A Reference From Andrej Bauer’s Recent Talk on Countable Reals

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Andrej Bauer gave a talk today in the topos institute colloquium (video here) announcing a proof that the dedekind reals can be countable in the absence of LEM and CC.

At roughly the 27 minute mark, he mentions a "nice constructive news post" by Ingo Blechschmidt in which he reviews "all the ways that don't work". This question, then, is twofold:

First, is there a link to this note somewhere? I would love to read it, but after some cursory googling I haven't been able to find it.

Second, and relatedly, is there an actual "constructive news" website where this note was posted? I know that a lot of discussions happen in Zulip chats, and I wouldn't be surprised to learn that this note was posted in one of those. It would be nice if there were a slightly more "traditional" place where new updates in constructive mathematics were posted, but again, some cursory googling hasn't brought anything up.

Thanks in advance, and sorry for the particularly soft question! ^_^

Best Answer

The constructive news google group doesn’t appear in Google search results, it seems. The post was from June 10, 2018, and is available at this link: https://groups.google.com/g/constructivenews/c/jSvzqu1LUis

Mathematical Definitions

You say that you are a physicist, so it's possible that you haven't been let in on the secret about mathematical definitions. If you have, skip this bit. If not, I'll tell you. (But, hush! It's a secret. Don't tell anyone else.)

I'll illustrate the point I wish to make with an example that I hope is familiar to you. What is the definition of a continuous map between metric spaces? I teach this, and I teach three definitions:

A function $f \colon M \to N$ is continuous if whenever $(x_n) \to x$ in $M$ then $(f(x_n)) \to f(x)$ in $N$. (Note: I chose metric spaces here, so sequences are sufficient.)
A function $f \colon M \to N$ is continuous if for every $x \in M$ and $\epsilon \gt 0$ then there is a $\delta \gt 0$ such that whenever $d_M(x,y) \lt \delta$, $d_N(f(x),f(y)) \lt \epsilon$.
A function $f \colon M \to N$ is continuous if whenever $U \subseteq N$ is open then $f^{-1}(U)$ is open in $M$.

These definitions are equivalent: they all agree which functions are continuous and which are not. So any statement made using one definition can be reformulated using another. But they have different uses, since they emphasise different aspects of what it means to be continuous. If you're interested in metric spaces because you use approximations then the first definition captures the idea of what you want to use: If I have an approximation of something, then after I hit it with a continuous function, it is still an approximation (of the image of the "something"). The second definition is actually the most practical when testing an explicit function for continuity: it's amenable to finding estimates and the like. Third is the most theoretically powerful: it's the first one we reach for when trying to prove theorems about continuous functions.

So just because a definition seems to be the "established" definition doesn't mean that that is the right way to think about it. Definitions are malleable, and we often use a different definition to the one that we truly believe is "right" simply because it is easier.

Topological Manifolds

Let me start with topological manifolds. I said at the beginning that the key here is that being a topological manifold is a property. That is, if I have a topological space then it either is or isn't a topological manifold. If it is, then I can use that fact when I study it; if it isn't, then I can't. If it is a topological manifold then I don't have to use that fact, but I'm likely to be making life difficult for myself if I don't. The key thing about a property is that if I ignore it, it is still there.

The people who study topological manifolds do so because many spaces of interest happen to be topological manifolds. If you change the definition, then those spaces will go on being locally Euclidean, and the people studying them will continue to use the fact that they are locally Euclidean, and all that will have changed is the language that they use. This is why I don't have much sympathy for your desire to change the definition.

Although most of the Euclidean structure doesn't have much topological influence on a topological manifold, it does provide a lot of useful tools in the analysis. For better or worse, Euclidean spaces are things that we simply know a lot about. So saying that a space is locally Euclidean means that we can use all our intuition and skills from the theory of Euclidean spaces to the study of the space. That's worth a lot, and you'd need to be very convincing to persuade people to give that up.

Now when thinking how to define a topological manifold, one encounters the question I alluded to in the above on definitions. Definitions come in all shapes and sizes. It's not clear from your question as to which definition you would like best. On the one hand, your dislike of the real line makes me think that you want the "pure" definition: the one that captures the soul of a topological manifold. I'll readily agree that the current definition is not that, it's more of the "body" type where it's easy to see how to use it. But the proto-definition that you give isn't that either: it's a mish-mash of topological concepts, each of which excludes a range of spaces, with the hope that in the end all you have left are the topological manifolds. I don't like that sort of definition, it's more of the $\epsilon$-$\delta$ type: has its place, but is neither the "soul" nor the "body".

However, what it feels most like is that you are playing that children's game where you have to explain what is an aardvark without using the words "aardvark", "anteater", "dictionary", or "pink panther".

