girl who is independent of ZFC :3 @monoidal-monoidoidoid - Tumblr Blog

if you ask a hypotheticalgirl if she exists she'll say "ummmm well maybe..."

Does anyone know if this result has an actual name in English? The German notes I'm working with call it "Verallgemeinerte Hauptraumzerlegung", but "Hauptraum" is already "generalized eigenspace" in English, so a direct translation leaves me with "Generalized generalized eigenspace decomposition"... I've found one resource that calls the spaces here "P_i-generalized eigenspaces", is that common?

For comparison, here's the result generally called "generalized eigenspace decomposition" in German.

yoneda-emma

I'm attending an "elementary geometry" lecture taught by a category theorist this semester and I've never felt so much like I was living through the chinese room hypothetical in real life

charl0ttan

wonderful bijective love

charl0ttan

what if all of me had a place both for and in all of you

ahno-nimus

it's a common misconception that five plus eight is thirteen. actually five plus eight equals the skull

evilwizard

SCARY MATH works in a whole different way that scientists don’t understand yet

yoneda-emma

> apply for a position doing exercise groups for a linear algebra lecture at university > find out at the first meeting that out of us six tutors three are trans > find out the next week that one of the remaining three is actually a closeted trans guy who has been waiting for someone to comment on the "he/him" sticker on his laptop so now it's actually four out of six :3

jabber-wock

zere are abelian groups???? zats so many groups

mathematikerin-freyja

Why are you doing algebraic geometry and not the superior math, which is convex analysis? Who needs sheafs anyway?! /j

Without the gentle but firm hand of category theory constantly resting on my lower back I feel unsupported and isolated in walking through the mathematical wilderness and my fundamentally nervous nature overwhelms me.

yoneda-emma

ordinals and cardinals truly are the coriolis force and/or entropy of set theory (in that they seem incomprehensible exactly until the moment you finally find an explanation willing to just lay out the math for you instead of purely relying on a nonsensical 'intuitive explanation')

yoneda-emma

so, first off, the standard construction of the natural numbers in modern set theory is that the number n is just the set n = {1, ..., n-1}, and then a < b just means 'a is an element of b'. did you know that the ordinal ω ('the smallest ordinal greater than all natural numbers') literally just *is* the set of all natural numbers? if you've internalized "a < b iff. a is an element of b", then that should be entirely obvious, since by definition ω has to be the smallest set containing all natural numbers! and then ω+1 is just {1,2,3,........, ω}, and so on! it's all just a very simple construction of either taking a set and adding the set itself as an element, or taking a union of the sets you've already built! (and then if you accept the axiom of choice you can just define a 'cardinal' to be an ordinal such that there is no bijection between that ordinal and any ordinal we had already built before that! in ZFC, cardinals are just a specific type of ordinal!)

second, the coriolis force is a force that anything moving on earth is subjected to, which deflects all movement on the northen hemisphere to the right and all movement on the southern hemisphere to the left. this concept has tormented me since high school geography, because every single 'intuitive' explanation only explains why north- or southward movement towards the north or south is deflected sideways, and entirely fails to explain why it also affects east- and westward movement. well, did you know that if you just write down "if you're standing on a rotating object, your velocity consists of your own movement plus the movement of the ground underneath you" and directly apply the product rule of calculus you just immediately get "acceleration = acceleration of the ground underneath you + centrifugal force + coriolis force"? the extra forces are entirely just because you get two separate terms when you take the derivative of a product!

and, lastly, did you know that entropy is just a measure of the likelihood that a given system of random variables ends up in a given state? the second law of thermodynamics ('in an isolated system, entropy never spontaneously decreases') is literally just 'if you roll six billion dice, you'll almost always get each result about one billion times, and you'll almost never get six billion sixes'!

yoneda-emma

ordinals and cardinals truly are the coriolis force and/or entropy of set theory (in that they seem incomprehensible exactly until the moment you finally find an explanation willing to just lay out the math for you instead of purely relying on a nonsensical 'intuitive explanation')

yoneda-emma

so, first off, the standard construction of the natural numbers in modern set theory is that the number n is just the set n = {1, ..., n-1}, and then a < b just means 'a is an element of b'. did you know that the ordinal ω ('the smallest ordinal greater than all natural numbers') literally just *is* the set of all natural numbers? if you've internalized "a < b iff. a is an element of b", then that should be entirely obvious, since by definition ω has to be the smallest set containing all natural numbers! and then ω+1 is just {1,2,3,........, ω}, and so on! it's all just a very simple construction of either taking a set and adding the set itself as an element, or taking a union of the sets you've already built! (and then if you accept the axiom of choice you can just define a 'cardinal' to be an ordinal such that there is no bijection between that ordinal and any ordinal we had already built before that! in ZFC, cardinals are just a specific type of ordinal!)

second, the coriolis force is a force that anything moving on earth is subjected to, which deflects all movement on the northen hemisphere to the right and all movement on the southern hemisphere to the left. this concept has tormented me since high school geography, because every single 'intuitive' explanation only explains why movement north or south is deflected sideways, and entirely fails to explain why it also affects movement to the east or west. well, did you know that if you just write down "if you're standing on a rotating object, your velocity consists of your own movement plus the movement of the ground underneath you" and directly apply the product rule of calculus you just immediately get "acceleration = acceleration of the ground underneath you + centrifugal force + coriolis force"? the extra forces are entirely just because you get two separate terms when you take the derivative of a product!

