You can do with numbers everything that you can do with sets, and the other way around
When I first ran into a remark at math-exchange that mentioned that set theory (ZF minus infinity) is bi-interpretable with number theory (PA), I thought that this result is quite amazing, because the axioms of set theory do not look at all like the axioms of number theory. In fact, at first glance, these two different theories have absolutely nothing to see with each other.
The Wikipedia page for bi-interpretability does not mention this example of bi-interpretability (it does not mention any other examples either.)
Set theory can do everything that number theory can do, because it is possible to represent numbers as sets and do arithmetic using set operations such as the set union and the set intersection. We can represent numbers as sets e.g. by using Von Neumann ordinals:
0 = { }, 1 = { { } }, 2 = { { }, { { } } }, ...
Just like in number theory, we only need to be able to add one to a number, because that will allow us to define all other arithmetic operators, i.e. addition, substraction, multiplication, and division. The generalized successor operation (="add one") is relatively simple. If we want to add one to a set, we just add the set to itself. For example:
{ a, b } ? 1
= { a, b } ? { { a, b } }
= { a, b, { a, b } }
If the set represents a number, then the result will be the number is exactly one larger, i.e. its successor. If we want to add two, we just repeat the operation. For example, in Von Neumann ordinal arithmetic, 1+2 looks like:
{ { } } + { { }, { { } } }
= ( { { } } ? { { { } } } ) + { { } }
= { { }, { { } } } + { { } }
= { { }, { { } } } ? { { { }, { { } } } }
= { { }, { { } }, { { }, { { } } } }
Expressions in this model of arithmetic interpretation leave a bit of a brainfuck impression, but it certainly works fine. So, the Von Neumann ordinals, [math]\Bbb{N}^{VN}[/math] along with arithmetic operators defined in term of set operations (+,-,x,/) ?(?,?) properly interpret number theory (PA):
PA ? [math]\Bbb{N}^{VN}[/math] (+,-,x,/) ?(?,?)
The Von Neumann ordinals are as much a legitimate model for PA as the standard model of natural numbers:
PA ? [math]\Bbb{N}[/math] (+,-,x,/)
The other way around works as well. Number theory can do everything set theory can do. According to the Wikipedia page on arithmetic sets, we can represent sets as number-theoretical predicates:
Quoting Wikipedia on arithmetical sets
Defining the union and intersection of arithmetical sets is actually the easy part. If the predicate [math]?_1(x)[/math] represents a first set and [math]?_2(x)[/math] a second one, then the union of both is [math]?_1(x) \vee ?_2(x) [/math] while the intersection will be [math]?_1(x) \wedge ?_2(x) [/math]. It is the translation from standard set-builder notation to arithmetic-set notation that is not as simple. As expected, the Wikipedia page on arithmetic sets does not even show one example of what arithmetic-set notation could look like.
For example, what does the set { 1, 5, "hello" } look like in arithmetic-set notation?
[math]?(x) := (x=\ulcorner 1\urcorner) \vee (x=\ulcorner 5\urcorner) \vee (x=\ulcorner hello \urcorner)[/math]
With [math]\ulcorner x\urcorner[/math] the Gödel encoding for x.
would be a good start, but it does not look much like standard PA arithmetic. We can make use, however, from the following considerations:
x [math]\vee[/math] y = x + y - xy
x [math]\wedge[/math] = xy
We can also consider that x=5 is equivalent to 1 - sgn(x-5)². Hence, after applying these considerations, the set-theoretical set { 1, 5, "hello" } can be represented as the following arithmetic set:
[math]?(x) := 1 - sgn( ( x - \ulcorner 1 \urcorner) ( x - \ulcorner 5 \urcorner) ( x - \ulcorner hello \urcorner) )²[/math]
In the end, I struggled the most with representing embedded set expressions in set builder notation, such as { 1, 6, { 4, 3} , 8 } in arithmetic-set notation. The outer set looks like:
[math]?(x) := 1 - sgn( ( x - \ulcorner 1 \urcorner) ( x - \ulcorner 6 \urcorner) ( x - \ulcorner ??? \urcorner) ( x - \ulcorner 8 \urcorner) )²[/math]
While the embedded set expression would look like:
[math]?(x) := 1 - sgn( ( x - \ulcorner 4 \urcorner) ( x - \ulcorner 3 \urcorner) )²[/math]
The solution to combine both into one expression may look simple, but it took me forever to realize that the solution had to be:
[math]?(x) := 1 - sgn( ( x - \ulcorner 1 \urcorner) ( x - \ulcorner 6 \urcorner) ( x - \ulcorner 1 - sgn( ( y - \ulcorner 4 \urcorner) ( y - \ulcorner 3 \urcorner) )² \urcorner) ( x - \ulcorner 8 \urcorner) )²[/math]
Hence, since it is possible to translate all (finite) set-theoretical expressions in set-builder notation into arithmetic-set notation, as well as reimplement all set operators as arithmetic operators, i.e. (?,?)?(+,-,x,/) , we can conclude that the arithmetic sets [math]S_{AR}[/math] interpret ZF-? in an equivalent way as its standard model S with (?,?):
ZF-? ? [math]S_{AR}[/math] (?,?)?(+,-,x,/)
ZF-? ? S (?,?)
Hence, the bi-interpretability of number theory and set theory can be expressed as the following four statements:
ZF-? ? S (?,?)
ZF-? ? [math]S_{AR}[/math] (?,?)?(+,-,x,/)
PA ? [math]\Bbb{N}[/math] (+,-,x,/)
PA ? [math]\Bbb{N}^{VN}[/math] (+,-,x,/) ?(?,?)
If you prefer a formal proof -- necessarily burdened by a larger bureaucracy of formalisms -- instead of the proof sketch/strategy laid out above, you can find it in the 2006 paper "On interpretations of arithmetic and set theory" by Richard Kaye and Tin Lok Wong.
The Wikipedia page for bi-interpretability does not mention this example of bi-interpretability (it does not mention any other examples either.)
