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1. • CommentRowNumber1.
   • CommentAuthorDavidRoberts
   • CommentTimeDec 20th 2011
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   Author: DavidRoberts
   Format: MarkdownItexConsider the well known fact: $Set_{\lt \lambda}$ is a model of ZFC for $\lambda$ inaccessible and we are working in the ambient context of a set theory with AC. But what is $Set_{\lt \lambda}$ when we don't assume AC in the ambient theory, where the relation $\le$ is not total? My naive guess is the collection of sets which admit an injection to $\lambda$ but no isomorphism, which is one generalisation of $\lt$ in this setting. The other generalisation involves sets which admit a surjection from $\lambda$ but no isomorphism. I think this may have been discussed somewhere, possibly in relation to [[universe in a topos]], but the discussion there only touches the two extremes: an elementary topos and $Set$ (with AC). Perhaps I just need to assume the existence of a universe as at [[universe in a topos]], but (shock! horror!) is there a material way to see how to do this?

   Consider the well known fact: $Set_{\lt \lambda}$ is a model of ZFC for $\lambda$ inaccessible and we are working in the ambient context of a set theory with AC. But what is $Set_{\lt \lambda}$ when we don’t assume AC in the ambient theory, where the relation $\le$ is not total? My naive guess is the collection of sets which admit an injection to $\lambda$ but no isomorphism, which is one generalisation of $\lt$ in this setting. The other generalisation involves sets which admit a surjection from $\lambda$ but no isomorphism.

   I think this may have been discussed somewhere, possibly in relation to universe in a topos, but the discussion there only touches the two extremes: an elementary topos and $Set$ (with AC).

   Perhaps I just need to assume the existence of a universe as at universe in a topos, but (shock! horror!) is there a material way to see how to do this?

2. • CommentRowNumber2.
   • CommentAuthorMike Shulman
   • CommentTimeDec 20th 2011
   • (edited Dec 20th 2011)
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   Author: Mike Shulman
   Format: MarkdownItexAre you asking what *should* the definition of "$Set_{\lt \lambda}$" be without AC? I would be inclined to say that in the absence of AC, the definition of "inaccessible" should be "a collection of cardinalities closed under X, Y, Z operations".

   Are you asking what should the definition of “$Set_{\lt \lambda}$” be without AC? I would be inclined to say that in the absence of AC, the definition of “inaccessible” should be “a collection of cardinalities closed under X, Y, Z operations”.

3. • CommentRowNumber3.
   • CommentAuthorDavidRoberts
   • CommentTimeDec 20th 2011
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   Author: DavidRoberts
   Format: MarkdownItexYes, that's what I'm asking. I also trying to get $Set_{\lt\lambda}$ to be a model of $ZF+ \neg AC$, so I don't think I can take it to consist only of cardinalities, assuming that cardinals are defined as certain ordinals. Actually, perhaps I just need to stop worrying and just say I have a model of $ZF+ \neg AC$ - this constitutes a set, hence an internal category in my ambient $Set$. Hmm, am I worrying over nothing?

   Yes, that’s what I’m asking. I also trying to get $Set_{\lt\lambda}$ to be a model of $ZF+ \neg AC$, so I don’t think I can take it to consist only of cardinalities, assuming that cardinals are defined as certain ordinals.

   Actually, perhaps I just need to stop worrying and just say I have a model of $ZF+ \neg AC$ - this constitutes a set, hence an internal category in my ambient $Set$. Hmm, am I worrying over nothing?

4. • CommentRowNumber4.
   • CommentAuthorTobyBartels
   • CommentTimeDec 20th 2011
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   Author: TobyBartels
   Format: MarkdownItexIf you don't accept $AC$, then you should never assume that [[cardinals]] are defined as certain ordinals; that's of the devil. Even though you're happy (in this context) with excluded middle, it seems that constructivists are the only people who've seriously dealt with how properties of a cardinal number $\lambda$ need to be replaced with properties of the class of cardinalities less than $\lambda$, characterising these properties on their own (and noting that only with $AC$ can one recover $\lambda$ from a class with these properties). There is some discussion at [a MathOverflow question that I asked](http://mathoverflow.net/questions/50143/cardinal-numbers-vs-collections-of-cardinal-numbers).

   If you don’t accept $AC$, then you should never assume that cardinals are defined as certain ordinals; that’s of the devil.

   Even though you’re happy (in this context) with excluded middle, it seems that constructivists are the only people who’ve seriously dealt with how properties of a cardinal number $\lambda$ need to be replaced with properties of the class of cardinalities less than $\lambda$, characterising these properties on their own (and noting that only with $AC$ can one recover $\lambda$ from a class with these properties). There is some discussion at a MathOverflow question that I asked.

5. • CommentRowNumber5.
   • CommentAuthorMike Shulman
   • CommentTimeDec 20th 2011
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   Author: Mike Shulman
   Format: MarkdownItexToby said what I meant — absent AC, "cardinality" has to mean "equipotence class of sets" or perhaps some representative thereof.