and, lastly, did you know that entropy is just a measure of the likelihood that a given system of random variables ends up in a given state? the second law of thermodynamics ('in an isolated system, entropy never spontaneously decreases') is literally just 'if you roll six billion dice, you'll almost always get each result about one billion times, and you'll almost never get six billion sixes'!

yoneda-emma

ordinals and cardinals truly are the coriolis force and/or entropy of set theory (in that they seem incomprehensible exactly until the moment you finally find an explanation willing to just lay out the math for you instead of purely relying on a nonsensical 'intuitive explanation')

yoneda-emma

To any set theory wizards following me: What's the least inconvenient way to add a hierarchy of universes to set theory, similar to the way dependent type theory does it? Specifically, I always felt that "resolving" Russels paradox by introducing proper classes and then only allowing quantification over sets was deeply unsatisfying, since on a philosophical level, this just pushes the issue up one step - I think it is intuitively clear that the "collection of all proper classes" should still be a valid object that exists, even if it can't be a proper class itself. I assume what I want is hyperclasses, but I'm having a real hard time finding basically any writing on them, and from what little I do find even these seem to usually just be considered as "collections of classes", and not a full hierarchy where an (n+1)-class is a collection of (n)-classes.... Are Grothendieck Universes (and/or Tarski Universes) already enough to solve this issue, even though they are formally just classified as sets? Also how come this isn't talked about more? I'd be fine switching out ZFC for some version of NBG or MK or even ETCS for this, but I do think that doing all of my future math in MLTT or HOTT because of this seems a bit unreasonable for now ^^

mystic-bone-mage

Admittedly this is not the sort of thing I think about in my daily set theoretic life, so take it with a set containing a grain of salt. However: what you might want are what I'll call α-sets. A set (in the sense of ZFC, or what have you) is called a 0-set. Having defined α-sets, an (α+1)-set is any collection of α-sets. Having defined α-sets for all α < γ limit, a γ-set is an α-set for some α < γ. For example, there is a 1-set of all 0-sets, a 2-set of all 1-sets, but no ω-set of all n-sets over all n < ω (that collection would be an (ω+1)-set).

I won't be considering the axiomatic framework of α-sets in depth right now, but if there's interest I can write something up another time (perhaps when I'm not on the toilet). I think the analogy with Grothendeick universes is a good one, but we're going to run into the same ontological issue here (ie, Grothendeick universes think they are proper classes, but we typically consider them sitting inside some bigger Grothendeick universe that thinks the smaller one is a set).

If we let V^α be the (α+1)-set of all α-sets, then for an inaccessible cardinal κ, we can perform essentially all of ordinary mathematics inside of V^{κ}. The emphasis is given to show that we've not actually solved the problem we were trying to address here; we have simply pushed it up κ many levels of abstraction, and for our effort we get to come up with a whole new axiomatic framework.

Okay, so maybe the solution to our issue will get spit out if we stop trying to have a "top node" to our hierarchy. Maybe considering the entire universe of α-sets for any α will give us what we want. If it doesn't, then I'm happy to tell you that the 1-set of all 0-ordinals is a 1-ordinal, that the (α+1)-set of all α-ordinals is an (α+1)-ordinal, and that we can do this construction over and over, transfinitely, considering α_β-sets indexed by the β-ordinals α. Of course, why stop there? Next we can consider α_{β_γ}-sets, where α is a β_γ-ordinal. We can do our transfinite iteration of transfinite iterations transfinitely many times! But that's where we should stop; I have a headache.

yoneda-emma

Ah, thank you! The whole "there is a 1-set of all 0-sets, a 2-set of all 1-sets, etc." is exactly what I was thinking myself (considering thats what agda does), and "Grothendieck universes think they are proper classes, but bigger Grothendieck universes think the smaller ones are sets" does sound reasonable as a resolution to "how do Groethendieck universes establish this hierarchy without going beyond proper classes" Your $V^α$ aren't directly related to Von Neumann universes, right? They just happen to use the same letter?

radicalideal

I think this is just the same as assuming there is a proper class of inaccessible cardinals {kappa_alpha}, and then an alpha-set is just an element of V_(kappa_alpha). Which is pretty tame as large cardinal axioms go.

yoneda-emma

Ah! I knew from Wikipedia that Grothendieck universes are equivalent to strongly inaccessible cardinals, but "an alpha-set is just an element of V_(kappa_alpha)" is a nice final puzzle piece, thank you!

janmusija

Here's a couple additional thoughts of mine you may find relevant:

If you care about minimizing consistency strength, "there is some definable class of ordinals (\kappa_\alpha) so that every set is contained in V_(\kappa_\alpha) for some \alpha" is of consistency strength well below an inaccessible cardinal. A very strong form of this property holds already within V_\kappa for \kappa strongly inaccessible. (the caveat here is that you need \kappa to be inaccessible if you want V_\kappa to be particularly nice in a second-order sense, so that it contains all of its own "small" subsets, rather than only e.g. all definable ones. Inaccessibles are definitely much nicer.)