Set theory can do everything that number theory can do, because it is possible to represent numbers as sets and do arithmetic using set operations such as the set union and the set intersection. We can represent numbers as sets e.g. by using Von Neumann ordinals:
0 = { }, 1 = { { } }, 2 = { { }, { { } } }, ...
Just like in number theory, we only need to be able to add one to a number, because that will allow us to define all other arithmetic operators, i.e. addition, substraction, multiplication, and division. The generalized successor operation (="add one") is relatively simple. If we want to add one to a set, we just add the set to itself. For example:
{ a, b } ? 1
= { a, b } ? { { a, b } }
= { a, b, { a, b } }
If the set represents a number, then the result will be the number is exactly one larger, i.e. its successor. If we want to add two, we just repeat the operation. For example, in Von Neumann ordinal arithmetic, 1+2 looks like:
{ { } } + { { }, { { } } }
= ( { { } } ? { { { } } } ) + { { } }
= { { }, { { } } } + { { } }
= { { }, { { } } } ? { { { }, { { } } } }
= { { }, { { } }, { { }, { { } } } }
Expressions in this model of arithmetic interpretation leave a bit of a brainfuck impression, but it certainly works fine. So, the Von Neumann ordinals, [math]\Bbb{N}^{VN}[/math] along with arithmetic operators defined in term of set operations (+,-,x,/) ?(?,?) properly interpret number theory (PA):
PA ? [math]\Bbb{N}^{VN}[/math] (+,-,x,/) ?(?,?)
The Von Neumann ordinals are as much a legitimate model for PA as the standard model of natural numbers:
PA ? [math]\Bbb{N}[/math] (+,-,x,/)
The other way around works as well. Number theory can do everything set theory can do. According to the Wikipedia page on arithmetic sets, we can represent sets as number-theoretical predicates:
Quoting Wikipedia on arithmetical sets
An arithmetical set is a set of natural numbers that can be defined by a formula of first-order Peano arithmetic ... by using Gödel numbers to represent elements of the set and declaring a subset of A to be arithmetical if the set of corresponding Gödel numbers is arithmetical.
Defining the union and intersection of arithmetical sets is actually the easy part. If the predicate [math]?_1(x)[/math] represents a first set and [math]?_2(x)[/math] a second one, then the union of both is [math]?_1(x) \vee ?_2(x) [/math] while the intersection will be [math]?_1(x) \wedge ?_2(x) [/math]. It is the translation from standard set-builder notation to arithmetic-set notation that is not as simple. As expected, the Wikipedia page on arithmetic sets does not even show one example of what arithmetic-set notation could look like.
For example, what does the set { 1, 5, "hello" } look like in arithmetic-set notation?
[math]?(x) := (x=\ulcorner 1\urcorner) \vee (x=\ulcorner 5\urcorner) \vee (x=\ulcorner hello \urcorner)[/math]
With [math]\ulcorner x\urcorner[/math] the Gödel encoding for x.
would be a good start, but it does not look much like standard PA arithmetic. We can make use, however, from the following considerations:
x [math]\vee[/math] y = x + y - xy
x [math]\wedge[/math] = xy
We can also consider that x=5 is equivalent to 1 - sgn(x-5)². Hence, after applying these considerations, the set-theoretical set { 1, 5, "hello" } can be represented as the following arithmetic set:
[math]?(x) := 1 - sgn( ( x - \ulcorner 1 \urcorner) ( x - \ulcorner 5 \urcorner) ( x - \ulcorner hello \urcorner) )²[/math]
In the end, I struggled the most with representing embedded set expressions in set builder notation, such as { 1, 6, { 4, 3} , 8 } in arithmetic-set notation. The outer set looks like:
[math]?(x) := 1 - sgn( ( x - \ulcorner 1 \urcorner) ( x - \ulcorner 6 \urcorner) ( x - \ulcorner ??? \urcorner) ( x - \ulcorner 8 \urcorner) )²[/math]
While the embedded set expression would look like:
[math]?(x) := 1 - sgn( ( x - \ulcorner 4 \urcorner) ( x - \ulcorner 3 \urcorner) )²[/math]
The solution to combine both into one expression may look simple, but it took me forever to realize that the solution had to be:
[math]?(x) := 1 - sgn( ( x - \ulcorner 1 \urcorner) ( x - \ulcorner 6 \urcorner) ( x - \ulcorner 1 - sgn( ( y - \ulcorner 4 \urcorner) ( y - \ulcorner 3 \urcorner) )² \urcorner) ( x - \ulcorner 8 \urcorner) )²[/math]
Hence, since it is possible to translate all (finite) set-theoretical expressions in set-builder notation into arithmetic-set notation, as well as reimplement all set operators as arithmetic operators, i.e. (?,?)?(+,-,x,/) , we can conclude that the arithmetic sets [math]S_{AR}[/math] interpret ZF-? in an equivalent way as its standard model S with (?,?):
ZF-? ? [math]S_{AR}[/math] (?,?)?(+,-,x,/)
ZF-? ? S (?,?)
Hence, the bi-interpretability of number theory and set theory can be expressed as the following four statements:
ZF-? ? S (?,?)
ZF-? ? [math]S_{AR}[/math] (?,?)?(+,-,x,/)
PA ? [math]\Bbb{N}[/math] (+,-,x,/)
PA ? [math]\Bbb{N}^{VN}[/math] (+,-,x,/) ?(?,?)
If you prefer a formal proof -- necessarily burdened by a larger bureaucracy of formalisms -- instead of the proof sketch/strategy laid out above, you can find it in the 2006 paper "On interpretations of arithmetic and set theory" by Richard Kaye and Tin Lok Wong.
Comments (79)
Thanks!
A couple of observations:
(1) I think the notion of a theory interpreting another can be made more intuitive by considering the relations between algebra and geometry. As is known at least since Descartes, there is a natural translation of basic algebraic operations (sum, product, subtraction, division, and extraction of roots) into geometric constructions involving ruler and compass, and vice-versa. At first, it may seem exotic that a discipline that is overtly about operations between numbers has anything to do with constructible figures and vice-versa, but once you study in detail the translation between these theories, they become more natural. In fact, the algebraic theory of field extensions even proved to be the natural setting for solving long-standing problems in geometry, such as the trisection of the angle, etc. The general notion of interpretation is basically a generalization of this idea.
