## July 19, 2021

### Large Sets 12.5

#### Posted by Mike Shulman

Previously: Part 12. Next: Part 13

Last time Tom told us about McLarty’s replacement axiom for ETCS, but mentioned that there are several other equivalent axioms in the literature, due to Osius, Lawvere, Cole, and myself. In this addendum I want to discuss and compare those other axioms, and explain why I prefer my own (which is essentially a modification of McLarty’s to a collection axiom rather than a replacement axiom).

Here is Osius’s axiom:

If $\Phi(X,A,m,B)$ is a formula, where $X,A,B$ are set-variables and $m:A\to X$ is a function-variable, then for any set $X$ there exists a set $Y$ such that for any set $A$ and monomorphism $m:A\rightarrowtail X$, if there exists a set $B$ such that $\Phi(X,A,m,B)$, then there exists such a $B$ with $B\le Y$.

Here is Lawvere’s axiom:

If $\Phi(A,B)$ is a formula with set-variables $A,B$ such that whenever $\Phi(A,B)$ and $\Phi(A,B')$ we have $B\cong B'$, then for any set $X$ there exists a set $Y$ such that $X\lt Y$ and for any $A$ and $B$ with $A\lt Y$ and $\Phi(A,B)$, we have $B\lt Y$.

Here is Cole’s:

If $\Phi(A,B)$ is a formula with set-variables $A,B$, then for any set $X$ there exists a set $Y$ such that for any set $A\lt X$, if there exists a set $B$ with $\Phi(A,B)$, then there exists such a $B$ with $B\lt Y$.

And here is mine, from Comparing material and structural set theories (a 2018 paper that revises the relevant half of the 2010 preprint):

If $\Phi(i,S)$ is a formula with an element-variable $i$ and a set-variable $S$, and $I$ is a set such that for all $i\in I$ there exists a set $S$ with $\Phi(i,S)$, then there exists a surjection $p:V\twoheadrightarrow I$ and a $V$-indexed family of sets $q:A\to V$, such that for all $v\in V$ we have $\Phi(p(v), q^{-1}(v))$.

Of these, Lawvere’s is the only one that I would call a reflection principle. According to the informal description on wikipedia of the latter:

Suppose that we have some collection A of methods for forming sets (for example, taking powersets, subsets, the axiom of replacement, and so on)…. we can introduce the following new principle for forming sets: “the collection of all sets obtained from some set by repeatedly applying all methods in the collection A is also a set”.

Lawvere’s axiom is essentially the specialization of this principle to a partial-functional formula $\Phi$: given a set $X$, the collection of all sets obtained by repeatedly applying $\Phi$ is bounded in cardinality by some set $Y$. More precisely, there is a set $Y$ larger than $X$ such that the collection of sets $\lt Y$ is closed under applying $\Phi$.

Cole’s axiom looks similar to Lawvere’s, but there is a crucial difference that $Y$ is only required to be larger than the sets $B$ obtained by applying $\Phi$ to sets $A$ that are smaller than $X$. In other words, it’s only a “one-step” closure. This is more like a replacement axiom than a reflection axiom, but since there is no uniqueness of $B$, it’s more of a collection axiom.

Informally, the ZF replacement axiom says that the image of (the elements of) a set under a functional formula is (the elements of) a set, while the ZF collection axiom says that the image of (the elements of) a set under any entire formula contains a set onto which the formula is still entire. Cole’s axiom says similarly that the image of the collection of sets smaller than some set $X$ under some entire formula contains the collection of sets smaller than some other set $Y$ onto which the formula is still entire.

In other words, while the ZF replacement and collection axioms are element-to-element, Cole’s collection axiom is set-to-set. (Although, admittedly, the distinction is blurry in ZF, where elements of sets are themselves also sets.) Similarly, McLarty’s replacement axiom is element-to-set: the image of the elements of a set under a functional formula is a family of sets indexed by a set. Finally, Osius’s axiom is subset-to-set collection, while mine is element-to-set collection.

A further difference is that Osius’s, Lawvere’s, and Cole’s axioms all “collect a bunch of sets” by finding them as the sets smaller than some given set, while McLarty’s and mine “collect a bunch of sets” by finding them as the elements of a family indexed by some other set (which, recall, in ETCS means the fibers of a function).