Higher large cardinal axioms can sort of inform how you would extend the notion of "how would you add a hierarchy of universes to set theory". For example, a (strongly) inaccessible limit of inaccessibles is equivalent to the idea of "a grothendieck universe which satisfies 'every set is contained in a grothendieck universe' (which I have heard phrased as the 'grothendieck axiom' somewhere I think)". A strongly mahlo cardinal \kappa corresponds to a grothendieck universe W = V_\kappa where for every function f:W->W, there is some smaller grothendieck universe W* in W so that f keeps members of W* inside W*. These are both pretty weak axioms, though, on the scale of large cardinal strength.

On that topic, I think one comment I have about this-- and this is admittedly probably less of a concern if you view these as like "a stratified hierarchy of universes" rather than "a number of small initial segments of the universe that happen to model ZFC"-- is that it does kind of raise the point of "okay, but the moment you are doing math with the assumption that there is a stratified tower of universes like this, you can no longer carry out 'essentially all math' that you might want within any piece of that tower."

yoneda-emma

It does kind of raise the point of "okay, but the moment you are doing math with the assumption that there is a stratified tower of universes like this, you can no longer carry out 'essentially all math' that you might want within any piece of that tower."

This is only true if we define 'all math' as 'anything you can do in whatever specific axiom system you chose', right? And you just mean that theres fewer statements we can prove compared to the vast number of new statements we can't prove? I'd define 'all math' as something more like 'everything derivable from *any* set of axioms', and surely from that perspective adding more layers to the tower only increases the amount of math we can theoretically carry out?

yoneda-emma

And the statement "given any amount of any objects of any kind, the collection of all of those objects is its own distinct object" just seems like something that is so obviously true in a platonic sense that it would feel really strange to me to do math without it

janmusija

Hm, what do you mean by this exactly? For example, it is clear that for an arbitrary set theoretic sentence φ, {all objects satisfying φ} is not in general going to be an object (i.e. a set, which is the only kind of object in the domain of discourse in the context of a model of set theory)-- this was exactly what got naïve set theories in trouble back in the day.

[to ensure that this doesn't get too hidden in a different branch of the reblog tree: FWIW I more or less entirely agree with your statement about the meaning of "all math". I mostly just intended this to illustrate a point to other potential readers]

yoneda-emma

I was using 'object' in a loose natural language sense here, assuming that some objects can be inherently different 'types' of things than others. To me, "sets are the only kind of object in set theory" just feels demonstrably wrong in a platonic sense - as soon as we start speaking about sets, 'the collection of all sets' just inherently becomes a nebulous thing floating around in the aether, and I think that whatever system of logic we are working in should inherently be able to talk and formally reason about it. Russels paradox tells us that {all objects satisfying φ} cannot be a part of whatever collection we defined 'objects' to be, but I think it is entirely natural (to the point that not doing so feels extremely strange to me, which is kind of why this post got made in the first place) to conclude that that just means that our notion of 'object' is not enough to reason in a satisfactorily formal way, that {all objects satisfying φ} must be a new type of thing, let's say a '2-object', and that there must be a hierarchy such that (n+1)-objects are the collection of things that {all n-objects satisfying φ} belongs to.

janmusija

Right, but this is already what set theory does in the first place.

I had a related discussion with some of my discord friends on this subject, and here's a paraphrase of some of the stuff I said, as it is relevant:

The impulse to add progressively higher tiers of classes to V is just really making V taller. Let "Ord" be the collection of all ("set") ordinals, then note Ord would be an ordinal if it were not a class. In particular, we can take "Ord + 1" to be the higher order class Ord u {Ord}, etc...

Now the usual V of all sets (or, in your terms, the 2-object consisting of all 1-objects) is V_Ord by foundation. The usual semantics of second-order set theory gives us P(V) = V_(Ord+1) (a 3-object); higher order theories give P(P(V)), P(P(P(V))), etc, which are V_(Ord+2), V_(Ord+3)...

From an external context, you have, for example, an inaccessible cardinal k. Now (V_k,V_(k+1)) is a model of second order ZFC.

If you want to have a really rich theory of high-level "classes", the best way to do that is not to stack a bunch of complex new stuff on top of V, but to just pick a cutoff k where members of Vk are what you consider "sets", and members of V(k+a) are a-classes. But then why even bother with the cutoff at all? There is nothing really special about it what you are considering sets and what you are considering classes, then, other than that the cutoff happens to happen at (e.g.) an inaccessible cardinal.

Thus, in some sense, higher order set theories don't really add anything terribly new-- you just do first order set theory and then stretch out the top.

In other words, I guess my point is: from this perspective, the notion that "you can quantify over sets, and not over classes" or "the only objects are sets" is not a restriction on what you can do with classes, but a guarantee that the rich and powerful theory you can perform with sets also extends to any other "object" (because when all true "objects" are sets, any object is subject to this theory of sets). Thus I would say that this is the richer and more complete universe, in a sense.

