the first place, then that is also not a formal system. Not everything is a system.

The primary dilemma of Gödel’s incompleteness theorems (as it is called) is not that some things can’t be proved, but we cannot prove that we cannot prove something exists. For example, in math, we can prove that 2 + 2 = 4. We can prove something that we can prove. So it can be proved that it can be proved that 2 +2 = 4. And I think George Boolos eloquently describes the rest.

“Now, two plus two is not five. And it can be proved that two plus two is not five. And it can be proved that it can be proved that two plus two is not five, and so on.

Thus: it can be proved that two plus two is not five. Can it be proved as well that two plus two is five? It would be a real blow to math, to say the least, if it could. If it could be proved that two plus two is five, then it could be proved that five is not five, and then there would be no claim that could not be proved, and math would be a lot of bunk.

So, we now want to ask, can it be proved that it can't be proved that two plus two is five? Here's the shock: no, it can't. Or, to hedge a bit: if it can be proved that it can't be proved that two plus two is five, then it can be proved as well that two plus two is five, and math is a lot of bunk. In fact, if math is not a lot of bunk, then no claim of the form "claim X can't be proved" can be proved.

So, if math is not a lot of bunk, then, though it can't be proved that two plus two is five, it can't be proved that it can't be proved that two plus two is five.”

In basic terms, for all systems, there is no “theory of everything.” That is, there is no consistent, complete system. Because, as the first theory states, systems that are consistent are incomplete. This was found by mapping all possible mathematical functions and proving that a formula G essential defined itself. You can’t define something by using itself, because then the solution is recursive. It is the same reason we do not define a word in the English language by using that word in the definition. If a set of axioms could prove it can never yield a contradiction, then it is complete, and there is a set of axioms that exist that metamathematically say “This set of axioms is consistent.” But if the set of axioms is incomplete, then there exist some formula F that cannot be proven. If this was the case, then G could be proven. Thus we have proof by contradiction.

And no, we cannot use Gödel’s incompleteness theorems to prove that Gödel’s incompleteness theorems are undecidable. Because first-order logic is semantically complete, but not syntactically complete. That is, we can prove the meaning of first order logic.

In general, it is not the best idea to apply mathematical concepts to things that can not be defined by mathematical statements, such as the behavior of children, learning, and the nature of relaxation. We could, of course, debate semantics on the matter, but if you want to disprove Gödel’s second incompleteness theorem, you need to answer why formula G is undecidable, yet true in his first incompleteness theorem, since the second incompleteness theorem is directly derived from there.

Last edited by GrandStrategos (2021-10-06 22:52:22)

AlexPontik · 2021-10-06 23:17:14

GrandStrategos wrote:

Simple answer: Systems exist in mathematical logic, and they cannot be applied to life in general, because there is no set of formulas that explain life.

If there is no set of formulas that explain life, then surely your explanation below, isn't that set, that explains life, is it?
But apart from that, if there is no set of formulas that explain life, then the books you have don't explain any part of life, to any degree...which is untrue.
A set of formulas can explain life, to life...this is why humans...use formulas...to explain life to themselves...

...or else people wouldn't be using formulas.

Now is there a statement, that completely and consistently defines life, regardless of who it is we are talking about?
Yes, life is something else than anything you can imagine ,this is why you have to live a life to realize that.

GrandStrategos wrote:

The key problem here seems to be a isunderstanding of what a formal system actually is. The child and the birds are NOT formal systems, because there is no set of axioms that fully defines them. Children cannot be defined by a set of axioms, because we cannot fully define the child. Physically, yes, but you cannot prove their mental state or their unconscious desires. “But the child can speak!” Yes, but you will receive an answer in the formal system of English, or whatever language they speak, because languages are formal systems of their own. We need a definitive set of axioms from the source.

A set of axioms that fully defines them, is their DNA, plus the laws of nature, which they cannot break.
DNA of a bird, plus laws of nature regarding birds, you got yourself a bird.
DNA of a human child, plus laws of nature regarding human children, you got yourself a child.

