Our best mathematical accounts of infinity exhibit a shocking lack of absoluteness—the very same set can be countable in one set-theoretic world, uncountable in another, and yet finite elsewhere.
I'm trying to write about Löwenheim-Skolem, just to mention it briefly for now (mostly through analogy). This post is great, I want to include a link for people that want more info if you don't mind!
Zermelo–Fraenkel set theory, combined with first-order logic, aimed to give a precise account about the nature of infinite collections, with the ultimate goal to settle all controversial questions originating from "naive" set theory.
About a century later, we find that models of this theory are malleable like clay. We are looking at those models from an outside perspective, where we have confidence if some specific set is "genuine finite", "genuine countable", or "genuine uncountable", and have a lot of fun constructing models that would disagree from the inside perspective.
But how can we justify our confidence? For a finite set, I can look at some explicit syntactical representation, e.g, {{},{{}}}, which is clearly finite. I am unable to do the same for some infinite set, yet it seems easy to think about some very specific countable infinite set, e.g. the set X of all finite Von-Neumann ordinals, and convince myself that X is indeed equinumerous to the "genuine natural numbers". I can futher convince myself that the set Y of all subsets of X is uncountable.
We can do all this using our naive intuitive, semantic notion of a set is, outside of any formal theory, just like Cantor did. But of course, nothing prevents us from formalizing these statements in first order logic, and derive a purely syntactical a proof from the ZF axioms. Thus, by completeness, all models (including the most lunatic ones) will agree on the fact that X is indeed countable, and Y is indeed uncountable.
There is no paradox here, since from my outside perspective, I see that each model will point at different "genuine sets" X' and Y', claiming that these sets would be exactly those that I refer to as X and Y in my proof.
Now here is the catch: We may think we have pretty good understanding what the "real" sets X,Y look like. But do we? I for my part have some mental model of X and Y in my mind, but I am unable write down an explict representation (as in using nested braces like for a finite set) to show it to you. Instead, the best I can do is to try and give a precise definition. Likely, the most precise way to do this would be using some formal notions from set theory. And here we are back to square one.
This poses a philosophical problem: Is there such a thing as the "real X,Y" ? Is there even such a thing as a "genuine countable" set?
The platonist answer would be, that there is indeed, and anything else is just the weakness of first order logic. This was pretty much my own view when I first heard about Skolem's paradox. Today, I am not so sure anymore. Maybe it is more appropriate to think of "countability" as some structural concept that entails specific consequences, and just dismiss our outsider perspective at all? In the end, what prevents me from replacing the term "as seen from my outside perspective" by "as seen from a sufficiently strong formal model"? But this would be pure denial of the fact that I am able to reason about these sets in a rather informal way.
Thank you for your thoughtful comment. My view is that Skolem's paradox shows us how someone, even a strong platonist, could be fundamentally mistaken about what they think of as "genuine" finite sets. We can imagine someone living inside one of these models M or W that I mention in the post, viewing a set as uncountable and convinced that it really is uncountable, even though from another perspective it is seen as countable, or worse, looking at a set that is thought to be clearly finite, even though from another perspective it is uncountable. I believe that when imagining finite sets we rarely consider extremely large sets explicitly, but rather understand their vastness in a schematic manner that is subject to the Skolem paradox. The paradox shows how we can create perfectly robust accounts of finiteness that have all the features that the platonists would want (namely, set theory from the perspective of a model W), and which nevertheless are still seen as wrong from the perspective of another foundational framework. Therefore, how can the platonist be convincing about the absolute nature of finite/infinite, if everything they might say could also be true in a context in which their conclusion is wrong?
I'm trying to write about Löwenheim-Skolem, just to mention it briefly for now (mostly through analogy). This post is great, I want to include a link for people that want more info if you don't mind!
Totally fine! And why don't you post a link here to your post, when it is ready?
Here it is hot off the press:
https://erincarmody.substack.com/p/bb2f390c-10b5-4231-9c2e-1e1a06cc912b
Let me know if any comments!
Thank you for this wonderful article!
Zermelo–Fraenkel set theory, combined with first-order logic, aimed to give a precise account about the nature of infinite collections, with the ultimate goal to settle all controversial questions originating from "naive" set theory.
About a century later, we find that models of this theory are malleable like clay. We are looking at those models from an outside perspective, where we have confidence if some specific set is "genuine finite", "genuine countable", or "genuine uncountable", and have a lot of fun constructing models that would disagree from the inside perspective.
But how can we justify our confidence? For a finite set, I can look at some explicit syntactical representation, e.g, {{},{{}}}, which is clearly finite. I am unable to do the same for some infinite set, yet it seems easy to think about some very specific countable infinite set, e.g. the set X of all finite Von-Neumann ordinals, and convince myself that X is indeed equinumerous to the "genuine natural numbers". I can futher convince myself that the set Y of all subsets of X is uncountable.
We can do all this using our naive intuitive, semantic notion of a set is, outside of any formal theory, just like Cantor did. But of course, nothing prevents us from formalizing these statements in first order logic, and derive a purely syntactical a proof from the ZF axioms. Thus, by completeness, all models (including the most lunatic ones) will agree on the fact that X is indeed countable, and Y is indeed uncountable.
There is no paradox here, since from my outside perspective, I see that each model will point at different "genuine sets" X' and Y', claiming that these sets would be exactly those that I refer to as X and Y in my proof.
Now here is the catch: We may think we have pretty good understanding what the "real" sets X,Y look like. But do we? I for my part have some mental model of X and Y in my mind, but I am unable write down an explict representation (as in using nested braces like for a finite set) to show it to you. Instead, the best I can do is to try and give a precise definition. Likely, the most precise way to do this would be using some formal notions from set theory. And here we are back to square one.
This poses a philosophical problem: Is there such a thing as the "real X,Y" ? Is there even such a thing as a "genuine countable" set?
The platonist answer would be, that there is indeed, and anything else is just the weakness of first order logic. This was pretty much my own view when I first heard about Skolem's paradox. Today, I am not so sure anymore. Maybe it is more appropriate to think of "countability" as some structural concept that entails specific consequences, and just dismiss our outsider perspective at all? In the end, what prevents me from replacing the term "as seen from my outside perspective" by "as seen from a sufficiently strong formal model"? But this would be pure denial of the fact that I am able to reason about these sets in a rather informal way.
Thank you for your thoughtful comment. My view is that Skolem's paradox shows us how someone, even a strong platonist, could be fundamentally mistaken about what they think of as "genuine" finite sets. We can imagine someone living inside one of these models M or W that I mention in the post, viewing a set as uncountable and convinced that it really is uncountable, even though from another perspective it is seen as countable, or worse, looking at a set that is thought to be clearly finite, even though from another perspective it is uncountable. I believe that when imagining finite sets we rarely consider extremely large sets explicitly, but rather understand their vastness in a schematic manner that is subject to the Skolem paradox. The paradox shows how we can create perfectly robust accounts of finiteness that have all the features that the platonists would want (namely, set theory from the perspective of a model W), and which nevertheless are still seen as wrong from the perspective of another foundational framework. Therefore, how can the platonist be convincing about the absolute nature of finite/infinite, if everything they might say could also be true in a context in which their conclusion is wrong?