Talk:Set-theoretic definition of natural numbers

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I added a discussion of the definition of Frege and Russell which works in naive set theory, type theory, and New Foundations.

Randall Holmes 18:02, 15 December 2005 (UTC)

[edit] This entry needs more and better content

Randall, I've given this article my usual polish but remain dissatisfied with it. The way you set out the Frege-Russell definition catches me off-guard. I do not have a sense that you have clearly demarcated Frege-Russell's way from Von Neumann's way. You did not mention that you were defining the Von Neumann ordinals, and that extracting the corresponding cardinals from these ordinals requires Choice. (The entry says this now, but only because I added it.) I am not confident that the two definitions in this article are killingly accurate, and maximal accuracy is important here, otherwise a poor reader might take Julius Caesar for a number, as per Frege's notorious worry! I'm adding a link to your wonderful 1998 text.

As I have told you before via another channel, the Frege-Russell way has always struck me as "just right." The fact that the resulting equivalence classes are not sets in ZFC does not damn Frege-Russell; rather it damns ZFC. A foundational system of mathematics should come with an elementary and intuitive definition of the finite cardinals, period. ZFC fails that test, NFU does not.202.36.179.65 18:13, 16 July 2006 (UTC)

[edit] Small edit

The description of the empty set in bracketts after the empty set symbol was a) wrong and b) unnecessary, so i replaced it with a link to the empty set page. I don't mean the description was incorrect it was just very poor english, and confusingly similar to saying "the set containing zero" which is of course incorrect. Triangl 01:12, 12 November 2006 (UTC)

[edit] Non-equivalence of "the empty set" and the symbol used before "(the empty set)"

The symbol used to describe the empty set is incorrect. {} denotes the empty set as does the symbol Ø. Then {Ø} is truly an non-empty set, because it is a set that contains the empty set. To formally define zero to be {Ø} might be true, but placing a link to the empty set after {Ø} is misleading. bradskins 22:21, 27 November 2006 (UTC)

I changed it, and you could have changed it as well. CMummert 11:54, 28 November 2006 (UTC)