   Toby said what I meant — absent AC, “cardinality” has to mean “equipotence class of sets” or perhaps some representative thereof.

6. • CommentRowNumber6.
   • CommentAuthorTobyBartels
   • CommentTimeDec 20th 2011
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   Author: TobyBartels
   Format: MarkdownItexBut there's another thing that Mike meant, which also needs to be said (and which Mike did say). So we really have *two* points to remember when dealing with large cardinals in the absence of full AC (and so a fortiori constructively): * A [[cardinal number]] is a bijection class of sets, which is not necessarily the collection of sets in bijection with some given ordinal number and so cannot necessarily be identified with the smallest such ordinal number. * A [[large cardinal]] is a downwards-closed collection of cardinal numbers, which is not necessarily the collection of cardinal numbers less than some given cardinal number and so cannot necessarily be identified with that cardinal number. Actually, even with AC, it is a mistake (albeit an understandable one) to identify a large cardinal (thought of as a collection of cardinal numbers) with the smallest cardinal number outside of it. The reason is that ‘downwards-closed’ ought to be interpreted as closed under taking quotient sets rather than subsets (and constructively, only quotient sets with respect to decidable equivalence relations). In particular, given any large cardinal $\kappa$, the collection of all *positive* cardinal numbers in $\kappa$ is also a large cardinal. So even with AC, to specify a large cardinal, you must specify both a cardinal number and a truth value. My evidence for the correctness of this definition is that there are constructions typically done with large cardinals that are also done with (at the very least) the collection $\{1\}$, as for example at [[familial regularity and exactness]]. (Of course, when one imposes closure conditions on a specific kind of large cardinal, one might still want to insist that $0$ belong to it.)

   But there’s another thing that Mike meant, which also needs to be said (and which Mike did say). So we really have two points to remember when dealing with large cardinals in the absence of full AC (and so a fortiori constructively):

   • A cardinal number is a bijection class of sets, which is not necessarily the collection of sets in bijection with some given ordinal number and so cannot necessarily be identified with the smallest such ordinal number.
   • A large cardinal is a downwards-closed collection of cardinal numbers, which is not necessarily the collection of cardinal numbers less than some given cardinal number and so cannot necessarily be identified with that cardinal number.

   Actually, even with AC, it is a mistake (albeit an understandable one) to identify a large cardinal (thought of as a collection of cardinal numbers) with the smallest cardinal number outside of it. The reason is that ‘downwards-closed’ ought to be interpreted as closed under taking quotient sets rather than subsets (and constructively, only quotient sets with respect to decidable equivalence relations). In particular, given any large cardinal $\kappa$, the collection of all positive cardinal numbers in $\kappa$ is also a large cardinal. So even with AC, to specify a large cardinal, you must specify both a cardinal number and a truth value. My evidence for the correctness of this definition is that there are constructions typically done with large cardinals that are also done with (at the very least) the collection $\{1\}$, as for example at familial regularity and exactness. (Of course, when one imposes closure conditions on a specific kind of large cardinal, one might still want to insist that $0$ belong to it.)

7. • CommentRowNumber7.
   • CommentAuthorTobyBartels
   • CommentTimeDec 20th 2011
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   Author: TobyBartels
   Format: MarkdownItexOur article [[large cardinal]] doesn't reflect the discussion above, but I'll wait to see if anybody objects to it before I do a major edit.

   Our article large cardinal doesn’t reflect the discussion above, but I’ll wait to see if anybody objects to it before I do a major edit.

8. • CommentRowNumber8.
   • CommentAuthorDavidRoberts
   • CommentTimeDec 20th 2011
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   Author: DavidRoberts
   Format: MarkdownItex@Toby - Sounds good to me, be my guest.

   @Toby - Sounds good to me, be my guest.

9. • CommentRowNumber9.
   • CommentAuthorMike Shulman
   • CommentTimeDec 20th 2011
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   Author: Mike Shulman
   Format: MarkdownItexI'm not sure I agree with that as a statement about *all* large cardinals. An inaccessible, certainly, is best defined as a down-closed collection of cardinalities, but it's not obvious to me that *all* types of large cardinals like weakly compact, measurable, etc. are best considered as a collection of cardinalities below them rather than as a single cardinality (or even a single well-ordering-type) with a particular property. It's also not obvious to me that down-closed should *always* mean closed under quotients rather than under subsets, even if in some cases it certainly does.

   I’m not sure I agree with that as a statement about all large cardinals. An inaccessible, certainly, is best defined as a down-closed collection of cardinalities, but it’s not obvious to me that all types of large cardinals like weakly compact, measurable, etc. are best considered as a collection of cardinalities below them rather than as a single cardinality (or even a single well-ordering-type) with a particular property. It’s also not obvious to me that down-closed should always mean closed under quotients rather than under subsets, even if in some cases it certainly does.

