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Google search yields the paper by RH Cowen called Generalizing König's infinity lemma. Due to my insufficient technical background, I am afraid, I cannot fully appreciate the paper.

Tout court, Cowen proves three theorems and concludes they are equivalent to prime ideal theorem of Boolean algebra.

Here is the first theorem which is generalization of KL:

Theorem 1 Let T be a collection of locally finite trees such that for any finite set of levels of T there is a consistent set of vertices piercing those levels. Then there is a consistent set of vertices piercing the entire set of levels of T.

Then he states:

Theorems 1-3 are equivalent to each other and to P.I., the prime ideal theorem for Boolean algebras; this is easy to establish using our results in [l] where some strong forms of Rado's selection lemma were shown to be equivalent to P.I

My questions are two-fold:

  • Does this paper show the connection between König's Lemma (KL) and primes through Prime Ideal Theorem (PIT)?

  • Has there been research done to prove primes' infinitude via KL?

Or am I completely off?

  • 3
    The prime ideal theorem is not really that closely related to prime numbers. The paper in your p.s. refers to a different result named after Konig.2011-12-28
  • 0
    @QiaochuYuan I realized it and edited it out.2011-12-28
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    @QiaochuYuan Regarding your first statement, would you then say it is a possible research avenue to attempt to prove between relation primes and KL?2011-12-28
  • 1
    Also (very minor quibble), you're mixing two baseball metaphors in your last sentence ("off base" and "out in left field").2011-12-28
  • 4
    Konig's Lemma (if it's the thing I'm thinking of) proves there's an infinite path in some kind of tree, but part of the hypothesis is that the tree is infinite in the first place. So it seems unlikely to me that you'd get a proof of the infinity of primes out of Konig. I could be wrong.2011-12-28

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