(2) I think you are confused about "arithmetical sets". As the wikipedia page explains, arithmetical sets are not sets which are coded by numbers, rather, they are sets of numbers definable in PA. For example, the set of the squares (i.e. the set {0, 1, 4, 9, 16, ... } is definable by the formula Ex(x * x = y), and hence is an arithmetical set. Similarly, every singleton of a natural number is also an arithmetical set. So considering the arithmetical sets will not give you an interpretation of ZF-Inf into PA. In order to achieve the latter, you need some way of coding the relation "x belongs to y" using only arithmetical predicates. As Kaye and Wong explain the paper you linked (p. 3), this is usually done by way of the Ackermann interpretation, namely "x belongs to y" is defined as "the xth digit of the binary expansion of y is 1". One then shows that, given this interpretation of "x belongs to y", all the axioms of ZF-Inf are satisfied. For instance, extensionality is trivially satisfied, since if w and z have the same binary expansion, they are the same number; the empty set is represented by the number 0; the set {a, b} is coded by 2^a + 2^b, etc. A more or less detailed (and tedious) proof that various set-theoretical concepts can be defined using this interpretation is given in section 1.(b) of this chapter (which is mentioned by K&W).
(3) Notice that bi-interpretability between two theories, T and T', is a stronger notion than mutual interpretability between T and T'. The latter only requires that there be an interpretation of T into T' and vice-versa, whereas the former requires that each interpretation be the inverse of the other.
(4) Incidentally, note also that our usual set theory, ZFC, goes far beyond first-order PA! So it's not exactly true that we can do with numbers everything we can do with sets. Rather, we can do with numbers everything we can do with the hereditarily finite sets!
So, in a universe with just 7 numbers, { 0, 1, 2, 3, 4, 5, 6 }, the subset { 2, 4 } is represented as 0010100. So, the binary representation is as long as the largest number in the subset. I don't see, however, how to represent { 2, { 3, 5} } in that scheme.
One model of the natural numbers are the string symbols that satisfy the regular language:
L = 0 | [1-9][0-9]*
We can pick any arbitrary number of symbols of at least two symbols, [math]\{ ?_0, ?_1 , ?_2, ..., ?_n\}[/math], and represent natural numbers as:
L = [math]?_0[/math] | [ [math]?_1[/math]-[math]?_n[/math] ] [ [math]?_0[/math]-[math]?_n[/math] ]*
L is a model of PA:
PA [math]\models[/math] ( [math]?_0[/math] | [ [math]?_1[/math]-[math]?_n[/math] ] [ [math]?_0[/math]-[math]?_n[/math] ]* ) { +, -, x , / }
We can also capture the Von Neumann ordinals and the constant expressions representing heriditarily finite sets as languages with (non-context free) grammars. In this approach, the Von Neumann ordinals interprete PA, simply because they are isomorphic under arithmetic with the language L mentioned above.
I think that the simplest way to argue that there is a language, i.e. a model, representing number-theoretical predicates that interprete ZF-? is to just create one, and then to argue that this language is isomorphic under arithmetic with the standard set-builder notation for set literals.
Kaye and Wong did it in another way, but I think that their approach is more complicated and less intuitive than an approach in which we treat models as formal languages.
Are you an engineer or a math major? You can actually discover alot of stuff about mathematical principles by studying how 3d engines are made such as Blender. A relationship can be made between any two objects/ideas if you attach a complex (or simple sometimes) equation to it. (one to one, linear, exponential, inverse exponential, logarithmic) and ofcourse you have your coefficients and constants in the equation which add even more depth to the relationship. Reality is similar to a 3d topographical map except reality is like hundreds of these 3d topographical maps all intertwined at the same time.
Most of the discussions on this forum are fairly banal and then you jump to something that can actually be proven and also at the same time has alot of depth.
A software engineer. Math is more of a hobby, actually. I would never have studied it at uni. Math is fun, intriguing, and surprising but not vocational enough as a profession.
Quoting christian2017
I am impossibly lousy at visual mathematics, especially, at the visual puzzling of classical Euclidean geometry ("Elements"). I can only do algebraic geometry, to a better extent, but I don't like it all that much either. I prefer symbol manipulation only. So, I pretty much never choose a math topic that is fundamentally visual. It reminds me too much of technical drawings we had to do at high school. I could never "see" the thing ...
Cool! You ever use Spring (a java technology)? I have to head out now. ttyl
I very rarely use Java, C# or C++, even though I should probably be more open concerning Java in Android development. I tend to use a combo of scripting languages along with C for native libraries.
I think you're losing sight of what it is we're after. We want an interpretation of ZF-Inf, not just any way to code random finite sets. I mention this because, given this goal, we should bear some things in mind:
(1) The universes of ZF-Inf are composed of sets, not random objects (that is, there are no urelements). So every object in the universe is built out of the empty set in a structured way. So your set {2, 4}, for example, is actually the set { { {}, {{}} }, { { }, { { } }, { { }, { { } } }, { {}, {{}}, { {}, {{}} } } (I may have missed some brackets there). This is important because our coding scheme will use this to its advantage.
(2) The universes of ZF-Inf are all infinite. This is clear from the fact that ZF-Inf has the power set axiom, so that there's no bound for the size of its sets.
(3) The interpretation is actually coding sets as numbers, so that to say that 3 belongs to 5 (say) is actually to say that the set coded by 3 belongs to the set coded by 5.
With this in mind, note that, under Ackermann's interpretation, the empty set is coded by 0, the singleton of the empty set (i.e. {{}}) is coded by 1, the number two is coded by 11 (i.e. by 3), the number three, by 111 (i.e. by 7), and so on and so forth for every von Neumann ordinal (note that I'm considering the leftmost digit as the 0th digit, the second leftmost digit as the 1st digit, etc.). On the other hand, the set {{{}}} is coded by 10, since it does not contain the empty set, but it does contain the singleton of the empty set.
