I'm trying to understand the difference between sets and proper classes. I found this PDF.

I'm having trouble in understand the following at page 115:enter image description here

It basically says that if a set of all cardinal numbers exist, then it has a cardinal number $k$. Then $k$ must be bigger than all cardinal numbers in the set of which it is the cardinal number, that is, the set of all cardinal numbers.

Why $k$ must be bigger? As I know, cardinal numbers are numbers related to sizes of sets. There's nothing about size in this definition.


marked as duplicate by Asaf Karagila cardinals Mar 6 '18 at 8:26

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This is a really badly written passage. As written, it suggests the following statement (or something along these lines):

If $X$ is a set of cardinals, then $\vert X\vert$ is greater than every element of $X$.

This is wildly false. E.g. the set $\{\aleph_{n+1}: n\in\mathbb{N}\}$ is countable, but all of its elements are uncountable!

Instead, we have to use something specific about this particular "set" of cardinals. This comes down to the axiom (scheme) of replacement, or more specifically transfinite recursion (which is a consequence of, and indeed equivalent to, replacement). Given any ordinal $\alpha$, we can "iterate" the cardinal successor operation along $\alpha$ (taking unions at limit stages), starting with (say) $\aleph_0$; e.g. if we take $\alpha=\omega+1$, we get the sequence $$\aleph_0, \aleph_1, \aleph_2, ..., \aleph_\omega.$$ Now suppose $\kappa$ is an arbitrary cardinal. Then ZFC proves the existence of $\kappa^+$, and by iterating the cardinal successor operation along $\kappa^+$ (viewed as an ordinal) we get a set $S_\kappa$ of distinct cardinals of cardinality $\kappa^+$; but $S_\kappa\subseteq Card$, so $Card$ has cardinality $\ge\kappa^+>\kappa$, and we're done.

It may be worth at this point thinking about what set theory would look like without replacement. If you drop replacement from ZFC (resp. ZF), you get Zermelo set theory with choice, ZC (resp. without choice, Z). This theory is extremely weak; for example, it can't even prove the existence of the ordinal $\omega+\omega$ or of the set $V_\omega$!

In fact, it gets worse. The set $V_{\omega+\omega}$ is a model of ZC (this isn't hard to check), and the only cardinals it contains are $0, 1, 2, 3, ...,\omega$. This means that in fact $V_{\omega+\omega}$ believes that there is a set of all cardinals! This tells us:

The statement "the class of all cardinals is not a set" requires the axiom (scheme) of replacement.

...OK, this is misleading since what's really going on is that the usual definition of "cardinal" is the wrong one for Zermelo set theory. However, there is still a sense in which Replacement is necessary for the class of cardinals to be "as big as it should be": in some models of ZC, such as $V_{\omega+\omega}^L$, there are only "set-many cardinals" in the sense that there is some definable surjection from some set to the class of cardinals.

My point here really is that there is more to the statement "the class of cardinals is not a set" than may immediately be clear from the quoted passage.

  • $\begingroup$ "Extremely weak" seems like a bit of an overstatement. I would add that in the context of ZC, the von Neumann definition of ordinals is not so useful but there are other definitions of ordinals (and therefore of cardinals) that work much better. $\endgroup$ – Eric Wofsey Mar 5 '18 at 3:15
  • $\begingroup$ I think that in the context of set theory, not even being able to prove that the collection of hereditarily finite sets is a set is indeed extremely weak (and this doesn't come down to a "right definition of ---" issue). But you're right that there are definitions of "cardinal" etc. that work better without replacement; my point was just that the statement in question is more subtle than the passage implies. $\endgroup$ – Noah Schweber Mar 5 '18 at 3:21
  • $\begingroup$ I do not think the last paragraph is a fair description. Obviously one does not identify cardinals and initial ordinals without replacement, and under the standard conventions in the lack of replacement, the class of cardinals is proper. $\endgroup$ – Andrés E. Caicedo Mar 5 '18 at 3:59
  • $\begingroup$ @AndrésE.Caicedo That's fair, the point I was trying to make was weaker; I've edited. $\endgroup$ – Noah Schweber Mar 5 '18 at 4:02
  • $\begingroup$ What's the difference between "false" and "wildly false"? (I'm not joking, or trying to be an ass; it seems like a genuinely interesting issue.) I think I understand "obviously false", "clearly false", "universally false", but I'm not sure what "wildly" is supposed to suggest. $\endgroup$ – MJD Mar 5 '18 at 19:44

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