Talk:Peano axioms

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[edit] Successor

Currently Successor function redirects here, yet its definition is deep in the middle of a section and rather brief. I feel this might not be enough explination for the layman to follow, and it could do with a little expansion. It might also be worth considering reverting Successor function back to a non redirect version. --Salix alba (talk) 16:31, 23 March 2007 (UTC)

[edit] Axiom 9

Axiom 9 is a little ambiguous about what K can be. Is it OK for K to contain numbers other than the naturals? --Salix alba (talk) 16:35, 23 March 2007 (UTC)

Sure; K can contain lots of other sets. The axiom is phrased that way because Peano phrased it that way. In the modern context of ZF, you could start with a set K and then use a separation axiom to extract the set of natural numbers in K, so it would be natural to restrict the axiom to sets that only contain natural numbers. But Peano was doing this in 1889 well before axiomatic set theory; the word set in axiom 9 should be read in a natural-language way. CMummert · talk 17:12, 23 March 2007 (UTC)

[edit] Peano arithmetic

This sections is lacking in historical info, who first formalised it in this way? The text quotes Kaye 1991, but mathworld has two references with the name before that date:[1]

• Kirby, L. and Paris, J. "Accessible Independence Results for Peano Arithmetic." Bull. London Math. Soc. 14, 285-293, 1982.
• Paris, J. and Harrington, L. "A Mathematical Incompleteness in Peano Arithmetic." In Handbook of Mathematical Logic (Ed. J. Barwise). Amsterdam, Netherlands: North-Holland, pp. 1133-1142, 1977.

I found that the formula

1. 

might need a little typesetting. It took a second reading to realise that z was not a fuction. --Salix alba (talk) 19:21, 23 March 2007 (UTC)

The idea that addition and multiplication of naturals can be obtained from just successor dates back to at least 1920 (Skolem, "Foundations of elementary arithmetic"). The idea of expressing arithmetic as a first-order rather than second-order theory was certainly known in the 1930s and probably before; it was a subject of interest because the Lowenheim-Skolem theorem had just been proved and its meaning was not yet understood. Kaye (1991) is just used as a contemporary standard reference; the subject of first-order arithmetic is extremely classical. You're right that the article would benefit from a paragraph on this. CMummert · talk 19:57, 23 March 2007 (UTC)