#huh!#much to think about...

yoneda-emma

Kind of basic question while the more in-depth post is still going on, just to make sure I'm not totally off base: The collection of all sets that don't contain themselves *does* form a proper class, right? In most well-founded set theories that collection should just be the proper class of all sets, but I assume that even without well-foundedness it still qualifies? And so does any other collection of the form {x : x is a set satisfying P}?

yoneda-emma

To any set theory wizards following me: What's the least inconvenient way to add a hierarchy of universes to set theory, similar to the way dependent type theory does it? Specifically, I always felt that "resolving" Russels paradox by introducing proper classes and then only allowing quantification over sets was deeply unsatisfying, since on a philosophical level, this just pushes the issue up one step - I think it is intuitively clear that the "collection of all proper classes" should still be a valid object that exists, even if it can't be a proper class itself. I assume what I want is hyperclasses, but I'm having a real hard time finding basically any writing on them, and from what little I do find even these seem to usually just be considered as "collections of classes", and not a full hierarchy where an (n+1)-class is a collection of (n)-classes.... Are Grothendieck Universes (and/or Tarski Universes) already enough to solve this issue, even though they are formally just classified as sets? Also how come this isn't talked about more? I'd be fine switching out ZFC for some version of NBG or MK or even ETCS for this, but I do think that doing all of my future math in MLTT or HOTT because of this seems a bit unreasonable for now ^^

mystic-bone-mage

Admittedly this is not the sort of thing I think about in my daily set theoretic life, so take it with a set containing a grain of salt. However: what you might want are what I'll call α-sets. A set (in the sense of ZFC, or what have you) is called a 0-set. Having defined α-sets, an (α+1)-set is any collection of α-sets. Having defined α-sets for all α < γ limit, a γ-set is an α-set for some α < γ. For example, there is a 1-set of all 0-sets, a 2-set of all 1-sets, but no ω-set of all n-sets over all n < ω (that collection would be an (ω+1)-set).

I won't be considering the axiomatic framework of α-sets in depth right now, but if there's interest I can write something up another time (perhaps when I'm not on the toilet). I think the analogy with Grothendeick universes is a good one, but we're going to run into the same ontological issue here (ie, Grothendeick universes think they are proper classes, but we typically consider them sitting inside some bigger Grothendeick universe that thinks the smaller one is a set).

If we let V^α be the (α+1)-set of all α-sets, then for an inaccessible cardinal κ, we can perform essentially all of ordinary mathematics inside of V^{κ}. The emphasis is given to show that we've not actually solved the problem we were trying to address here; we have simply pushed it up κ many levels of abstraction, and for our effort we get to come up with a whole new axiomatic framework.

Okay, so maybe the solution to our issue will get spit out if we stop trying to have a "top node" to our hierarchy. Maybe considering the entire universe of α-sets for any α will give us what we want. If it doesn't, then I'm happy to tell you that the 1-set of all 0-ordinals is a 1-ordinal, that the (α+1)-set of all α-ordinals is an (α+1)-ordinal, and that we can do this construction over and over, transfinitely, considering α_β-sets indexed by the β-ordinals α. Of course, why stop there? Next we can consider α_{β_γ}-sets, where α is a β_γ-ordinal. We can do our transfinite iteration of transfinite iterations transfinitely many times! But that's where we should stop; I have a headache.

yoneda-emma

Ah, thank you! The whole "there is a 1-set of all 0-sets, a 2-set of all 1-sets, etc." is exactly what I was thinking myself (considering thats what agda does), and "Grothendieck universes think they are proper classes, but bigger Grothendieck universes think the smaller ones are sets" does sound reasonable as a resolution to "how do Groethendieck universes establish this hierarchy without going beyond proper classes" Your $V^α$ aren't directly related to Von Neumann universes, right? They just happen to use the same letter?

radicalideal

I think this is just the same as assuming there is a proper class of inaccessible cardinals {kappa_alpha}, and then an alpha-set is just an element of V_(kappa_alpha). Which is pretty tame as large cardinal axioms go.

yoneda-emma

Ah! I knew from Wikipedia that Grothendieck universes are equivalent to strongly inaccessible cardinals, but "an alpha-set is just an element of V_(kappa_alpha)" is a nice final puzzle piece, thank you!

janmusija

Here's a couple additional thoughts of mine you may find relevant:

If you care about minimizing consistency strength, "there is some definable class of ordinals (\kappa_\alpha) so that every set is contained in V_(\kappa_\alpha) for some \alpha" is of consistency strength well below an inaccessible cardinal. A very strong form of this property holds already within V_\kappa for \kappa strongly inaccessible. (the caveat here is that you need \kappa to be inaccessible if you want V_\kappa to be particularly nice in a second-order sense, so that it contains all of its own "small" subsets, rather than only e.g. all definable ones. Inaccessibles are definitely much nicer.)

Higher large cardinal axioms can sort of inform how you would extend the notion of "how would you add a hierarchy of universes to set theory". For example, a (strongly) inaccessible limit of inaccessibles is equivalent to the idea of "a grothendieck universe which satisfies 'every set is contained in a grothendieck universe' (which I have heard phrased as the 'grothendieck axiom' somewhere I think)". A strongly mahlo cardinal \kappa corresponds to a grothendieck universe W = V_\kappa where for every function f:W->W, there is some smaller grothendieck universe W* in W so that f keeps members of W* inside W*. These are both pretty weak axioms, though, on the scale of large cardinal strength.