All of these axioms are equivalent over ETCS. Which is best? Well, aesthetics differ, but you won’t be surprised to hear that I prefer my own. Let me try to explain why.

First, I think the “family” approach is a better match for the actual practice of mathematics, where we use families of sets all the time. The collection of all sets smaller than some given one is interesting in set theory, but doesn’t pop up very frequently in, say, topology or algebra. This argues for McLarty’s axiom and mine over those of Osius, Lawvere, and Cole.

Secondly, the family approach generalizes better to weaker theories. To my knowledge, from the Osius-Lawvere-Cole axioms we can only deduce a family-style axiom in the presence of something like the powerset axiom, so that for any $X$ there is an indexed family of sets (namely the family of subsets of $X$) containing a set of each isomorphism class $\lt X$.

For these reasons, I think McLarty’s axiom and mine are both preferable to those of Osius-Lawvere-Cole. For comparing McLarty’s axiom and mine, we have to think about the difference between replacement and collection.

To start with, collection axioms also generalize to weaker theories better than replacement axioms. While collection and replacement axioms are equivalent over ETCS or Zermelo set theory, in weaker theories collection axioms tend to be strictly stronger, and the extra strength is often important. This is especially true in ZF-like theories, whose replacement axioms insist that the formula $\Phi$ be functional up to strict equality of sets, while in mathematical practice we rarely characterize a set absolutely uniquely, only uniquely up to isomorphism. In the presence of the other axioms of ZF you can play games to turn uniqueness-up-to-iso into literal uniqueness, but the trick can fail in weaker theories, so it’s better to assume a collection axiom from the get-go.

This issue doesn’t arise for McLarty’s axiom in ETCS, which does only require uniqueness up to isomorphism. But this observation leads to another reason I prefer my axiom, which is that there’s something very odd about McLarty’s. Usually in category theory when something is unique, it’s unique up to unique isomorphism, but McLarty only asks for uniqueness up to non-unique isomorphism. A little thought should suffice to observe why this must be so: when something is unique up to unique isomorphism, the isomorphism is generally only unique such that it commutes with or preserves some other given structure. But in McLarty’s axiom there is no other structure, so it will very rarely happen that the isomorphism is literally unique — only when the sets $S$ have no nonidentity automorphisms. To my knowledge the only place this happens in practice is the proof that Tom gave that replacement implies separation, since empty sets and terminal sets have no nonidentity automorphisms.

However, this odd feature of McLarty’s axiom means that it essentially incorporates some amount of the axiom of choice. Intuitively, if for each $i\in I$ there is a set $S$ such that $\Phi(i,S)$, then in order to produce the desired family $p:X\to I$ we have to choose, for each $i\in I$, a particular set $S_i$, and then define $X = \sum_i S_i$. Often in category theory we find ourselves “choosing” objects that are unique up to unique isomorphism, and there’s a sense in which such choices are not a “real” use of the axiom of choice; but when the isomorphisms are not unique, there really is some choice involved.

I don’t know of an “intrinsic” way to argue for this (such as a proof in ETCS-minus-choice that McLarty’s axiom implies some amount of choice). But one thing we can say is that the category of sets in ZF does not (at least, not obviously) satisfy McLarty’s axiom.

Why? To prove a categorical axiom like this from the ZF replacement axiom, we need to make a membership-based set that’s strictly unique. We can do this with a varation of Scott’s trick: given $\Phi(i,S)$, let $\Psi(i,V)$ assert that $V = V_\alpha$, where $\alpha$ is the least von Neumann ordinal such that there exists an $S\in V_\alpha$ such that $\Phi(i,S)$. Then $V$ is strictly unique given $i$, and so we can use ZF replacement. However, this doesn’t give us an $I$-indexed family of sets $S_i$ such that $\Phi(i,S_i)$; it gives us an $I$-indexed family of families of sets $(S_{i j})_{j\in J_i}$ such that $\Phi(i,S_{i j})$ for each $i,j$. Here $J_i$ is the set of all $S\in V_\alpha$ such that $\Phi(i,S)$. Thus what we’ve actually proven is my collection axiom, with $J = \sum_i J_i$. (This proof can also be factored through a proof that the ZF replacement axiom implies the ZF collection axiom.) The only way I know of to get McLarty’s axiom from here is to apply the axiom of choice to the surjection $p:J\twoheadrightarrow I$.