Note that simply having a coding scheme is not nearly enough for an interpretation (let alone bi-interpretation). You also need to show that (i) the elements in this coding scheme are all definable in the theory that is doing the interpretation and that (ii) all the axioms of the target theory are provable under this coding scheme. These are not trivial matters and some ingenuity is required to see that everything works smoothly (see the chapter by Hájek and Pudlák that I linked in the previous chapter to see how it is done).
A shape is a powerful set.
Otherwise you're literally barking at space.
I know the route of all base 4 mathematics, by it's shape.
That is the standard model and therefore the standard formal language of ZF-?, essentially unique only up to isomorphism. Therefore, { 2, 4 } is a legitimate expression in an alternative model and formal language that is isomorphic under set calculus to the standard model.
As a side note, given the bi-interpretability with PA, ZF-? is subject to Löwenheim-Skolem, and is therefore not categorical.
The BNF grammar of the standard model would be:
The BNF grammar is not sufficient to entirely validate set expressions because it is not capable of expressing that the set cannot contain duplicates. Hence, the formal language of the standard set model is not context-free. So, there is also need for a function that would be able to do the following:
remove_duplicates({ a, a, b, b }) = { a, b }
Quoting Nagase
Agreed.
It is just that it does not contain induction-completed cardinals ? of infinity.
Quoting Nagase
It is obvious that the strategy that you prefer, works absolutely fine. It has been the general idea to work like that since the nineties anyway.
However, that does not mean that my strategy would not work too.
I just designed a formal language that produces legitimate number-theoretical predicates and that is isomorphic with the standard ZF-? language/model under the standard set operations (?,?). I like my own approach much better than the standard approach, if only, because it is much simpler.
You say:
First, it's not clear what is for languages to be isomorphic. A model M is isomorphic to a model N iff there is a bijection between their respective domains that respects the interpretation of the non-logical symbols. What does it mean for languages to be isomorphic? Second, it's not clear what is the "standard ZF-? language under the standard set operations (?,?)". The "standard ZF-? language" is the language whose only non-logical symbol is the set membership symbol, "?". What does it mean for this language to be "under the standard set operations (?,?)"?
More importantly, you claimed in your first post that your procedure was meant to express the bi-interpretability of PA and ZF-Inf. What I'm saying is that this is very far from the truth. You have not shown how to define the relevant notions in PA (i.e. you have not shown that PA proves that your definitions are well-defined). You have not shown that, using your definitions, we can prove the axioms of ZF-Inf. And finally, you have also not shown that your "interpretations" are inverses, which is crucial for bi-interpretability.
Finally, I'm confused by your use of the sign function. For any x, sgn(x) is either 1, 0, or -1, corresponding to the cases x>0, x=0, x<-1. So there are only three possible values for 1-sgn(x), namely 0, 1, 2. Hence this term can only code at best three possible objects.
A formal language is a class of sentences, just like a model-theoretical model is. Such classes can be isomorphic. This is very possible.
For example, the class of decimal representations of natural numbers is isomorphic under arithmetic to the class of binary representations of natural numbers. The translation function ? is definable and even trivially computable. Example:
[math]?(4) = 100[/math]
[math]?^{-1}(100) = 4[/math]
I also tend to believe that a model-theoretical model is a formal language, and that a formal language is a model-theoretical model (of sorts). It is a similar view as expressed by the Curry-Howard correspondence (CHC): "A proof is a program, and a program is a proof (of sorts)." Still, the bureaucracy of formalisms to actually prove the CHC is daunting and full of subtleties. I would certainly not be able to do it alone.
As I remarked previously, the standard model of natural numbers can be completely captured by a really simple regular expression (=a regular language), one example being (=decimal):
[math]\Bbb{N}[/math] = 0 | [1-9][0-9]*
There really is a bijection between [math]\Bbb{N}[/math] and the Von Neumann ordinals [math]\Bbb{N}^{VN}[/math], while arithmetic clearly is homomorphic in both directions. That is the reason why [math]\Bbb{N}^{VN}[/math] interpretes PA using arithmetic expressed in terms of set operations; simply because [math]\Bbb{N}[/math] already does. Given the isomorphism with the standard model [math]\Bbb{N}[/math], there is no need to further prove that [math]\Bbb{N}^{VN}[/math] truly interpretes PA.
The bijection really exists, because the translation function ? can be defined and is even trivially computable:
[math]\forall k \in \Bbb{N}^{VN}, \exists j \in \Bbb{N} : j = ?(k) [/math]
[math]\forall j \in \Bbb{N}, \exists k \in \Bbb{N}^{VN} : k = ?^{-1}(j) [/math]
In the expressions above, [math]\Bbb{N}^{VN}[/math] and [math]\Bbb{N}[/math] are collections of strings. So, 9 is represented by the string "9" in [math]\Bbb{N}[/math] or by the binary string "1001" in base 2 (or by other strings in other bases, depending on the choice of base).
Quoting Nagase
The proof sketch in my OP is obviously not a complete formal proof.
Still, if two models are essentially the same up to isomorphism, you can reuse the proof that the definitions for the first model are well-defined to argue that the ones for second model are too, since both models are isomorphic.
Quoting Nagase
The predicate ?(x) only defines a set membership function. For example:
?(x) := 1 - sgn( (x-4)(x-9)(x-12) )²
?(x)=1 for { 4,9,2 } but not for any other value. For example, 8 is not a member of the set:
?(8) = 1 - sgn( (8-4)(8-9)(8-12) )²
= 1 - sgn( (4)(-1)(-4) )²
= 1 - sgn(16)²
= 1 - 1²
=0
You can try for any value k not in { 4,9,2 } and you will find that ?(k) = 0, simply because the sgn(...)² subexpression will in that case always evaluate to 1, leading to a result that is 1 - 1 = 0.