On that topic, I think one comment I have about this-- and this is admittedly probably less of a concern if you view these as like "a stratified hierarchy of universes" rather than "a number of small initial segments of the universe that happen to model ZFC"-- is that it does kind of raise the point of "okay, but the moment you are doing math with the assumption that there is a stratified tower of universes like this, you can no longer carry out 'essentially all math' that you might want within any piece of that tower."

yoneda-emma

It does kind of raise the point of "okay, but the moment you are doing math with the assumption that there is a stratified tower of universes like this, you can no longer carry out 'essentially all math' that you might want within any piece of that tower."

This is only true if we define 'all math' as 'anything you can do in whatever specific axiom system you chose', right? And you just mean that theres fewer statements we can prove compared to the vast number of new statements we can't prove? I'd define 'all math' as something more like 'everything derivable from *any* set of axioms', and surely from that perspective adding more layers to the tower only increases the amount of math we can theoretically carry out?

yoneda-emma

And the statement "given any amount of any objects of any kind, the collection of all of those objects is its own distinct object" just seems like something that is so obviously true in a platonic sense that it would feel really strange to me to do math without it

janmusija

Hm, what do you mean by this exactly? For example, it is clear that for an arbitrary set theoretic sentence φ, {all objects satisfying φ} is not in general going to be an object (i.e. a set, which is the only kind of object in the domain of discourse in the context of a model of set theory)-- this was exactly what got naïve set theories in trouble back in the day.

[to ensure that this doesn't get too hidden in a different branch of the reblog tree: FWIW I more or less entirely agree with your statement about the meaning of "all math". I mostly just intended this to illustrate a point to other potential readers]

yoneda-emma

I was using 'object' in a loose natural language sense here, assuming that some objects can be inherently different 'types' of things than others. To me, "sets are the only kind of object in set theory" just feels demonstrably wrong in a platonic sense - as soon as we start speaking about sets, 'the collection of all sets' just inherently becomes a nebulous thing floating around in the aether, and I think that whatever system of logic we are working in should inherently be able to talk and formally reason about it. Russels paradox tells us that {all objects satisfying φ} cannot be a part of whatever collection we defined 'objects' to be, but I think it is entirely natural (to the point that not doing so feels extremely strange to me, which is kind of why this post got made in the first place) to conclude that that just means that our notion of 'object' is not enough to reason in a satisfactorily formal way, that {all objects satisfying φ} must be a new type of thing, let's say a '2-object', and that there must be a hierarchy such that (n+1)-objects are the collection of things that {all n-objects satisfying φ} belongs to.

yoneda-emma

To any set theory wizards following me: What's the least inconvenient way to add a hierarchy of universes to set theory, similar to the way dependent type theory does it? Specifically, I always felt that "resolving" Russels paradox by introducing proper classes and then only allowing quantification over sets was deeply unsatisfying, since on a philosophical level, this just pushes the issue up one step - I think it is intuitively clear that the "collection of all proper classes" should still be a valid object that exists, even if it can't be a proper class itself. I assume what I want is hyperclasses, but I'm having a real hard time finding basically any writing on them, and from what little I do find even these seem to usually just be considered as "collections of classes", and not a full hierarchy where an (n+1)-class is a collection of (n)-classes.... Are Grothendieck Universes (and/or Tarski Universes) already enough to solve this issue, even though they are formally just classified as sets? Also how come this isn't talked about more? I'd be fine switching out ZFC for some version of NBG or MK or even ETCS for this, but I do think that doing all of my future math in MLTT or HOTT because of this seems a bit unreasonable for now ^^

mystic-bone-mage

Admittedly this is not the sort of thing I think about in my daily set theoretic life, so take it with a set containing a grain of salt. However: what you might want are what I'll call α-sets. A set (in the sense of ZFC, or what have you) is called a 0-set. Having defined α-sets, an (α+1)-set is any collection of α-sets. Having defined α-sets for all α < γ limit, a γ-set is an α-set for some α < γ. For example, there is a 1-set of all 0-sets, a 2-set of all 1-sets, but no ω-set of all n-sets over all n < ω (that collection would be an (ω+1)-set).

I won't be considering the axiomatic framework of α-sets in depth right now, but if there's interest I can write something up another time (perhaps when I'm not on the toilet). I think the analogy with Grothendeick universes is a good one, but we're going to run into the same ontological issue here (ie, Grothendeick universes think they are proper classes, but we typically consider them sitting inside some bigger Grothendeick universe that thinks the smaller one is a set).

If we let V^α be the (α+1)-set of all α-sets, then for an inaccessible cardinal κ, we can perform essentially all of ordinary mathematics inside of V^{κ}. The emphasis is given to show that we've not actually solved the problem we were trying to address here; we have simply pushed it up κ many levels of abstraction, and for our effort we get to come up with a whole new axiomatic framework.

