Math Is Fun Forum

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.