Hence, the number-theoretical predicate function ?(x) is capable of returning 1 when the x is in the set, and 0 when it is not. This means that ?(x) effectively represents a set within arithmetic. Since set arithmetic (?,?) can be implemented in terms of ([math]\vee , \wedge[/math]) on these predicates, it is always a model for some set theory.
Given its bijection with the standard set-builder notation capable of representing sets in ZF-?, it is isomorphic with the standard model for ZF-? under set arithmetic, and therefore a model for ZF-?.
Quoting alcontali
Whatever you read on a forum, it's not ZF minus infinity that is, in a certain sense (a qualification I'll leave tacit henceforth), equivalent with first order (a qualification I'll leave tacit henceforth) PA. Rather it is ZF minus infinity plus the negation of infinity that is equivalent with PA.
Quoting alcontali
No, PA has successor (adding one), addition, and multiplication as primitive.
Quoting alcontali
We take the union of the set with the singleton of the set.
Quoting alcontali
The finite (Von Neumann) ordinals ARE the universe for the standard model of PA.
Quoting alcontali
That seems to be your main point. And it is wildly and clearly incorrect. PA cannot prove the existence of an infinite set; and, crucially for mathematics, PA does not provide for the construction of real numbers.
Quoting alcontali
The quote you adduced says 'arithmetical sets' can be represented that way. Not sets in general.
Quoting alcontali
No, the equivalence of PA and FINITE set theory, aka known as the theory of hereditarily finite sets.
Quoting alcontali
Not if you're talking about basic mathematical logic. A language is not a set of sentences. A language is a set of symbols and arity functions. A theory is a set of sentences closed under deduction. A model is not a set of sentences. A model is a certain kind function on the set of symbols of a language.
Some authors use 'ZF-Inf' to really mean ZF with infinity replaced by the negation of infinity. But better practice is to notate as '(ZF-Inf)+~Inf' to ward against confusions such as this by another poster:
"Quoting alcontali
'ZF minus Inf' there is incorrect and seems to come from 'ZF-Inf', when it should be 'ZF minus infinity plus the negation of infinity' or 'ZF with infinity replaced by the negation of infinity'.
Unless I'm overlooking something, for any theory, and for any cardinality, there is a model of the theory such that the universe of the model has a member of that cardinality.
The original paper by Richard Kaye and Tin Lok Wong says:
Quoting On interpretations of arithmetic and set theory
In my impression, it means that the induction-complete notion of infinity ("axiom of infinity") is not part of the ZF?inf theory. The idea to call it "ZF?inf" is apparently even older than this particular paper.
Quoting GrandMinnow
Addition and multiplication are defined in terms of the successor function:
Quoting Wikipedia on Peano's axioms
In my impression, addition and multiplication (and their inverses) are not counted as being part of Peano's axioms.
Quoting GrandMinnow
ZF?inf can't either.
Quoting GrandMinnow
Yes, but that should have been clear from the context. The set theory at hand is ZF?inf and not ZFC.
Quoting GrandMinnow
I was talking about languages as seen through the lens of computer science, i.e. regular languages, context-free languages, and so on. They are not exactly the same as what is meant by language in mathematical logic. Still, they are covered by the same term, i.e. "language". For example, the natural numbers as a model can be characterized by a simple regular language.
Quoting GrandMinnow
A model is a set of sentences in a particular "language" (as defined by computer science). For example, the natural numbers are the set of sentences characterized by a class of isomorphic regular expressions.
Yes, it exactly supports what I said. The theory in question is not ZF minus Inf; rather it is ZF minus infinity plus the negation of infinity.
Quoting alcontali
In second order PA. But in first order PA, addition and multiplication are primitive. And in this context of equivalence with (ZF-Inf)+Inf, we're talking about first order PA.
Quoting alcontali
Your impression stems from your lack of distinguishing first order from second order.
Quoting alcontali
Your usage and understanding of the concepts all around is so abysmally sloppy that context does not excuse you. When you write "set theory" but mean "the theory of herditarily finite sets" then you need to write "the theory of hereditarily finite sets" and not "set theory".
Quoting alcontali
You said "model theoretic" in the exact passage. Granted, you did go into other areas in the discussion, so you might think again that context justifies, but your posting in all this is so jumbled that no one can be expected to sort through your widely misleading terminological departures. Unless you stipulate clearly that you have something very different in mind, you should stay consistent with the terminology of model theory, especially when you call it 'model theoretic'.
Quoting alcontali
There might be such a notion in another branch of study, but not in ordinary mathematical discussion. By glibly throwing around terminology the way you do, it is, extraordinarily difficult to take your posts in a way that is not confusing let alone terribly misleading.
"You can do with numbers everything that you can do with sets, and the other way around"
I thought that this was clear by clearly mentioning what set theory it is about. "The set theory" in my post is indeed "the theory of hereditarily finite sets".
Quoting GrandMinnow
I was not aware of the fact that the meaning of the terms "language" and "sentence" in computer science would be so out of place in mathematical logic. I am quite surprised by that, but I can see that it is probably true. It is mostly a question of properly qualifying these terms.
As I see it, a model-theoretical model is a grammar (as understood in computer science) along with a set of operators that interprets a theory which is a set axioms (written in again another grammar). I find it a lot easier to understand the concept in these terms.
In fact, I find it quite amazing that the collection of sentences (not logic ones but computer-science ones) constrained by a particular grammar can interpret a set of axioms. You get two unrelated formalisms that are to an important extent equivalent. I find that surprising and intriguing.
That does not mean that I would be interested in everything that ever gets investigated in model theory, such as, for example, the model for the real numbers and so on. I cannot do anything with real numbers, because they are generally not computable.
So, my post was about my understanding about the subject. It was not meant to be a particularly standard take on the matter, or an elaboration on the numerous formalisms and subtleties involved in the formal proof. For that purpose, the reader should rather use the original paper by Richard Kaye and Tin Lok Wong.
In terms of computability, it is not incorrect. Computability cannot handle full ZFC. In computability, everything is fundamentally just a natural number; even its pseudo-real numbers. Hence, in all practical terms, you cannot even do anything with what ZFC can do more than ZF-inf, because computers cannot handle that.