Okay, so maybe the solution to our issue will get spit out if we stop trying to have a "top node" to our hierarchy. Maybe considering the entire universe of α-sets for any α will give us what we want. If it doesn't, then I'm happy to tell you that the 1-set of all 0-ordinals is a 1-ordinal, that the (α+1)-set of all α-ordinals is an (α+1)-ordinal, and that we can do this construction over and over, transfinitely, considering α_β-sets indexed by the β-ordinals α. Of course, why stop there? Next we can consider α_{β_γ}-sets, where α is a β_γ-ordinal. We can do our transfinite iteration of transfinite iterations transfinitely many times! But that's where we should stop; I have a headache.

yoneda-emma

Ah, thank you! The whole "there is a 1-set of all 0-sets, a 2-set of all 1-sets, etc." is exactly what I was thinking myself (considering thats what agda does), and "Grothendieck universes think they are proper classes, but bigger Grothendieck universes think the smaller ones are sets" does sound reasonable as a resolution to "how do Groethendieck universes establish this hierarchy without going beyond proper classes" Your $V^α$ aren't directly related to Von Neumann universes, right? They just happen to use the same letter?

radicalideal

I think this is just the same as assuming there is a proper class of inaccessible cardinals {kappa_alpha}, and then an alpha-set is just an element of V_(kappa_alpha). Which is pretty tame as large cardinal axioms go.

yoneda-emma

Ah! I knew from Wikipedia that Grothendieck universes are equivalent to strongly inaccessible cardinals, but "an alpha-set is just an element of V_(kappa_alpha)" is a nice final puzzle piece, thank you!

janmusija

Here's a couple additional thoughts of mine you may find relevant:

If you care about minimizing consistency strength, "there is some definable class of ordinals (\kappa_\alpha) so that every set is contained in V_(\kappa_\alpha) for some \alpha" is of consistency strength well below an inaccessible cardinal. A very strong form of this property holds already within V_\kappa for \kappa strongly inaccessible. (the caveat here is that you need \kappa to be inaccessible if you want V_\kappa to be particularly nice in a second-order sense, so that it contains all of its own "small" subsets, rather than only e.g. all definable ones. Inaccessibles are definitely much nicer.)

Higher large cardinal axioms can sort of inform how you would extend the notion of "how would you add a hierarchy of universes to set theory". For example, a (strongly) inaccessible limit of inaccessibles is equivalent to the idea of "a grothendieck universe which satisfies 'every set is contained in a grothendieck universe' (which I have heard phrased as the 'grothendieck axiom' somewhere I think)". A strongly mahlo cardinal \kappa corresponds to a grothendieck universe W = V_\kappa where for every function f:W->W, there is some smaller grothendieck universe W* in W so that f keeps members of W* inside W*. These are both pretty weak axioms, though, on the scale of large cardinal strength.

On that topic, I think one comment I have about this-- and this is admittedly probably less of a concern if you view these as like "a stratified hierarchy of universes" rather than "a number of small initial segments of the universe that happen to model ZFC"-- is that it does kind of raise the point of "okay, but the moment you are doing math with the assumption that there is a stratified tower of universes like this, you can no longer carry out 'essentially all math' that you might want within any piece of that tower."

yoneda-emma

It does kind of raise the point of "okay, but the moment you are doing math with the assumption that there is a stratified tower of universes like this, you can no longer carry out 'essentially all math' that you might want within any piece of that tower."

This is only true if we define 'all math' as 'anything you can do in whatever specific axiom system you chose', right? And you just mean that theres fewer statements we can prove compared to the vast number of new statements we can't prove? I'd define 'all math' as something more like 'everything derivable from *any* set of axioms', and surely from that perspective adding more layers to the tower only increases the amount of math we can theoretically carry out?

yoneda-emma

And the statement "given any amount of any objects of any kind, the collection of all of those objects is its own distinct object" just seems like something that is so obviously true in a platonic sense that it would feel really strange to me to do math without it

yoneda-emma

To any set theory wizards following me: What's the least inconvenient way to add a hierarchy of universes to set theory, similar to the way dependent type theory does it? Specifically, I always felt that "resolving" Russels paradox by introducing proper classes and then only allowing quantification over sets was deeply unsatisfying, since on a philosophical level, this just pushes the issue up one step - I think it is intuitively clear that the "collection of all proper classes" should still be a valid object that exists, even if it can't be a proper class itself. I assume what I want is hyperclasses, but I'm having a real hard time finding basically any writing on them, and from what little I do find even these seem to usually just be considered as "collections of classes", and not a full hierarchy where an (n+1)-class is a collection of (n)-classes.... Are Grothendieck Universes (and/or Tarski Universes) already enough to solve this issue, even though they are formally just classified as sets? Also how come this isn't talked about more? I'd be fine switching out ZFC for some version of NBG or MK or even ETCS for this, but I do think that doing all of my future math in MLTT or HOTT because of this seems a bit unreasonable for now ^^

mystic-bone-mage

Admittedly this is not the sort of thing I think about in my daily set theoretic life, so take it with a set containing a grain of salt. However: what you might want are what I'll call α-sets. A set (in the sense of ZFC, or what have you) is called a 0-set. Having defined α-sets, an (α+1)-set is any collection of α-sets. Having defined α-sets for all α < γ limit, a γ-set is an α-set for some α < γ. For example, there is a 1-set of all 0-sets, a 2-set of all 1-sets, but no ω-set of all n-sets over all n < ω (that collection would be an (ω+1)-set).

I won't be considering the axiomatic framework of α-sets in depth right now, but if there's interest I can write something up another time (perhaps when I'm not on the toilet). I think the analogy with Grothendeick universes is a good one, but we're going to run into the same ontological issue here (ie, Grothendeick universes think they are proper classes, but we typically consider them sitting inside some bigger Grothendeick universe that thinks the smaller one is a set).

