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Writing 1970 after E's theorem is a bit misleading. The theorem is from 1964, it was the content of his Ph.D. thesis. It was published in 1970, in the first volume of the Annals of Math. Logic. (but it was fiarly available after 1964, many libraries still have the thesis)
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The Easton article claims (p. 141), that the proof is valid in Peano
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