5 ms·

I 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 natu

I 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 13y ago
How then are the naturals defined? By the Von Neumann construction? (0, {0}, {0, {0}}, ...)
- cscurmudgeon 13y ago
  I 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 13y ago
    What 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 13y ago
      It 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...
deleted 13y ago
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SatvikBeri 13y ago
Corrected link: http://en.wikipedia.org/wiki/Goodstein%27s_theorem http://en.wikipedia.org/wiki/Goodstein%27s_theorem
- cscurmudgeon 13y ago
  Thanks!
danbruc 13y ago
That comes down to the (philosophical) question if natural numbers existed before the first human started thinking about them. Depending on this the Peano axioms either just describe a part of the universe or they introduce something new. (I don't think it matters that natural numbers were known before the Peano axioms were formulated - before that we just defined natural numbers in a less formal way.)

strangestchild 13y ago

How then are the naturals defined? By the Von Neumann construction? (0, {0}, {0, {0}}, ...)

cscurmudgeon 13y ago
I 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 13y ago
  What 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 13y ago
    It 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...

deleted 13y ago

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SatvikBeri 13y ago

Corrected link: http://en.wikipedia.org/wiki/Goodstein%27s_theorem http://en.wikipedia.org/wiki/Goodstein%27s_theorem

cscurmudgeon 13y ago
Thanks!

danbruc 13y ago

That comes down to the (philosophical) question if natural numbers existed before the first human started thinking about them. Depending on this the Peano axioms either just describe a part of the universe or they introduce something new. (I don't think it matters that natural numbers were known before the Peano axioms were formulated - before that we just defined natural numbers in a less formal way.)