10. • CommentRowNumber10.
    • CommentAuthorDavidRoberts
    • CommentTimeDec 20th 2011
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    Author: DavidRoberts
    Format: MarkdownItexCertainly cardinals defined by game-theoretic means would be tricky to define this way.

    Certainly cardinals defined by game-theoretic means would be tricky to define this way.

11. • CommentRowNumber11.
    • CommentAuthorDavidRoberts
    • CommentTimeDec 21st 2011
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    Author: DavidRoberts
    Format: MarkdownItexReally all I am after is a category $S$ of ZF-sets internal to a category $Set$ of ZF-sets, albeit one with a specific set violating AC (the one from Cohen's second model, essentially the 'countable pairs of socks' set). Anyway, maybe I need to go and think hard for a bit....

    Really all I am after is a category $S$ of ZF-sets internal to a category $Set$ of ZF-sets, albeit one with a specific set violating AC (the one from Cohen’s second model, essentially the ’countable pairs of socks’ set).

    Anyway, maybe I need to go and think hard for a bit….

12. • CommentRowNumber12.
    • CommentAuthorTobyBartels
    • CommentTimeDec 22nd 2011
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    Author: TobyBartels
    Format: MarkdownItexYes, in #3 where you write >Actually, perhaps I just need to stop worrying and just say I have a model of $ZF+ \neg AC$ - this constitutes a set, hence an internal category in my ambient $Set$. Hmm, am I worrying over nothing? this is probably exactly what you should say, unless you specifically want to link to established work on large cardinals (in which case you still have to be very careful to check whether their results use AC).

    Yes, in #3 where you write

    Actually, perhaps I just need to stop worrying and just say I have a model of $ZF+ \neg AC$ - this constitutes a set, hence an internal category in my ambient $Set$. Hmm, am I worrying over nothing?

    this is probably exactly what you should say, unless you specifically want to link to established work on large cardinals (in which case you still have to be very careful to check whether their results use AC).

13. • CommentRowNumber13.
    • CommentAuthorTobyBartels
    • CommentTimeDec 22nd 2011
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    Author: TobyBartels
    Format: MarkdownItex@ Mike #9: Weakly compact cardinals fit right in, and I even mentioned them in my MO question: A __weakly compact cardinal__ is an inaccessible cardinal $K$ (to be thought of as a collection of cardinal numbers) such that the height of a tree belongs to $K$ whenever every level's width belongs to $K$ and every branch's length belongs to $K$. Measurable cardinals are more interesting.

    @ Mike #9: Weakly compact cardinals fit right in, and I even mentioned them in my MO question: A weakly compact cardinal is an inaccessible cardinal $K$ (to be thought of as a collection of cardinal numbers) such that the height of a tree belongs to $K$ whenever every level’s width belongs to $K$ and every branch’s length belongs to $K$. Measurable cardinals are more interesting.

14. • CommentRowNumber14.
    • CommentAuthorDavidRoberts
    • CommentTimeDec 22nd 2011
    • (edited Dec 22nd 2011)
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    Author: DavidRoberts
    Format: MarkdownItex@Toby #12 - I suppose assuming the existence of a model is a large cardinal axiom of a sorts, but no, I don't need to specifically link to large cardinals (at this point, and hopefully never). I don't know anything about assuming existence of models if $ZF + \neg AC$ is the ambient set theory. Otherwise I may just have to go down the completely sheaf-theoretic road, which is ok.

    @Toby #12 - I suppose assuming the existence of a model is a large cardinal axiom of a sorts, but no, I don’t need to specifically link to large cardinals (at this point, and hopefully never). I don’t know anything about assuming existence of models if $ZF + \neg AC$ is the ambient set theory. Otherwise I may just have to go down the completely sheaf-theoretic road, which is ok.

15. • CommentRowNumber15.
    • CommentAuthorMike Shulman
    • CommentTimeDec 22nd 2011
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    Author: Mike Shulman
    Format: MarkdownItex@Toby 13: Sure. The point I was making is that it's not clear to me that *all* large cardinals fit that framework; certainly some that are larger than inaccessibles do. Maybe the page [[large cardinal]] could point out that sometimes, or even "often", this is the case, but just not assert that it is *always* the case. (Actually, it's not even clear to me what a good constructive definition of "measurable cardinal" would be. Maybe the right thing to look at is elementary embeddings directly, which might not correspond to any sort of cardinality condition.)

    @Toby 13: Sure. The point I was making is that it’s not clear to me that all large cardinals fit that framework; certainly some that are larger than inaccessibles do. Maybe the page large cardinal could point out that sometimes, or even “often”, this is the case, but just not assert that it is always the case. (Actually, it’s not even clear to me what a good constructive definition of “measurable cardinal” would be. Maybe the right thing to look at is elementary embeddings directly, which might not correspond to any sort of cardinality condition.)