GrandStrategos wrote:

The same goes for the birds; why does one bird go in one direction to find food, while another chooses a completely new one?
There is no axiom that describes what direction it goes in and whether it finds food.

There is, it is an axiom for life that life is free to make choices, from the beginning of life.

GrandStrategos wrote:

This is certainly false, because children and birds do not all share the exact same sequences of genes as others in their set.

When taking DNA, all the set of genes, which define a healthy species, are part of the set of genes which make life possible in reality.

GrandStrategos wrote:

The third example can follow a similar line of logic. Humans do not formulate sentences before speaking by using their own “system” because humans cannot be defined by a set of axioms, and are thus not a system. Rather, we use the formal system of the English language, specifically using the formal language defined by English, specifically its syntax (how it is spelled) and its semantics (what it means). If human were individual formal systems, then everybody would have their own language, since systems cannot learn from other systems.

Systems cannot learn from other systems, is wrong at best...a waste of my time and effort replying to you at worst...as you as a system cannot learn from other systems...

Regarding systems, you writing down that "humans cannot be defined by a set of axioms" is an axiom also genius...you should have considered that before you attempted such a tedious and long reply...

GrandStrategos wrote:

There is another thing that makes your first 3 examples non sensical, and that is the fact that all 3 examples show living beings learning, not proving. Implying that a child has proven how to walk, or a bird to fly, or a human to speak logically all imply that they had it since birth, which is false. And you can’t prove learning, since humans, animals, and other living beings interpret knowledge in different ways.

Genius...focus in your next reply, as I am not going to waste more of my time and effort with your random walks in your head...

In order to learn something, genius, replying to me with such certainty, not having spend any time and effort to think, you have to prove to yourself what you learned...otherwise you haven't proved to yourself genius that you are making sense before you say something...

"Implying that a child has proven how to walk, or a bird to fly, or a human to speak logically all imply that they had it since birth, which is false"
Healthy children can learn how to walk from birth.

Healthy birds can learn how to fly from birth.

Both are made able to be able to prove that to themselves from birth, that happens through their instincts, meaning children have the instinct to try to walk, and birds have the instinct to try to fly.

If you here, still don't understand, do you think that children don't have the instinct to fly from birth, and can prove this to themselves...once they learn how to walk?

GrandStrategos wrote:

And as stated before, a bird has no axioms to even form a proof about itself. The child cannot prove himself, mainly because he does not understand the axioms of himself, because that requires knowledge of what he is. Children do not even begin to question their identity as a person.

The genetic code defines a living being.
The genetic code of a species defines a species.

Children begin to question their identity as a person, even before they can talk...this is why they do talk to you, to tell you who they are and what they want.

GrandStrategos wrote:

Physics is a formal system. However, what you describe is not physics. What you describe is the scientific method. Replace “physics” with another science and the implied point still stands. Physics is not the research papers, but the actual meat and norms; the formulas, the variables, and the rules that we believe it follows. Physics has axioms such as Newton’s Laws of Motion. A mistake here is that you are equating consistency to consistency in results. The system of physics on paper is everything we know about motion. When we discover a new part of physics, the system is not inconsistent, we just discover something that may or may not change our understanding of the system of physics.

It is an axiom up to now that reality seems to happen in the same way it happened before, reality is something else than anything anyone can imagine.
Even if reality is something else than anything anyone can imagine, everyone can still say that.
Anyone disagreeing with reality being something else than anything anyone can imagine, claim to be then one for who reality it that one imagines.

And if you are that one, genius, for who reality is what you imagined, and you had a problem with me writing straight to your face here, that reality is something else than anything anyone can imagine, why didn't you come earlier to tell me about it?
Wasn't reality something you genius imagined, so me telling you that reality is something else than anything anyone can imagine is part of your imagination?
And if that is so, why am I wasting my time and effort with your thoughtless, smartass wannabe answer....