Again, a couple of observations:
(1) If the objective was to just express the finite arithmetical sets, and not interpret ZF-Inf, I don't understand why you don't simply use the first idea that came to your mind. That is, if the set is, say, {2, 3, 4}, then we can express that using the PA formula x=SS0 v x=SSS0 v x=SSSS0. No need to use a sgn function.
(2) In any case, again, note that being able to express a finite set is not the same as being able to interpret ZF-Inf. In particular, expressing a set is achieved by finding a suitable formula that is true only of the members of the set, whereas interpreting ZF-Inf requires one to find terms to stand for sets, and a suitable relation between such terms that captures the notion of belonging. For example, under the Ackermann interpretation, we can name the set {} by 0, the set {{}} by 1, and then we can say that "{ } belongs to {{}}" by way of "the 0th digit of 1 is 1". On the other hand, your supposed interpretation cannot say "the set {} belongs to the set {{}}", because you have neither provided a name for the relevant sets nor provided a suitable relation that captures the set membership relation.
(3) I'm not particularly convinced that a model is a "set of sentences". Consider any non-standard model of PA. How can we capture this non-standard model in a set of sentences? Certainly it is not by taking the set of all sentences true in it, because it is easy to show that there are non-isomorphic, elementarily equivalent non-standard models of PA. So the model outstrips our (first-order) expressive resources.
I joined when the other forum went down, but I haven't been much active. I generally post only when I have some spare time from university work...
Quoting Wikipedia on Löwenheim–Skolem theorem
A nonstandard model is created by throwing a symbol ? into the fray; the result of which is the creation of lots of new sentences in such extended grammar.
Well, it is true that if a first-order theory has a model of a given infinite cardinality, then it has models of every infinite cardinality. But a theory may have only one finite model, say of cardinality n for some natural number n. In that case, it may have no infinite models at all.
Then you're much better saying it rather than depending a very confused "context".
Quoting alcontali
If your usage is is indeed standard computer science, then, yes, the terminology of computer science is radically different from basic mathematical logic. If you insist on using the terminology differently from the way it is used in ordinary discussions about hereditarily finite sets and PA, then you need to clearly state your system of terminology from your own basics rather than, without due specification, mixing it up with ordinary usage.
This is to miss the point. You can pick two non-standard models of the same cardinality and which satisfy the same sentences, but which are nevertheless distinct. So no first-order sentence, indeed not even a set of such sentences, will distinguish them.
Incidentally, note that the procedure you quoted for creating non-standard models is not achieved by introducing just one symbol, k, but by introducing k many symbols; this k can be any infinite cardinal.
The headline reads onto itself and is incorrect. You do somewhat correct it in your post (though, even there, you mistate by saying "ZF minus infinity" rather than "ZF with infinity replaced by the axiom of infinity").
I'm not referring to the cardinality of the universe. I'm referring to the cardinality of each member of the universe.
Well, if a given model has cardinality n, then it has no members of cardinality n+1... unless you're just saying that we can always replace one of the elements of model by a set of any cardinality, treated as a black box. Is this what you have in mind?
This is a notion of non-standard model not in model theory (in mathematical logic) but in computer science?
It is clearly not the notion in model theory.
I only get to figure that out by actually doing it. If I don't do it, then I will never figure out what the problems are, not even the problems with the terminology.
Furthermore, I do not investigate all topics in computer science, many of which are not necessarily interesting to me. An example of how the term "formal language" may appear in this realm:
Quoting Wikipedia on automata theory
The meaning of "formal language" in automata theory is not necessarily the same as in mathematical logic:
Quoting Wikipedia on formal languages
In fact, now I see that I should probably use the term "word" instead of "sentence" for natural numbers, but also not necessarily, because in terminology compiler construction they would still be sentences ("The automaton accepts a sentence"). The term sentence in mathematical logic is probably the most closely related to "those strings for which some decision procedure produces the answer YES".
Your improvised public "figuring out", with confusions conflating terminology from two different areas of stusy, results in posts that are misinformation.
Quoting alcontali
That is very incorrect as far as mathematical logic is concerned.
That would be impossible in computer science.
Computer science can fundamentally not handle infinite cardinalities, because they are generally not computable. They are sometimes handled symbolically, though.
Quoting Wikipedia on NaN
I do not believe that there is any guarantee that this NaN symbol makes sense in a model-theoretical sense. These symbols may show up, but it really depends on the implementation standard what exactly they mean. Furthermore, that symbol is mostly considered to be the result of a bug, which should not even show up ...
Then it's model theory. And your statement about it is plainly incorrect.
The confusion is merely in the lack of formality; but that was actually expected by the reader. Furthermore, what you write, does not necessarily help anybody to understand the subject either.
Many infinite-size models cannot be handled by any grammar, because their sentences are necessarily countable, while these models may not have a countable cardinality. It can only work for something like ZF-inf, of countable cardinality.
So, no, I actually agree, throwing that symbol into the fray, won't make it work.
Very much not merely a lack of formality. It is dreadful confusion and misinformation.
Quoting alcontali
I try to write correct statements, as best I can within the limits of posts. Much of that is merely point blank stating what is correct without necessarily giving an explanation, let alone an entire explanation. I cannot, in the space of posts, wind backwards through the trail of concepts to primitives for every concept. But at least, most importantly, I try not to write misinformation.
For actual understanding, one needs to read the basic textbooks in the subject, not just posts in a forum.
Adding a symbol is not relevant, not due to whatever you said about sizes of models, but rather more simply that it is not even involved in the notion of a non-standard model.
That's computer science, or a special bland of computer science and model theory that you are working out live?
I predict all formats of 1-9 creates a number more like 8 on averages.
If I used the universe as a computer and list all categories a pattern would emerge of a pyramid common and uncommon shape, creating links with numbers. There would be the times that it's normative and times that it isn't.
It's a pattern we can notice about even more oddly defined natural number 1-9.
In not using the universe as a judge you would lose track of symmetrical nature of physical, written list. Writing randomly about a page would create too much harmony.