If we let V^α be the (α+1)-set of all α-sets, then for an inaccessible cardinal κ, we can perform essentially all of ordinary mathematics inside of V^{κ}. The emphasis is given to show that we've not actually solved the problem we were trying to address here; we have simply pushed it up κ many levels of abstraction, and for our effort we get to come up with a whole new axiomatic framework.

Okay, so maybe the solution to our issue will get spit out if we stop trying to have a "top node" to our hierarchy. Maybe considering the entire universe of α-sets for any α will give us what we want. If it doesn't, then I'm happy to tell you that the 1-set of all 0-ordinals is a 1-ordinal, that the (α+1)-set of all α-ordinals is an (α+1)-ordinal, and that we can do this construction over and over, transfinitely, considering α_β-sets indexed by the β-ordinals α. Of course, why stop there? Next we can consider α_{β_γ}-sets, where α is a β_γ-ordinal. We can do our transfinite iteration of transfinite iterations transfinitely many times! But that's where we should stop; I have a headache.

yoneda-emma

Ah, thank you! The whole "there is a 1-set of all 0-sets, a 2-set of all 1-sets, etc." is exactly what I was thinking myself (considering thats what agda does), and "Grothendieck universes think they are proper classes, but bigger Grothendieck universes think the smaller ones are sets" does sound reasonable as a resolution to "how do Groethendieck universes establish this hierarchy without going beyond proper classes" Your $V^α$ aren't directly related to Von Neumann universes, right? They just happen to use the same letter?

ouroborussy

I also don't do exactly this, but I am in set theory so might be able to shed some light.

The V^alpha looking similar to the von Neumann hierarchy isn't by accident. In fact if I remember correctly Godel's phrasing of the von Neumann hierarchy itself was that one adds "new types" at every stage.

The only mathematical difference between a sequence of Von Neumann ranks

V_alpha -> V_{alpha+1} -> ...

and the metatheoretic sequence

set -> class -> hyperclass -> ...

is that in the latter, "the universe of sets" has some sorts of closure properties beyond simply being some V_alpha. So it could be modelled perfectly well by positing some sequence of

V_alpha -> V_{alpha+1} -> ...

where the V_alpha is special enough to look like a universe of sets.

This is exactly what a Grothendieck universe seeks to do (a Grothendieck universe is a V_alpha that's a model of a theory a bit stronger than ZFC).

V_alpha is in our domain of discourse, so "is a set", but everything you could do with a theory of all sets could be done in V_alpha, and everything you could do with a theory of classes could be done in V_alpha+1, etc.

Using these, you can do any proof you'd like from MK or NBG by simply working with some V_alpha as your model (with second order quantifiers ranging over V_alpha+1). A theory of hyperclasses would work with the theory where you have access to another sort of variable that ranges over V_alpha+2.

In some sense these approaches (using MK or HoTT or hyperclasses vs using ZFC with universes) are completely equivalent (as long as you've made sure V_alpha behaves exactly like whatever "universe of sets" means to you, and as long as you have sufficiently many universes in the case of HoTT). The ZFC+universes approach has the benefit of connecting you to a largish body of set theoretic literature and of being a first order theory (never having to create "classes" or "hyperclasses" and redevelop the same theory for them), while the MK has sort of an ontological benefit of declaring classes to be really different things than sets. The maths is basically identical.

--------

One thing I'd like to add, which was sort of the point I was making with this. The mental jump we take when we say "class of sets", "hyperclass of all classes", etc. is to declare the universe of sets/classes/etc a new Thing that we can discuss. A new object of our ontology. Both approaches

1) use MK, leaving the "first order universe behind"

2) use a Grothendieck universe V_alpha, leaving the "universe *under alpha* behind"

are doing essentially the same thing.

-------

The least inconvenient way to do it is actually highly debatable. There are a few modern theories (ZFC+countably many universes, feferman set theory, broad infinity, etc) that occupy a reasonable position, in that they

1) subsume the universes needed for HoTT, MK (in a technical sense what I'm saying is you can build models of HoTT and MK in these theories)

2) are well(ish) understood

3) are convenient in the sense of being well(ish) motivated. See for instance:

I would say however that the theory of large cardinals motivates far more universes than are presumed in any of these theories (or far more *than* universes as a concept altogether) so the most convenient way to introduce universes remains open depending on what convenient means to you. Happy to discuss more if you like!

yoneda-emma

Thank you for the extra insights! "Countably many universes" definitely does seem like the most sensible option intuitively

yoneda-emma

To any set theory wizards following me: What's the least inconvenient way to add a hierarchy of universes to set theory, similar to the way dependent type theory does it? Specifically, I always felt that "resolving" Russels paradox by introducing proper classes and then only allowing quantification over sets was deeply unsatisfying, since on a philosophical level, this just pushes the issue up one step - I think it is intuitively clear that the "collection of all proper classes" should still be a valid object that exists, even if it can't be a proper class itself. I assume what I want is hyperclasses, but I'm having a real hard time finding basically any writing on them, and from what little I do find even these seem to usually just be considered as "collections of classes", and not a full hierarchy where an (n+1)-class is a collection of (n)-classes.... Are Grothendieck Universes (and/or Tarski Universes) already enough to solve this issue, even though they are formally just classified as sets? Also how come this isn't talked about more? I'd be fine switching out ZFC for some version of NBG or MK or even ETCS for this, but I do think that doing all of my future math in MLTT or HOTT because of this seems a bit unreasonable for now ^^