Thus, again, if we care about weaker theories without choice, then a collection axiom is more appropriate. In my paper I showed that in ETCS-minus-choice, my collection axiom implies a “unique isomorphism” version of McLarty’s axiom: given any “additional structure” that a set could have, if for each $i\in I$ there is a set $S$ with that structure such that $\Phi(i,S)$, and such an $S$ is unique up to unique isomorphism preserving the structure, then there is a family $p:X\to I$ with that structure “fiberwise over $I$” such that $\Phi(i,p^{-1}(i))$ for each $i$. (The same is true in even weaker theories than ETCS-minus-choice, as long as my “collection of sets” axiom is augmented by a similar “collection of functions” axiom.)

One might argue that this latter “unique isomorphism” replacement property is a better axiom. For instance, it’s more justified by the intuitive idea of replacement; collection is a bit harder to explain from that perspective. And I’m sympathetic to that point of view. But on the other hand, giving a precise formulation of “additional structure” is kind of technical and involved, whereas the collection axiom can be stated much more simply. And the passage to a cover in the collection axiom isn’t that strange if you’re familiar with Kripke-Joyal semantics — but that’s a story for another day.

Finally, sometimes it happens that one really does need collection, not just replacement. For instance, suppose we’re working with categories, or higher categories, or homotopical structures, built out of sets, and we’ve constructed an “operation” on them that’s unique up to some kind of equivalence, like a projective resolution or fibrant replacement. Now suppose we also want apply this “operation” to families of such structures all at once. Such a construction won’t be unique even up to non-unique isomorphism, so we can’t even apply McLarty’s replacement axiom. But with collection, we can build a family of results for each object in a family of structures, and then try to put them together in an appropriate way.

This wouldn’t necessarily be an argument for assuming a collection axiom directly if it were easy to prove collection from replacement. However, I don’t know of a structural proof that collection implies replacement! The only way I know of to do it is ZF-theoretic, using a $V_\alpha$ argument as above. So if you want to prove in ETCS that McLarty’s axiom implies mine, as far as I know the only way to do it is to first show that ETCS+R is equivalent to ZFC and then run the ZFC proof. (By contrast, McLarty’s axiom follows easily from mine plus choice.) This is another reason why I prefer to assume collection to start from.

Posted at July 19, 2021 6:35 PM UTC

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### Re: Large Sets 12.5

Very nice! Among other things, it’s great to have all those statements of replacement side by side and in comparable notation, so that we can compare and contrast them.

(In my copy of Osius’s paper, some parts are scanned so badly that it’s only just possible to read them — including, unfortunately, the part where he discusses replacement.)

Posted by: Tom Leinster on July 20, 2021 1:43 PM | Permalink | Reply to this

### Re: Large Sets 12.5

The same is true of mine. Perhaps it was the same scan. (-:

Posted by: Mike Shulman on July 20, 2021 1:47 PM | Permalink | Reply to this

### Re: Large Sets 12.5

Typo, I think: in the statement of your own replacement axiom, the codomain of $p$ should be $I$.

Posted by: Tom Leinster on July 20, 2021 4:27 PM | Permalink | Reply to this

### Re: Large Sets 12.5

Fixed, thanks.

Posted by: Mike Shulman on July 20, 2021 5:56 PM | Permalink | Reply to this

### Re: Large Sets 12.5

one thing we can say is that the category of sets in ZF does not (at least, not obviously) satisfy McLarty’s axiom.

I wasn’t aware of that.

As McLarty emphasizes that his version of replacement is not motivated by the ZF axiom (but rather by Cantor’s original proposal), I suppose he wouldn’t view that as a drawback. But independently of what’s seen as good or bad, what you say here is an interesting mathematical point.

Posted by: Tom Leinster on July 20, 2021 5:40 PM | Permalink | Reply to this

### Re: Large Sets 12.5

It looks like McLarty’s axiom is equivalent over ZF to Cardinal Representatives, Form 97 in “Consequences of the Axiom of Choice”: For every set $A$ there is a function $c$ with domain $P(A)$ such that for all $x,y\in P(A),$ (i) $c(x)=c(y)\iff x\cong y$ and (ii) $c(x)\cong x.$ Jech’s “Axiom of Choice” Theorem 11.3 shows that this fails in a certain model of ZF.