I can imagine numbers swirling in, they happen to have this special relationship.
True positivity.
8 when defined, but thought of in pespective with base 8 planting 8 in base 4, because base 8 is possible ( A good view of all sectors of number if positivity not implied).
A view inside the square as a symbol or math.
Base 8 is just that number out qualifies base 4, and it's written sequence can be created with an answer, and in base 8 this answer has a locale.
Control over energy, knowledge is significance.
The method of equalization.
Did you know? Game producers have been making critical engine mistakes. 3D is not done like it's thought.
Experienced 3D has negative and postive depth, we understand it wrong. There is no perfectly centred cube. When you make game engines you focus more on an upgradable net.
Quoting Wikipedia on construction of nonstandard models
This is typically represented by adding a symbol ?. If such model were of countable size then adding that symbol would be enough to represent these additional elements as: ... ?-2, ?-1, ?, ?+1, ?+2, ...
In her slides, Victoria Gitman explains this literally as:
slide 10: "Expand LA by adding a constant c to obtain the language LA? = (+,·,<,0,1,c)."
slide 11: "M has c+ 1,c+ 2,c+ 3 ,... as well as c?1,c?2,c?3,...."
So, what is wrong with her slides?
Again, as I have said already, it would be enough to expand the grammar of the model with a constant, if the size of the model remained countable. It does not work for larger models because grammar cannot represent uncountable models.
Quoting GrandMinnow
Look, if you do not want to make constructive remarks, then why do you make any remarks at all?
No, you have it very wrong.
The universe has additional members. We don't add symbols that engender sentences not already in the theory.
You are completely confused about this and other topics in mathematical logic. Please stop and instead first read an introductory textbook in the subject.
Quoting alcontali
My remarks have corrected your misinformation. That is constructive. And a further constructive remark is that you need to read a basic textbook in the subject instead of spreading wanton misinformation about it.
Anyway my question stands: In the earlier passage, what are you talking about? Model theory, computer science or your own blend of them?
And I have no idea what you mean by w-1 and w-2 in the context of saying w-1, w-2, w, w+1, w+2. Do you have a definition of some kind of operation of ordinal subtraction that allows subtracting 1 or 2 from a limit ordinal? What exact set or ordinal type do you mean by w-1 and w-2? Actually, nevermind, better you should instead get a book on working in the first order precicate calculus, then an intro book on set theory and then an intro book on mathematical logic.
That you reference those slides to misunderstand them, is a function of relying on various Wikipedia articles and Google searches for various presentation, out of context of a systematic treatment such as in a good textbook.
I asked you what is wrong with Victoria Gitman's slides.
Quoting GrandMinnow
You can find that literally in her slides. Did you even look at them?
Quoting GrandMinnow
I like her slides.
Unlike your vitriolic remarks, they add to the understanding of the topic. In fact, I do not care about what you may think about her slides, because she turns out to be useful while you are not. Her representation of ... ?-2, ?-1, ?, ?+1, ?+2, ... is also quite clarifying, unlike what you are doing.
Quoting GrandMinnow
It is my OP. I posted the topic. If you do not like it, feel free to unceremoniously fuck off, will you?
And I looked at the Wikipedia page you referenced. The method of adding a constant is a step in proving the existence of non-standard models of PA. But the actual non-standard model is on a language that does not include the constant.
And what you wrote is not, as you are claiming, literally what she wrote. She mentions an arbitrary constant ‘c’. But you introduced in particular ‘w’ (omega), so I would think you have in mind omega (the set of natural numbers).
So, by w-1, do you mean some kind of ordinal subtraction, or something else?
It’s not vitriolic to point out that you are posting misinformation and that you would do better to read a textbook.
That you posted a topic doesn’t provide that people should not correct your misinformation nor advise that you wouldn’t be so prone to confusions if you read a textbook.
She extends the language of the model in slide 10: "Expand LA by adding a constant c to obtain the language LA? = (+,·,<,0,1,c)."
The term "language" in her slides is something that can have an uncountable number of sentences. Her model cannot have a CS grammar, unlike the standard model, because CS grammars can only represent a countable number of sentences.
Quoting GrandMinnow
Ask Victoria Gitman about what she means with that term in her slide. I think that I understand it, but better ask her instead.
Quoting GrandMinnow
Since you are not much useful in correcting errors, besides merely criticizing other people, your contribution is simply worthless and just annoying. I am not an expert on model theory. I just gave my understanding on something it investigates. What you are doing, however, is completely worthless. In what sense do you believe that your vitriolic remarks would be helpful in any way? If they successfully prevent people from discussing the topic, will you have achieved your nefarious goals? You are not here to discuss anything. You are here to point out that you believe that you know the subject better than anybody else here, mostly by putting them down. Don't you see that nobody needs or even appreciates your presence? Your obnoxious attitude turns you into a worthless individual whom nobody likes to be around.
The axioms of PA do not have the added constant. There is a difference between (1) a proof of existence through a detour via another language and (2) an actual model of PA.
What I wrote there (and with similar remarks on this particular point) may be too strict. It belies that there may be a less narrow notion of 'model of'.
Narrow:
A model M of a theory T is model for the language of T, such that every sentence in T is true in M (a theory being a set of sentences closed under deduction). So a model of PA is model for the language of PA (which does not include an added constant 'c'). So a model M that, for example, Gitman proves to exist is not itself for the language of PA and therefore not a model of PA, but then we restrict such a model to the language of PA and that restriction is a model of PA.
Less Narrow:
We could allow that a model M of a theory T is model for a language that includes the language of T, such that every sentence in T is true in M. So a model M that, for example, Gitman first proves to exist is itself a model of PA, and we don't need to then restrict the the model.
Quoting alcontali
She doesn't preclude uncountable languages in general. But where she proves the existence of non-standard models of PA, the languages mentioned are countable languages. Uncountability does not have a role in it.