Think genius, next time you reply to me, everything that you have written up to now, has been just a waste of time and effort.

AlexPontik · 2021-10-06 23:19:12

The next one replying, put time and effort before you do so, I am not going to waste my time with wannabes here...

Bob · 2021-10-07 05:58:02

Ok. think on this:

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GrandStrategos · 2021-10-07 06:55:08

Again, formal systems can’t be children, birds, or humans. Formal systems are abstract structures used for inferring theorems from axioms according to a set of rules. An abstract structure is a hypostatic abstraction that is defined by a set of laws, properties and relationships in a way that is logically if not always historically independent of the structure of contingent experiences, for example, those involving physical objects. So, in the context of first-order logic, formal systems can’t be physical objects.

Reality is a bit more debated, but regardless of its status, it cannot be used to disprove the second incompleteness theorem because:

Assume F is a consistent formalized system which contains elementary arithmetic. Then
F ⊬Cons(F).

Specifically, this basic arithmetic has to be Robinson arithmetic, since that is the qualification needed to pass the first theorem, which is necessary for the second. We can talk about what it means to “prove” but unless you can map it to Robinson arithmetic, then you argument does not work.

AlexPontik · 2021-10-18 19:47:40

GrandStrategos wrote:

Again, formal systems can’t be children, birds, or humans. Formal systems are abstract structures used for inferring theorems from axioms according to a set of rules. An abstract structure is a hypostatic abstraction that is defined by a set of laws, properties and relationships in a way that is logically if not always historically independent of the structure of contingent experiences, for example, those involving physical objects. So, in the context of first-order logic, formal systems can’t be physical objects.

do you think the more words you add...the more sense you make?

So children, birds or humans are not formal systems to you...

well...let's say that in this conversation I am not interested in what a formal system is, as it seems to you this is up to debate...
I am just interested in what a REAL SYSTEM is and children, birds and humans,

are quite more real than the language your are writing here no?

GrandStrategos wrote:

Reality is a bit more debated, but regardless of its status, it cannot be used to disprove the second incompleteness theorem because:
Assume F is a consistent formalized system which contains elementary arithmetic. Then
F ⊬Cons(F).
Specifically, this basic arithmetic has to be Robinson arithmetic, since that is the qualification needed to pass the first theorem, which is necessary for the second. We can talk about what it means to “prove” but unless you can map it to Robinson arithmetic, then you argument does not work.

Assume that you are trying to write incomprehensibly, and not try to explain yourself to others...
Do you think others will spend time and effort reading what you write?

Reality is not for you, neither for Godel himself was, something that you can imagine...it also isn't something I can imagine as for me reality is something else than anything I can imagine, and that I don't see any reason to debate with anyone...why?

Because if for you reality was something that you imagined, and it was a problem for you for me to tell you, in such a straight manner, that reality is something else than anything anyone can imagine, if reality for you was something that you imagined, why didn't you REALLY come a bit earlier to tell me about it.

Do you have any idea how simple it would be , if you simply appeared on my door and said "you're thinking to post something on a math forum regarding a famous proof, aren't you? Well, you're wrong and here's why..."
Even without the why, you coming earlier than I thought to post on my door would have been enough to convince me, but now...

what is it that you really don't understand about reality?

Math Is Fun Forum

#1 2021-10-01 02:36:30

[Math Challenge]: Prove me wrong.

#2 2021-10-06 09:57:19

Re: [Math Challenge]: Prove me wrong.

#3 2021-10-06 23:17:14

Re: [Math Challenge]: Prove me wrong.

#4 2021-10-06 23:19:12

Re: [Math Challenge]: Prove me wrong.

#5 2021-10-07 05:58:02

Re: [Math Challenge]: Prove me wrong.

#6 2021-10-07 06:55:08

Re: [Math Challenge]: Prove me wrong.

#7 2021-10-18 19:47:40

Re: [Math Challenge]: Prove me wrong.

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