Posted by: Harry West on July 20, 2021 6:54 PM | Permalink | Reply to this

### Re: Large Sets 12.5

Wow, thanks! So McLarty’s axiom really does incorporate “some amount of choice”, at least in the sense that it is unprovable in ZF.

Posted by: Mike Shulman on July 20, 2021 7:02 PM | Permalink | Reply to this

### Re: Large Sets 12.5

I went to add a link to this discussion to the nlab page on the replacement axiom and was reminded that there was a lengthy discussion of replacement axioms on the categories mailing list back in 2008, in case anyone is interested in the history.

Posted by: Mike Shulman on July 20, 2021 7:20 PM | Permalink | Reply to this

### Re: Large Sets 12.5

And here’s Todd’s answer to a MathOverflow question of Andrej Bauer, “Where in ordinary math do we need unbounded separation and replacement?” It refers back to a Café conversation between Todd and Mike, which in turn links to the nLab page on transfinite construction of free algebras.

Posted by: Tom Leinster on July 21, 2021 1:45 PM | Permalink | Reply to this

### Colin

Michael, could you say exactly what you mean by my version of replacement? As far as I can see, my published 2004 version is just a notational variant of your 2008 version. We just use different names for the relevant indexed families of sets, and their fibers.

Posted by: Colin S McLarty on July 21, 2021 6:12 PM | Permalink | Reply to this

### Re: Colin

Hi Colin, nice to see you here! By your axiom I mean the one that Tom quoted in the last post:

Let $I$ be a set and $\Phi(i,S)$ a formula such that for every $i \in I$ there exists a set $S$ such that $\Phi(i,S)$ and any other $S'$ such that $\Phi(i,S')$ is bijective to $S$. Then there exist a set $X$ and a function $p: X \to I$ such that $\Phi(i, p^{-1}(i))$ for all $i \in I$.

This is essentially the same as your axiom from p48 of “Exploring categorical structuralism”. It’s different from mine that I quoted above:

If $\Phi(i,S)$ is a formula with an element-variable $i$ and a set-variable $S$, and $I$ is a set such that for all $i\in I$ there exists a set $S$ with $\Phi(i,S)$, then there exists a surjection $p:V\twoheadrightarrow I$ and a $V$-indexed family of sets $q:A\to V$, such that for all $v\in V$ we have $\Phi(p(v), q^{-1}(v))$.

The difference is that mine doesn’t ask for uniqueness of $S$ but does allow passage to a cover $p$.

Posted by: Mike Shulman on July 21, 2021 7:04 PM | Permalink | Reply to this

### Colin

Ah, yes. But of course my version only asks for uniqueness of $S$ up to isomorphism, since my version of ETCS can only ever describe a set up to isomorphism. And I think this answers your question of how to derive your version from mine: just use the identity function on $I$ as a cover. It also refutes any concern that my version might imply something not probable in ZFC, since the ZFC version of replacement gives uniqueness tout court, not just up to isomorphism.

Posted by: Colin S McLarty on July 21, 2021 7:40 PM | Permalink | Reply to this

### Colin

Oh, I see I missed the importance of non-uniqueness. I will have to think more about this.

Is there a way to just delete a reply?

Posted by: Colin S McLarty on July 21, 2021 8:16 PM | Permalink | Reply to this

### Re: Colin

For comparing yours to mine, the problem is that $S$ in my axiom is not assumed to be unique even up to isomorphism. So you can’t just apply your axiom to the same $\Phi$.

For comparing to ZF, in both axioms uniqueness is an assumption, not a conclusion. So your axiom is stronger than the ZF axiom, since it has a weaker assumption (uniqueness only up to isomorphism). Harry West suggested above that your axiom implies Cardinal Representatives, which is unprovable in ZF.

Posted by: Mike Shulman on July 21, 2021 7:45 PM | Permalink | Reply to this

### Colin

Henry West’s suggestion is correct: For any set $A$ define the set $K$ of cardinality equivalence classes of subsets of $A$. Let the relation $\Phi$ relate each element $y\in K$ to each of its elements. That is to each subset $x\subseteq A$ with $x\in y$. Then my replacement axiom gives an indexed family over $K$ where the fiber over any $y\in K$ has the cardinality of each of the $x\in y$.

Posted by: Colin S McLarty on July 21, 2021 9:30 PM | Permalink | Reply to this

### Colin

On the other hand, ETCS already has the whole Axiom of Choice. I have no practical interest in set theories without choice. (The case of a well-pointed topos without choice is for me a fine part of general topos theory, even an especially valuable theoretical fact, but with no motivation to make it a foundation for mathematics.)