mystic-bone-mage

Admittedly this is not the sort of thing I think about in my daily set theoretic life, so take it with a set containing a grain of salt. However: what you might want are what I'll call α-sets. A set (in the sense of ZFC, or what have you) is called a 0-set. Having defined α-sets, an (α+1)-set is any collection of α-sets. Having defined α-sets for all α < γ limit, a γ-set is an α-set for some α < γ. For example, there is a 1-set of all 0-sets, a 2-set of all 1-sets, but no ω-set of all n-sets over all n < ω (that collection would be an (ω+1)-set).

I won't be considering the axiomatic framework of α-sets in depth right now, but if there's interest I can write something up another time (perhaps when I'm not on the toilet). I think the analogy with Grothendeick universes is a good one, but we're going to run into the same ontological issue here (ie, Grothendeick universes think they are proper classes, but we typically consider them sitting inside some bigger Grothendeick universe that thinks the smaller one is a set).

If we let V^α be the (α+1)-set of all α-sets, then for an inaccessible cardinal κ, we can perform essentially all of ordinary mathematics inside of V^{κ}. The emphasis is given to show that we've not actually solved the problem we were trying to address here; we have simply pushed it up κ many levels of abstraction, and for our effort we get to come up with a whole new axiomatic framework.

Okay, so maybe the solution to our issue will get spit out if we stop trying to have a "top node" to our hierarchy. Maybe considering the entire universe of α-sets for any α will give us what we want. If it doesn't, then I'm happy to tell you that the 1-set of all 0-ordinals is a 1-ordinal, that the (α+1)-set of all α-ordinals is an (α+1)-ordinal, and that we can do this construction over and over, transfinitely, considering α_β-sets indexed by the β-ordinals α. Of course, why stop there? Next we can consider α_{β_γ}-sets, where α is a β_γ-ordinal. We can do our transfinite iteration of transfinite iterations transfinitely many times! But that's where we should stop; I have a headache.

yoneda-emma

Ah, thank you! The whole "there is a 1-set of all 0-sets, a 2-set of all 1-sets, etc." is exactly what I was thinking myself (considering thats what agda does), and "Grothendieck universes think they are proper classes, but bigger Grothendieck universes think the smaller ones are sets" does sound reasonable as a resolution to "how do Groethendieck universes establish this hierarchy without going beyond proper classes" Your $V^α$ aren't directly related to Von Neumann universes, right? They just happen to use the same letter?

radicalideal

I think this is just the same as assuming there is a proper class of inaccessible cardinals {kappa_alpha}, and then an alpha-set is just an element of V_(kappa_alpha). Which is pretty tame as large cardinal axioms go.

yoneda-emma

Ah! I knew from Wikipedia that Grothendieck universes are equivalent to strongly inaccessible cardinals, but "an alpha-set is just an element of V_(kappa_alpha)" is a nice final puzzle piece, thank you!

janmusija

Here's a couple additional thoughts of mine you may find relevant:

If you care about minimizing consistency strength, "there is some definable class of ordinals (\kappa_\alpha) so that every set is contained in V_(\kappa_\alpha) for some \alpha" is of consistency strength well below an inaccessible cardinal. A very strong form of this property holds already within V_\kappa for \kappa strongly inaccessible. (the caveat here is that you need \kappa to be inaccessible if you want V_\kappa to be particularly nice in a second-order sense, so that it contains all of its own "small" subsets, rather than only e.g. all definable ones. Inaccessibles are definitely much nicer.)

Higher large cardinal axioms can sort of inform how you would extend the notion of "how would you add a hierarchy of universes to set theory". For example, a (strongly) inaccessible limit of inaccessibles is equivalent to the idea of "a grothendieck universe which satisfies 'every set is contained in a grothendieck universe' (which I have heard phrased as the 'grothendieck axiom' somewhere I think)". A strongly mahlo cardinal \kappa corresponds to a grothendieck universe W = V_\kappa where for every function f:W->W, there is some smaller grothendieck universe W* in W so that f keeps members of W* inside W*. These are both pretty weak axioms, though, on the scale of large cardinal strength.

On that topic, I think one comment I have about this-- and this is admittedly probably less of a concern if you view these as like "a stratified hierarchy of universes" rather than "a number of small initial segments of the universe that happen to model ZFC"-- is that it does kind of raise the point of "okay, but the moment you are doing math with the assumption that there is a stratified tower of universes like this, you can no longer carry out 'essentially all math' that you might want within any piece of that tower."

yoneda-emma

It does kind of raise the point of "okay, but the moment you are doing math with the assumption that there is a stratified tower of universes like this, you can no longer carry out 'essentially all math' that you might want within any piece of that tower."

This is only true if we define 'all math' as 'anything you can do in whatever specific axiom system you chose', right? And you just mean that theres fewer statements we can prove compared to the vast number of new statements we can't prove? I'd define 'all math' as something more like 'everything derivable from *any* set of axioms', and surely from that perspective adding more layers to the tower only increases the amount of math we can theoretically carry out?

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