Timeline for Set theories without "junk" theorems?
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23 events
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| Jul 27, 2020 at 1:57 | comment | added | Toby Bartels | @Tom Leinster : Agreed that this is a poor medium, and we could move to chat (or the nforum) if you want. I guess that while (1) is true as far as I know, you can hardly say anything worthwhile in ETCS without abuse of (or extremely detailed) notation (and this is also true of ZFC, just not quite as much). The question is whether the notation that we adopt, whatever that is, to allow us to say what we want, also allows us to say ‘junk’ that we don't want. I see no difference between ETCS and ZFC in this regard. | |
| Jul 25, 2020 at 23:54 | comment | added | Tom Leinster | MO comment threads are a crappy medium for conversations like this, so while I've enjoyed your stimulating thoughts, Toby, I'm going to make this my last comment. So: (1) No one has been able to come up with an example of a "junk" theorem in ETCS except by abusing notation. (2) As a practical matter, notation must sometimes be abused - but wisely! (3) In ETCS, it's important to distinguish between a set $R$ and a subset $i:R\to S$ (e.g. $\mathbb{R}\to 2^\mathbb{Q}$); both are defined up to iso, but iso means different things for each. In ETCS, "subset" is structure rather than property! | |
| Jul 25, 2020 at 22:23 | comment | added | Toby Bartels | (And notice that every one of these statements involves an abuse of notation in ETCS, so that is not the source of the problem. Sorry, now I'm done.) | |
| Jul 25, 2020 at 22:22 | comment | added | Toby Bartels | And ETCS is the same; because of how the Axiom of Infinity is written in the two set theories, we don't have this particular junk theorem, but we still have the other ones. Write $3\in\{q\in\mathbb{Q}\;|\;3<\pi\}$ or $3<\pi$, and nobody minds; but write $3\in\pi$, and it's a problem. Yet these all mean the same thing if you define real numbers as lower cuts of rational numbers, a definition that is equally natural in any set theory that (like both ZFC and ETCS) doesn't axiomatize the real numbers directly. | |
| Jul 25, 2020 at 22:17 | comment | added | Toby Bartels | What I really disagree with is your claim that abuse of notation is orthogonal to junk theorems. To the contrary, I think that notation is key. Nobody objects to $\{\{\},\{\{\}\}\}\in\{\{\},\{\{\}\},\{\{\},\{\{\}\}\}\}$, which is obvious (if you don't get lost in the braces), and nobody objects to $2<3$ (which my 4-year-old understands); they only object to $2\in3$. In ZFC with the von Neumann natural numbers, all of these mean the same thing; it's how you write this that determines whether or not it looks like junk. | |
| Jul 25, 2020 at 22:11 | comment | added | Toby Bartels | I agree that the junk theorems that people complain about have non-junky content. The content of $3_{\mathbb{Q}} \in \pi_{\mathbb{R}}$ is that $3 < \pi$. The content of $2_{\mathbb{N}} \in 3_{\mathbb{N}}$ in ZFC is that $2 < 3$. (Is $\in$ always secretly $<$? No, such as if you define real numbers as upper cuts, but it does seem pretty common.) Given a formal definition with apparent junk theorems, you can always tease out what they really mean, and that's something with real content. So I agree about that. But I don't think that this means that junk theorems are entirely unproblematic. | |
| Jul 25, 2020 at 22:01 | comment | added | Toby Bartels | We should be able to agree on most things in mathematics, but let's try to find the areas of disagreement. There is a canonical membership relation from $\mathbb{Q}$ to $\mathbf{2}^{\mathbb{Q}}$, and this pulls back to a relation from $\mathbb{Q}$ to $R$ given any function $R \to \mathbf{2}^{\mathbb{Q}}$; so, if you define $\mathbb{R}$ as a subset of $\mathbf{2}^{\mathbb{Q}}$, then there is a canonical membership relation from $\mathbb{Q}$ to $\mathbb{R}$. And if you define $\mathbb{R}$ using lower cuts, then the rational number $3$ is related to the real number $\pi$ under this relation. | |