What my Replacement importantly incorporates, when compared with your Collection, is a case of the Scott Trick as you note. I think it is a very good idea to use Replacement directly, rather than Collection plus Scott Trick. It is simpler and more natural.

And I seriously do not want to adjoin any version of the whole Axiom of Foundation which is normally used in ZF to justify the Scott Trick. The Axiom of Foundation is famously unused in mathematics outside of research set theory, and dropping it is no actual loss even for that research: All of ZFC remains available within my ETCS+Replacement, in the form of the theory of well-founded extensional trees and Mostowski’s collapsing lemma. Indeed the entire ZFC theory of forcing is available here that way. And note ZFC forcing itself already relies centrally on Mostowski collapse–it is no foreign concept to ZFC theorists. These things are just not put into the ETCS+R axioms directly.

Posted by: Colin S McLarty on July 21, 2021 11:48 PM | Permalink | Reply to this

### Scott’s trick in ETCS+[McLarty’s R]

Hi Colin,

thanks for chiming in. Cardinal Representatives seems to me to be a weak instance of Scott’s trick, is the whole thing available in ETCS+R (as you define it)? This seems somewhat unlikely, and also slightly tricky to sort out, given that proper classes (or rather, things as large as the whole category of sets) aren’t a thing inside ETCS.

Posted by: David Roberts on July 22, 2021 1:19 AM | Permalink | Reply to this

### Colin

Hi David,

Cardinal Representatives as Henry West states it is not about proper classes. The global problem of selecting one representative set for every cardinality in the universe of sets is about a proper class, of course. But this version of Cardinal Representatives is the local problem of taking any set $A$, and choosing one set of each cardinality up to that of $A$.

That can be done in ETCS, using the Axiom of Choice without using Replacement. But it can also be done, as West suggested, in the fragment of ETCS+R without Choice, by using Replacement.

Posted by: Colin S McLarty on July 22, 2021 2:37 AM | Permalink | Reply to this

### Re: Scott’s trick in ETCS+[McLarty’s R]

Ah, ok. I checked in Jech’s Axiom of Choice, since it was on my shelf, and saw that Theorem 11.3 mentions a global cardinality function. My mistake!

Posted by: David Roberts on July 22, 2021 2:51 AM | Permalink | Reply to this

### Colin

No, part of my 11:48 note is still backwards. This much was correct: To get my Replacement incorporates some Choice as you said, in the sense that to get my Replacement from Your Collection takes Choice (which ETCS has anyway).

But here is the correction: Your Collection (and not my Replacement) incorporates part of the Scott Trick, in the sense that something like Scott must be used to get my Replacement from your Collection (i.e. some way of taking a potentially proper class of relata to a given $x$, and getting just a set of them).

Anyway, for me the key remains: Replacement (as I give it) is used in some math outside of research set theory (albeit few results actually need it). The Scott Trick (let alone the whole Axiom of Foundation) is not used outside research set theory.

And as I said in 11:48 is correct: the whole idea of Foundation remains available in my ETCS+R anyway since that theory suffices to define well-founded extensional trees and prove Mostowski Collapse.

Posted by: Colin S McLarty on July 22, 2021 1:53 AM | Permalink | Reply to this

### Re: Colin

The Scott Trick (let alone the whole Axiom of Foundation) is not used outside research set theory.

That’s ironic, since I only asked because I have a theorem (joint with Martti Karvonen) in pure category theory in the absence of Choice, about when a category is concretisable via a condition of Isbell. Freyd’s original proof uses global choice, well-ordering the class of all sets. We can remove Choice, but having Scott’s trick (or at least a filtration of the objects of $\mathbf{Set}$ into sets, indexed by a well-order) seems essential to the proof. We do not need ZFC, and work in algebraic set theory instead, positing a boolean well-pointed pretopos as the category of classes with the usual properties.

Posted by: David Roberts on July 22, 2021 2:48 AM | Permalink | Reply to this

### Re: Colin

Your Collection (and not my Replacement) incorporates part of the Scott Trick, in the sense that something like Scott must be used to get my Replacement from your Collection (i.e. some way of taking a potentially proper class of relata to a given $x$, and getting just a set of them).

