Is there any literature, especially when doing mathematics without the axiom of choice, that discusses using collections of cardinal numbers in place of individual cardinal numbers, when discussing cardinal numbers with certain properties?

That's a little vague, and I will presently give a few motivational examples that should help to clarify what I'm thinking of. But first, a quick definition of the translation between cardinal numbers and collections of cardinal numbers assuming the axiom of choice: If K is a cardinal number, then {L | L < K} is a collection of cardinals which is small and downward-closed. Conversely, if C is a collection of cardinals which is small and downward-closed, then there exists a unique cardinal K (to wit, the smallest cardinal that does not belong to C) such that C = {L | L < K}. So the correspondence between these two perspectives is straightforward —if we assume choice.

Now I'll consider some types of cardinal numbers. A weak limit cardinal is an infinite cardinal K such that L+ < K whenever L < K. A strong limit cardinal is an infinite cardinal K such that 2L < K whenever L < K. A regular cardinal is an infinite cardinal K such that Σi ∈ I Li < K whenever |I| < K and each Li < K. An inaccessible cardinal is an uncountable regular limit cardinal. A weakly compact cardinal is an inaccessible cardinal K such that the height of a tree is less than K whenever every level has width less than K and every branch has length less than K (the tree property). Etc.

Although these are all properties of an individual cardinal number K, they all refer to that cardinal only through the collection of smaller cardinal numbers. (The exceptions are the adjectives ‘infinite’ and ‘uncountable’, which are really only there to rule out trivial cases.)

If you don't assume the axiom of choice, then it's easy to still consider these conditions on a small, downward-closed collection of cardinals, but now this is more general than a collection of the form {L | L < K}. If you go beyond doubting the axiom of choice and drop the law of excluded middle (so doing constructive mathematics, in a moderate sense), you can get more flexibility by messing with the interpretation of ‘downward-closed’; for example, the collection {0, 1, 2, …} of all finite (in the strictest sense) cardinal numbers is closed under taking decidable sub-cardinals but not arbitrary sub-cardinals and so gives a ‘regular’ collection of cardinals which is different from {L | L < ℵ0}.

I'm interested in understanding what regular cardinals and inaccessible cardinals should be in constructive mathematics. (Bigger than that, I don't really even understand them in classical mathematics.) It seems to me that they have to be collections of cardinals rather than individual cardinal numbers. But this perspective already makes sense in classical mathematics, especially without the axiom of choice. Has anybody studied this?

  • $\begingroup$ Under the Axiom of Choice, all infinite cardinals satisfy your definition of "weak limit cardinal". $\endgroup$ – user5810 Dec 22 '10 at 9:47
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    $\begingroup$ I assume by L + 1 he means the successor cardinal. $\endgroup$ – arsmath Dec 22 '10 at 9:51
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    $\begingroup$ @Ricky: I agree with arsmath; I think Toby used L + 1 to mean successor cardinal since he talks about collections of cardinals. However, in that case, it becomes a little confusing because he also mentions that the cardinality of the set of cardinals less than $C$ under AC is $|C|$ which would be false for successor and singular cardinals $C$. $\endgroup$ – Jason Dec 22 '10 at 11:22
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    $\begingroup$ "The successor of an infinite cardinal" is not a well-defined notion without choice (there may be several immediate successors, there may be cardinals larger than L without minima). L+1 may be strictly larger than L even if L is infinite, precisely if L is Dedekind finite. If the issue is to pick a larger cardinal than L, that typically gives us a successor of L, then there is a canonical choice: Let κ be the first ordinal that does not inject into L, and consider L+κ. This suggestion works fine in the sense that strong limit cardinals are then weak limit cardinals as well. $\endgroup$ – Andrés E. Caicedo Dec 22 '10 at 16:48
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    $\begingroup$ I think "K = {L | L < K} if we interpret cardinal numbers as von Neumann ordinals" is only true if L ranges over ordinals, not cardinals as previously in the paragraph. $\endgroup$ – Mike Shulman Dec 22 '10 at 20:07

As mentioned, in constructive set theory we generally talk about large sets rather than large cardinals.

The rough idea is: a regular set is basically a transitive model of Replacement. An inaccessible set is basically a regular model of Exponentiation: if $a \in X$ and $b \in X$ then ${}^ab \in X$ (absent the law of the Excluded Middle this does not imply $Pow(a) \in X$). For the details I am glossing over, see e.g. Aczel and Rathjen's Notes on constructive set theory section 10.

It is provable in ZFC that the inaccessible sets in the constructive sense are exactly the $V_\kappa$ with $\kappa$ a strong inaccessible cardinal in the classical sense.

Of course we may ask what, in the name of the good lord that made the integers, is a "regular set", concretely, and I just happen to have a paper on arXiv on that (section 10). I have some private scribblings on extending this to inaccessibles. (It is entirely sane to believe in inaccessible sets but not $Pow(\mathbb{N})$...)

Aczel and Rathjen's survey also includes a definition for Mahlo sets - basically an inaccessible $X$ such that for any function $f\in {}^XX$ there is an inaccessible $Y \in X$ which is closed under $f$ (not quite accurate - see the paper for details). Over ZFC these are again the $V_\mu$ for $\mu$ a strongly Mahlo cardinal.

The paper Mike Shulman cited gives even larger set notions. These are in the context of impredicative set theory (IZF rather than CZF).

  • $\begingroup$ Do you mean second-order replacement in the second paragraph? (Otherwise, the claim in the third paragraph is generally false. If $\kappa$ is the first strong inaccessible, then there are plenty of $\alpha<\kappa$ such that $V_\alpha \prec V_\kappa$; these all satisfy first-order replacement and exponentiation.) $\endgroup$ – François G. Dorais Jan 3 '11 at 13:31
  • $\begingroup$ Maybe I glossed over too much... Yes, and actually, second-order Strong Collection. $\endgroup$ – Daniel Mehkeri Jan 3 '11 at 22:19
  • $\begingroup$ Thanks! I mean to look up Friedman & Scedrov (Mike's answer) too, but your answers are online, which is even better. (Your answers are also in a weaker foundation, which is also good.) $\endgroup$ – Toby Bartels Jan 8 '11 at 17:15

One reference I know of is "Large sets in intuitionistic set theory" by Friedman and Scedrov (APAL 27, 1984), which is about large-cardinal properties. However, for the reasons you cite, it phrases them in terms of "large sets" rather than "large cardinals."


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