| Jul 23, 2020 at 17:15 | comment | added | Tom Leinster | It seems to me that the question of when mathematicians choose to abuse notation, and when it's safe to do so, is separate. We all try to abuse notation to just the right degree, and we constantly make value judgements about what that is. But that's orthogonal to the question of "junk" theorems in ETCS. (Also: I find the emotive term "junk" unhelpful. All mathematical statements have content; it's a question of what it is.) | |
| Jul 23, 2020 at 17:06 | comment | added | Tom Leinster | So in ETCS, there is no canonical membership relation on $\mathbb{R} \times \mathbb{Q}$ enabling one to ask "is $3 \in \pi$?". As of course you know (and I'm aware you do know all this!), the sets $\mathbb{R}$ and $\mathbb{Q}$ are only defined up to isomorphism, like everything else in ETCS. In order to get such a relation, you have to make an explicit choice of embedding $i: \mathbb{R} \to 2^\mathbb{Q}$. Sure, you could choose to abuse notation to obscure your choice of $i$ from your readers, but why would you? | |
| Jul 23, 2020 at 17:00 | comment | added | Tom Leinster | @TobyBartels I'm not sure we disagree very much. I agree that, in your setting, $3 \in \pi$ is a (loosely-stated) theorem. But it's not a "junk" theorem, because the accurate statement is $3\in(j:\pi\to\mathbb{Q})$ where $j$ is the map implementing the Dedekind cut. That has a perfectly sensible meaning. The point is that having defined $\mathbb{R}$ via the subobject $i:\mathbb{R}\to2^\mathbb{Q}$, we're free to treat it as an abstract set (as we almost always do) or as equipped with the map $i$ to make it into a subobject of $2^\mathbb{Q}$ (which we only do when interested in Dedekind cuts). | |
| Jul 23, 2020 at 7:31 | comment | added | Toby Bartels | In particular, if $A$ is $\mathbb{Q}$, $\mathcal{T}$ is the Scott topology under $\geq$, $x$ is $3$, and $G$ is $]-\infty,\pi[$, then $3 \in G$. But if we represent real numbers as lower subsets of $\mathbb{Q}$, then $\mathcal{T}$ is the set of extended real numbers (or extended lower real numbers if I'm being constructivist), $G$ is $\pi$, and whatever abuse of notation allowed us to write $3 \in G$ before now allows us to write $3 \in \pi$. | |
| Jul 23, 2020 at 7:31 | comment | added | Toby Bartels | I mean, suppose that I'm trying to do topology, and I say that a topological structure on a set $A$ is a collection of subsets of $A$ satisfying certain conditions, and then I start writing things like ‘Let $A$ be a set, let $\mathcal{T}$ be a topology on $A$, let $x$ be an element of $A$, let $G$ be an element of $\mathcal{T}$, and consider whether $x \in G$’, then you might object that this is an abuse of notation, but whether you write it that way or add pedantic decorations, this is a thing that mathematicians talk about. | |
| Jul 23, 2020 at 7:03 | comment | added | Toby Bartels | @TomLeinster : If you're going to use ETCS to do ordinary mathematics, then you've got to adopt some kind of notation (or abuse thereof) to have collections of concrete sets, where the elements of some abstract set are interpreted as subsets of some other abstract set to which elements of the latter abstract set might belong. In whatever notation you use for that, $3 \in \pi$ is a theorem (assuming that you construct real numbers as lower sets of rational numbers). | |
| Jul 23, 2020 at 0:26 | comment | added | Tom Leinster | 8 years on, I maintain that there are no "junk theorems" in ETCS. The examples mentioned by @TobyBartels and Steven G only look like junk because of abuses of notation. For Toby's example of $3\in_{\mathbb{Q}}\pi$, conscientious notation would be $3 \in_{\mathbb{Q}}(j: \pi\to\mathbb{Q})$, where $j$ is your preferred injection implementing the Dedekind cut. But there's nothing junky there. It's a perfectly meaningful statement about a particular map $j$. While "$3 \in \pi$" looks like junk, the proper form "$3 \in j$" plainly isn't. Similar comments apply to Steven's example. | |