An alternative is to come up with a definition that is actually different in that it doesn't completely agree with the current definition. In that case, your work is harder. You have to show why the new definition is better than the old one. The most convincing arguments would be either that your definition allows you to do more, or that it allows you to consider more spaces. But these are unlikely to both hold. If you allow more spaces, you probably lose out on abilities; if you find new tools, then you probably can't apply them to all the current topological manifolds. If you really want to do this then your best bet is not to mention topological manifolds at all, but to invent a wholly new concept, say "Topological foldimen" and simply say, "Topological manifolds that are X are foldimen, and foldimen that are Y are manifolds.". Then hope that there are plenty of interesting foldimen out there.

Smooth Manifolds

Smooth manifolds, on the other hand, are much more malleable. This is because being a smooth manifold is something a little bit extra. The standard definition of a smooth manifold starts with a topological manifold and then adds a little extra on top. Now, forgetting that extra does mean something. If you forget it, it goes away, and you can't be sure what it was.

As an illustration, if I have two topological manifolds, $X$ and $Y$, then the question "Is $X$ homemorphic to $Y$?" has the same answer if I remember that they are manifolds or not. But if I say that they are smooth manifolds, then the question "Is $X$ diffeomorphic to $Y$?" depends completely on their being smooth manifolds.

This actually gives us some room to manoeuvre. Because we need to construct the extra structure, we can consider different constructions. However, this is where your dislike of the reals counts against us. We cannot construct something from nothing: we need to start with something. There are many possible answers as to what that "something" is, but they all boil down to identifying certain spaces as "known" meaning that we decide what the structure for those spaces should be. Since they are "known", and everything else will be defined relative to them, they should be spaces that we really do know about. It's hard to get spaces that are more well-known than the Euclidean spaces. Certainly when calculus is concerned. All of the examples of this that I've seen have used Euclidean spaces, or "nice" subsets thereof.

So these are our "known" spaces, which we will use to define what a "smooth structure" means for "unknown" spaces. This is where we have some flexibility in the definition, and here is where we can get rid of that annoying "local stuff". Maybe, just maybe, we don't need to have actual charts and can get away with something weaker.

Actually, we can. No "maybe" about it. But the problem is that by weakening the definition, we end up with more things than we might like to admit to the hallowed halls of manifolddom. Nonetheless, there is some merit in pursuing this line as it separates out the construction aspect (of what "smooth" means) from the property aspect (of what a "manifold" is).

As I said, there are many approaches at this point. I'm going to outline one just so that you have one in mind. There are others, and this is not the place to evaluate them. If you don't like this one, the rest still carries through. I just want to be sure that you have one picture in your mind.

Here it is: it rests on the slogan that manifolds are all about smooth curves. If we both look at a manifold, we should agree on which curves in it are smooth and which are not. So one way to specify a "smooth space" is to give a list of all its smooth curves. One probably wants some conditions, but this can all be made precise. Thus a "smooth space" is a space in which we all agree on smooth curves.

As I said, there are other approaches, but whatever they are they still give us a list of the smooth curves. This will be important in a minute.

Now it turns out that this admits far more than just manifolds. In fact, too much. There are really weird spaces in our list now, and we'd like to get rid of them.

Remember the curves? Good. We're going to use them. Using the curves, one can define the tangent space of a "smooth space". Just the same as for a smooth manifold: derivatives of curves. Unlike smooth manifolds, this needn't be locally trivial, nor even the fibres vector spaces. There's also a notion of a topological tangent bundle, which is related to neighbourhoods of the diagonal.

Here's the definition:

A smooth manifold is a smooth space whose smooth and topological tangent bundles agree.

Now, I'm not 100% sure that this is exactly the same as "smooth manifold", but certainly all smooth manifolds fit this description, and it excludes smooth manifolds-with-boundary, but there might be the odd pathological case that this doesn't exclude. Nonetheless, it is a very powerful description: it implies that the tangent spaces are actual vector spaces - we didn't assume that, if you remember.

Conclusion

If you want to mess with definitions, go ahead: it's fun. But be careful that you know what type of definition you are aiming for. There are reasons that the current definitions are what they are and the reasons tend to be pragmatic: over the years, we've found certain definitions useful and others not. There is a lot of dead wood, and there are always new insights that shed new light on old concepts, but if you want to replace an old definition, then remember that it's been there for a while, working hard, and will strongly resist all attempts to "retire" it from service!

Looking for a recent reference for a result of Shelah

After some googling a proof of the consistency of $\mathfrak b<\mathfrak s$, which seems to be following Shelah's construction, can be found as theorem 4.5 in Uri Abraham's chapter of the handbook.

Best Answer

Related Solutions

[Math] Defining a manifold without reference to the reals

Mathematical Definitions

Topological Manifolds

Smooth Manifolds

Conclusion

Looking for a recent reference for a result of Shelah

Related Question