I assume you mean “…to get your Collection from my Replacement”. It’s true that (as I said in the last paragraph of the post) some sort of Scott trick seems to be necessary to prove any sort of collection from any sort of replacement. And I agree that we shouldn’t have to recourse to foundation and Scott-like tricks when doing structural set theory.

But I don’t think it is correct to conclude from this that “collection incorporates the Scott trick”. I would conclude instead that replacement is too weak, so that to get the necessary and useful axiom of collection from it one has to invoke something like the Scott trick. Collection is the axiom one needs and wants; the fact that one needs foundation to prove it from replacement is an indictment, not of collection, but of replacement.

Posted by: Mike Shulman on July 22, 2021 5:09 AM | Permalink | Reply to this

### Colin

This discussion has been extremely helpful to me in working through these issues. But when you say “Collection is the axiom one needs and wants” I must ask who wants it for what?

Above, you mention research on weaker fragments of set theory as using Collection. I know of that mostly through metamathematics of Peano Arithmetic, Reverse Mathematics, and Adrien Mathias’s work related to MacLane set theory. Those indeed often use Collection. But all of those are explicitly about the comparative proof theory of a great many different axiom systems. None of those subjects are offered as giving an axiom system that should really be your foundation for mathematics. And the provability results in them hold no matter what foundation you actually believe (as long as it includes at least, say, Peano Arithmetic so that a good agile version of Godel Completeness is provable).

On the ZF front, I certainly agree current research on, say, the Continuum Hypothesis, needs to have Collection – but that research also needs the Axiom of Foundation so the distinction between Collection and Replacement is nugatory there.

Indeed, today’s ZF set theorists agree research on CH needs to have the denial of the Axiom of Constructibility (either as an axiom or, more likely, as a consequence of axioms). But I do not think you want to adopt that as an axiom in your set theory. Am I right?

I do not believe any textbook before me right now on algebra, topology, or analysis needs Collection. They often use Replacement in a lazy way – where Separation would have sufficed with a bit of care– and sometimes they really use Replacement.

Those textbooks are in fact my reasons for discussing Replacement in ETCS. Do you find uses of Collection on that level?

I wonder how the condition that Dave Roberts talks about, in his post marked July 22, 2021 2:48 AM, relates to Replacement and Collection.

Posted by: Colin S McLarty on July 22, 2021 6:05 AM | Permalink | Reply to this

### Re: Colin

Constructive mathematicians are a class of people who do actual mathematics in weaker axiom systems (rather than just studying their metatheory), and in general they’ve found that they do need collection. In particular, the foundational axiom systems called “Intuitionistic Zermelo-Frankel” and “Constructive Zermelo-Frankel” do explicitly include collection.

As far as uses of collection in classical mathematics, I gave one example in the penultimate paragraph of the post. Although one could probably take issue with it in various ways.

I do want to emphasize, though, that in the grand scheme of things I think what we’re discussing here is pretty minor. Your replacement axiom is, in my opinion, much better than the Osius-Lawvere-Cole axioms, and an entirely appropriate replacement axiom for ETCS (which, as you say, already includes choice anyway). It’s also well motivated (I agree that it states Cantor’s idea plainly in categorical language) and practically useful for most of the (few) places where a classical mathematician needs replacement.

I would say my axiom is a fairly small modification of yours — basically the “obvious” way to turn it from a replacement axiom into a collection axiom — and I prefer it mainly because I like to think about weaker theories, where collection is more needed and where even the small amount of choice incorporated in your axiom should be avoided. I wrote this post because Tom mentioned the existence of other replacement-like axioms for ETCS without stating any of them, and I wanted to take the opportunity to write about them all and, yes, advertise mine a bit for the benefit of a like-minded audience. But I realize that by emphasizing the differences between the two axioms more than the similarities, and making unnecessarily provocative statements about what “one” needs and wants, I may have created the impression of more conflict/disagreement than really exists, and if so I apologize.

Posted by: Mike Shulman on July 22, 2021 6:47 AM | Permalink | Reply to this

### Colin

Certainly you must not apologize, and perhaps I should for getting too wrapped up (besides for perpetrating some hasty errors). The real valuable point is not to choose one axiomatization but to get clear on all the possibilities, and what they mean. This discussion has been great for me that way.

Posted by: Colin S McLarty on July 22, 2021 11:38 PM | Permalink | Reply to this

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