| Mar 26, 2018 at 21:31 | comment | added | Carl Mummert | On the other hand, when we define an element of a set $X$ to be a map $1 \to X$, then we can find $\pi$ as a map $1 \to \mathbb{R}$ and we can then ask "is $\pi$ surjective?" Any time we define an object of some kind to be an object of a second kind, we will have junk theorems unless we are unable to ask any questions about objects of the second kind that cannot be asked about objects of the first kind. | |
| Mar 26, 2018 at 21:29 | comment | added | Carl Mummert | In second-order arithmetic, we have both an $S$ unary function and a $\in$ binary relation, but $1 \in S$ is not a well formed formula, because each side of $\in$ must be a term of type 0, and $S$ on its own is not a term at all. The same would happen in the extension of ZFC with a larger signature - $S$ on its own would not be a term. @Guillaume Brunerie | |
| Mar 13, 2012 at 6:33 | comment | added | Toby Bartels | Tom, in ETCS there are two distinct meanings of $x \in y$. In one of these, $y$ is an abstract set (an object of the category $Set$) and $x$ is an element of $y$ as you said. In the other, there is some abstract set $z$ lying around but unmentioned, $x$ is an element of $z$, and $y$ is a subset of $z$ (meaning a monomorphism to $z$). This latter sense may be written $x \in_z y$, but ordinary mathematical practice abused the notation. If real numbers are encoded in ETCS as lower subsets of rational numbers, then $2_\mathbb{Q} \in_\mathbb{Q} 3_\mathbb{R}$ is a theorem. | |
| Mar 12, 2012 at 21:00 | comment | added | Steven Gubkin | (although this particular example has a nice interpretation: r(q) =0 if r>q and r(q) = 1 if r<=q, say.) | |
| Mar 12, 2012 at 20:43 | comment | added | Steven Gubkin | @Tom - You are right, but there will still be junk theorems about the embeddings of one set into another. For example, if you are representing the real numbers as a monomorphism to 2^Q, you are going to have theorems about "evaluating a real number at a rational", which wouldn't make sense to most people out of context. | |
| Mar 12, 2012 at 15:37 | comment | added | Tom Leinster | @Toby: I'm not sure what you mean. I suspect you and I have different systems in mind. In ETCS, I'd define an element of a set $X$ to be a map $1 \to X$. So for "$3 \in \pi$" to make sense, $\pi$ would have to be a set and $3$ would have to be a map $1 \to \pi$. And this isn't the case. @abo: for the same reason, I don't see any evidence of junk theorems in ETCS. | |
| Mar 11, 2012 at 16:26 | comment | added | Toby Bartels | The language about natural numbers isn't in ETCS to avoid junk statements about natural numbers but to serve in an axiom of infinity. But you make a good point, that there can still be junk statements involving things like real numbers. For example, if a real number is defined as a lower set of rational numbers, then we have such junk theorems as $3 \in \pi$. We even have $2 \in 3$, where here $2$ is the rational number $2$ and $3$ is the real number $3$. (The abuse of language here is essentially the same as in material set theory.) | |
| Mar 11, 2012 at 15:21 | comment | added | Guillaume Brunerie | "The same effect could be achieved, by modifying ZFC by introducing the same primitives and assuming, on top of the normal ZFC axioms, the Peano Axioms." No, even if you add $0$, $\mathbb{N}$ and $S$ as primitives, you will still be able to ask whether $\mathbb{N}\in S$ which is a "junk question" (and its answer will be a junk theorem) | |
| Mar 11, 2012 at 15:10 | history | answered | abo | CC BY-SA 3.0 |