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It is absolutely mind-blowing how much time and passion humans have invested in understanding the consequences of nine simple statements...and how hard they res
by danbruc 14y ago
It is absolutely mind-blowing how much time and passion humans have invested in understanding the consequences of nine simple statements...and how hard they resist against unveiling their secrets.
0∈ℕ
∀x∈ℕ:S(x)∈ℕ
∀x∈ℕ:S(x)≠0
∀x,y∈ℕ:S(x)=S(y) ⇒ x=y
0∈X∧∀x∈ℕ:(x∈X ⇒ S(x)∈X) ⇒ ℕ⊆X
x + 0 := x
x + S(y) := S(x + y)
x · 0 := 0
x · S(y) := x · y + x
These are the Peano axioms [1] defining the natural numbers with addition and multiplication. And for at least 3000 years humans are tying to figure out properties of these natural numbers, all consequences of these definition, and the end is not in sight.
[1] https://en.wikipedia.org/wiki/Peano_axioms https://en.wikipedia.org/wiki/Peano_axioms
- dmvaldman 14y agoIt's not so surprising. This could also be summarized as "God created the integers." But really the process is reversed: given arithmetic, what is the smallest axiom set to give us the power to prove our theorems. It's like, given all code, what is the smallest language we can use to express it. Which is sequences of 0s and 1s.
- zwegner 14y ago> It's like, given all code, what is the smallest language we can use to express it. Which is sequences of 0s and 1s. This should be of interest: https://en.wikipedia.org/wiki/Binary_lambda_calculus https://en.wikipedia.org/wiki/Binary_lambda_calculus
- cscurmudgeon 14y agoI wouldn't say they define the natural numbers. A more apt phrasing would be "describe the natural numbers." They are more like the laws of physics for the natural numbers. Also, there are easily understandable theorems about the natural numbers not provable from the Peano axioms. http://en.wikipedia.org/wiki/Goodsteins_theorem http://en.wikipedia.org/wiki/Goodsteins_theorem
- strangestchild 14y agoHow then are the naturals defined? By the Von Neumann construction? (0, {0}, {0, {0}}, ...)
- cscurmudgeon 14y agoI would say that the Von Neumann construction and PA try to capture the natural numbers. It is a philosophical stance which can be called platonism. What you say is called the formalist stance.
- strangestchild 14y agoWhat I mean is that since Goodstein's theorem is provably true for the naturals, but is not a consequence of the Peano axioms, then the definition of the naturals used to demonstrate Goodstein's must be strictly stronger than the Peano axioms themselves. I was wondering what this definition might be. I'm familiar with the distinction between formalism and Platonism, although I still haven't made my mind up yet :)
- danbruc 14y agoIt is a consequence of the Peano arithmetic (Peano axioms plus definition of addition and multiplication), it is just not provable in this system. Gödel's first incompleteness theorem [1] states this fact, that no theory above a certain expressiveness (read as can express natural numbers with addition and multiplication) can be consistent and complete. Assuming Peano arithmetic is consistent, it can not be complete and complete means you can prove all true facts expressible in the system within the system itself. The (standard) proof of Goodstein's theorem uses ordinal numbers [2] which are outside of Peano arithmetic and the Kirby–Paris theorem proves that there is no proof inside Peano arithmetic [3]. [1] http://en.wikipedia.org/wiki/G%C3%B6dels_incompleteness_theorems http://en.wikipedia.org/wiki/G%C3%B6dels_incompleteness_theo... [2] http://en.wikipedia.org/wiki/Ordinal_number http://en.wikipedia.org/wiki/Ordinal_number [3] http://en.wikipedia.org/wiki/Goodsteins_theorem#Proof_of_Goodstein.27s_theorem http://en.wikipedia.org/wiki/Goodsteins_theorem#Proof_of_Goo...
- Houshalter 14y agoTo be fair there are a lot of possible systems from which you could derive infinite complexity. There are an infinite number of true statements that can be discovered after all. Most are not particularly interesting or useful. You could come up with proofs for more and more specific behaviors that occur in turing machines under specific conditions. Or cellular automata. Or any number of arbitrary systems. But most wouldn't be relevant to the real world. The cool thing about numbers is that there are a lot of possible applications to the real world. But there are also an infinite number of trails you could follow with mathematics that lead nowhere useful at all. Automated Mathematician tried to explore mathematics and find interesting concepts the same way mathematicians do and using similar heuristics. And it got fairly far, discovering numbers and primes and Goldbach's conjecture. But eventually it just started discovering useless concepts which had no apparent relevance to anything and were pointlessly complex or specific.
- hasenj 14y agoI'm pretty sure these math efforst existed long before the Peano axioms were formulated.
- sliverstorm 14y agoJust imagine the day that one of Peano's axioms is discovered to not hold true under all circumstances. Perhaps in a parallel dimension or something.