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edited Dec 8, 2009 at 19:48

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I'm not sure why you say this. I am right now looking at the following paper of Ax:

MR0229613 (37 #5187) Ax, James The elementary theory of finite fields. Ann. of Math. (2) 88 1968 239--271.

The first sentence is: "In this paper, we prove the decidability of the theory of finite fields and the theory of p-adic fields." He goes on in the introduction to explain exactly what this means: given a statement E in the language of rings, he gives an algorithm for determining the set of prime powers q such that E holds in the finite field of order q. He doesn't say anything about dependence on CH, and I don't see how this could possibly be the case ("decidability given CH" is a far cry from decidability!).

The only mention of CH in the introduction to the paper is in Theorem B, which states that a certain isomorphism of ultraproducts holds given CH. This makes more sense, since cardinality questions come into play in the isomorphism of elementarily equivalent objects.

I'm not sure why you say this. I am right now looking at the following paper of Ax

MR0229613 (37 #5187) Ax, James The elementary theory of finite fields. Ann. of Math. (2) 88 1968 239--271.

The first sentence is: "In this paper, we prove the decidability of the theory of finite fields and the theory of p-adic fields." He goes on in the introduction to explain exactly what this means: given a statement E in the language of rings, he gives an algorithm for determining the set of prime powers q such that E holds in the finite field of order q. He doesn't say anything about dependence on CH, and I don't see how this could possibly be the case ("decidability given CH" is a far cry from decidability!).

The only mention of CH in the introduction to the paper is in Theorem B, which states that a certain isomorphism of ultraproducts holds given CH. This makes more sense, since cardinality questions come into play in the isomorphism of elementarily equivalent objects.

I'm not sure why you say this. I am right now looking at the following paper of Ax:

MR0229613 (37 #5187) Ax, James The elementary theory of finite fields. Ann. of Math. (2) 88 1968 239--271.

The first sentence is: "In this paper, we prove the decidability of the theory of finite fields and the theory of p-adic fields." He goes on in the introduction to explain exactly what this means: given a statement E in the language of rings, he gives an algorithm for determining the set of prime powers q such that E holds in the finite field of order q. He doesn't say anything about dependence on CH, and I don't see how this could possibly be the case ("decidability given CH" is a far cry from decidability!).

The only mention of CH in the introduction to the paper is in Theorem B, which states that a certain isomorphism of ultraproducts holds given CH. This makes more sense, since cardinality questions come into play in the isomorphism of elementarily equivalent objects.

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created Dec 8, 2009 at 19:37

65.4k
12
241
381

I'm not sure why you say this. I am right now looking at the following paper of Ax

MR0229613 (37 #5187) Ax, James The elementary theory of finite fields. Ann. of Math. (2) 88 1968 239--271.

The first sentence is: "In this paper, we prove the decidability of the theory of finite fields and the theory of p-adic fields." He goes on in the introduction to explain exactly what this means: given a statement E in the language of rings, he gives an algorithm for determining the set of prime powers q such that E holds in the finite field of order q. He doesn't say anything about dependence on CH, and I don't see how this could possibly be the case ("decidability given CH" is a far cry from decidability!).

The only mention of CH in the introduction to the paper is in Theorem B, which states that a certain isomorphism of ultraproducts holds given CH. This makes more sense, since cardinality questions come into play in the isomorphism of elementarily equivalent objects.