Talk:Cardinal assignment

This article should be merged with Von Neumann cardinal assignment. JRSpriggs (talk) 20:14, 8 February 2010 (UTC)

When Moschovakis defined the "Problem of Cardinal Assignment" (4.20 in his book), the first condition is that ⁠${\displaystyle A=_{c}\vert A\vert }$⁠. This is satisfied by the von Neumann cardinal assignment and the trivial weak cardinal assignment ⁠${\displaystyle \vert A\vert =A}$⁠, but not by Scott cardinals. Moschovakis introduced Frege cardinals in 12.41, and claimed that ⁠${\displaystyle A=_{c}\vert A\vert }$⁠ is fundamental in Cantor's view of cardinals but not necessary in Frege's approach. He defined Scott cardinals in Problem x12.46 but never called them a "cardinal assignment".

How should we define the term "cardinal assignment"? Is there any other literature that uses this term? Bbbbbbbbba (talk) 11:38, 4 February 2026 (UTC)

@JRSpriggs @Farkle Griffen Since you have participated in the discussion about merging this article into cardinal number, do you have any opinions about this? Bbbbbbbbba (talk) 01:30, 5 February 2026 (UTC)

You're right that I can't find any other major authors using the phrase "cardinal assignment". If we had to define it, the best definition would probably be in terms of Hume's principle, which is a topic covered plenty of sources. Something along the lines of: "Cardinal assignment is the problem of defining a function ${\displaystyle X\mapsto |X|}$, such that ${\displaystyle A}$ is equinumerous with ${\displaystyle B}$ if and only if ${\displaystyle |A|=|B|}$, for any sets A and B. Some authors also include the requirement that ${\displaystyle |X|\sim X}$."

The second requirement isn't necessarily uncommon, but it is far less important than the first. – Farkle Griffen (talk) 19:38, 5 February 2026 (UTC)

If no other authors uses the phrase "cardinal assignment", then it maybe makes sense to use that phrase specifically for Moschovakis' version which requires ⁠${\displaystyle \vert X\vert \sim X}$⁠, and use something else (like "cardinality function" currently used in Cardinal number) for the more general definition. Bbbbbbbbba (talk) 16:49, 8 February 2026 (UTC)

If no other authors use the phrase, I would worry that an article dedicated to Moschovakis' definition wouldn't meet Wikipedia's notability guideline. – Farkle Griffen (talk) 19:04, 8 February 2026 (UTC)

This would be a further reason to merge this article to cardinal number. Now that I've thought more about it, I do feel that Moschovakis' concept of cardinal assignment is a not-quite-elegant way to try to reconcile the "weak cardinal assignment" approach (where we define ⁠${\displaystyle \vert A\vert \equiv A}$⁠ and use ⁠${\displaystyle \sim }$⁠ as ⁠${\displaystyle =_{c}}$⁠) and the "strong cardinal assignment" approach (where we use von Neumann cardinals and use ⁠${\displaystyle =}$⁠ as ⁠${\displaystyle =_{c}}$⁠). However, I think there is some value in acknowledging the "weak cardinal assignment" approach, especially since I believe the "strong cardinal assignment" approach is not yet known to be viable in the Zermelo set theory without replacement. Bbbbbbbbba (talk) 13:04, 14 February 2026 (UTC)