Talk:Cardinal assignment

Merger

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

Scott cardinals do not satisfy Moschovakis' original definition of "cardinal assignment"

When Moschovakis defined the "Problem of Cardinal Assignment" (4.20 in his book), the first condition is that ⁠$A=_{c}\vert A\vert$⁠. This is satisfied by the von Neumann cardinal assignment and the trivial weak cardinal assignment ⁠$\vert A\vert =A$⁠, but not by Scott cardinals. Moschovakis introduced Frege cardinals in 12.41, and claimed that ⁠$A=_{c}\vert A\vert$⁠ 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 $X\mapsto |X|$, such that $A$ is equinumerous with $B$ if and only if $|A|=|B|$, for any sets A and B. Some authors also include the requirement that $|X|\sim X$."

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 ⁠$\vert X\vert \sim X$⁠, 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 ⁠$\vert A\vert \equiv A$⁠ and use ⁠$\sim$⁠ as ⁠$=_{c}$⁠) and the "strong cardinal assignment" approach (where we use von Neumann cardinals and use ⁠$=$⁠ as ⁠$=_{c}$⁠). 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)