This is the point where I somehow backtracked on what Gitman wrote, while I initially thought it was actually really good. I really ran off with the idea that a model is a language. It clarified the difference between a theory, i.e. a set of axioms, and a model, i.e. a CS-like grammar describing sentences. I still think of things like that for countable models, but I have to leave it open if the concept properly applies to models that are larger than that, such the nonstandard models of PA.
Countability is often assumed to be an inherent property of any language in other arguments such as in Richard's paradox:
Quoting Wikipedia on Richard's paradox
The argument is clearly very similar to Cantor's diagonal argument. In the explanation above, language itself is clearly considered countable.
I think that it would be useful for Gitman to explain why such uncountable nonstandard model would still be a legitimate language.
I currently see it more as some kind of dense cloud, centred around a choice for ?, of number symbols/strings, that cannot be adequately captured by the concept of language.
Writers might differ on the definition of 'theory', but it turns roughly the same with whatever adjustments we need:.
(1) A theory is a set of sentences closed under deduction. The theorems of the theory are the members of the theory.
That definition has the advantage that we can consider theories without having a particular set of axioms in mind. For example, given a model M, we have Th(M), which is the set of sentences true in M. So an axiomatization doesn't have to be specified.
(2) A theory is the set of theorems derivable from a particular set of axioms.
With that definition, given a set of axioms S, the theory axiomatized by S is the set of sentences derivable from S.
Quoting alcontali
I don't claim that is not a notion in computer science, but in context such as Gitman, which is set_theoretic/mathematical_logic/model_theory, that is not how the concept of a model is formulated.
Quoting alcontali
The particular nonstandard models of PA that Gitman is discussing are countable models. There are countable nonstandard models of PA and uncountable nonstandard models of PA. And, in the first definition of 'model of' I mentioned in the earlier post, the language is countable, and with the second definition, an uncountable language for a model of PA can be restricted to a countable language.
Quoting alcontali
A model is not a language.
An uncountable model is one in which the universe for the model is an uncountable set. Every theory that has an infinite model has both countable and uncountable models, even as the language for the theory is countable.
Well, Gitman clearly mentions that she considers a model to be (some kind of) language:
slide 10: "Expand LA by adding a constant c to obtain the language LA? = (+,·,<,0,1,c)."
So, that is why I first said something similar: "Just add a Löwenheim-Skolem ? symbol to the model's language."
Quoting Wikipedia on Löwenheim-Skolem
These infinite cardinals are clearly an element in Cantor's [math]\aleph[/math] (or [math]\beth[/math]) sequence.
Therefore, Gitman's "language" does not always have a countable number of sentences. So, what kind of "language" is it supposed to be?
She's adding a constant to the LANGUAGE of the theory. The model is a model for the language. The model is not a language.
There is a language. Then there are two things.
(1) A model for the language.
(2) A theory in the language.
Then the model is a model of the theory iff (the theory is in the language, and the model is a model for that language, and every sentence in the theory is true in the model).
Yes, as I said, if a theory has an infinite model, then it has both countable and uncountable models.
Quoting alcontali
That's a non sequitur. The set of sentences is countable but, if there is an infinite model of the theory, then there are countable and uncountable models of the theory.
And Gitman goes on to discuss particular countable models.
The language, the theory, and the model are separate, but related things.
If you like, I can recommend introductory textbooks in this subject that will explain all this for you, step-by-step, in greater detail and context than I can give in posts.
Quoting GrandMinnow
Quoting Wikipedia on the term signature
Yes, I see now, what Gitman wrote, is an extension of the signature, and therefore of the language of the theory. If it is a countable model then the extension will also show up in its (CS) definable language, but that is clearly not possible for nonstandard models of PA.
Quoting GrandMinnow
There does not seem to be much that can be downloaded, but if you have any links to good material with examples, feel free to post them. From the paucity of materials online, the subject does not seem to be that popular, actually.
Exactly.
Quoting alcontali
I don't know about online, but here is a course of books I highly recommend, in order of study:
* Logic: Techniques of Formal Reasoning - Kalish, Montague, Mar
This tells you how to make sure that your mathematical arguments are within the first order predicate calculus. But if you feel that your mathematical reasoning is not prone to such things as mistakes with quantifiers and improper instantiations of variables, then you can skip this book. Though I still highly recommend it to make really sure you're always on firm ground.
* Elements Of Set Theory - Enderton
From the previous book, you will have a good grasp of proof in the first order predicate calculus you need for a rigorous study of set theory.
* A Mathematical Introduction To Logic - Enderton
From the previous book, you will know the set theory you need for a rigorous study of mathematical logic.
It looks like worthless dead-tree material. I don't do that. There is no longer a need to cut trees in order to publish information. There is also no need any longer to physically sit a room for that purpose. We do not need these dinosaurs. If they do not put it online, then someone will put something better online.
With a view on staying relevant, maybe you and your friends can start reading HTML for dummies?
But did you even look to see whether these might also be online?
What friends? You said I don't have any.
In that case, read "HTML for dummies" alone.
Look, the academia are stuck in a highly inefficient way of transmitting information while charging an arm and a leg in the process. Now, this subject is not vocational at all. So, it is indeed fun as a hobby, but the only thing you can professionally do with it, is teaching in the academia, and then also get stuck in highly inefficient ways of doing things while further bankrupting an already corrupted youth. I only use online resources for my hobbies. In fact, I only use online resources for professional activities too. So, we are not going to get anywhere discussing these things any further, because we simply live in two different worlds, while I more or less despise yours. So, no, we are simply not going to do things your way, and there is no need to argue over that. We'd better agree to disagree.
notation: for a function f, let f’y= the x such that fx =y.
Let M be a model for the language L.
Let S be any set whatsoever with the same cardinality as the universe U for M. Doesn’t matter the cardinalities of the members of S. Let f be a bijection from U onto S.
Then, for example, suppose ‘H’ is a relation symbol in L, and suppose M maps ‘H’ to the relation R in UxU.
Let M* be the model with universe S, and, for example, M* maps the symbol ‘H’ to {
So M and M* are isomorphic.
Or did I make a mistake?
Happily, public universities are free in Brazil, so it is possible here to obtain higher education without incurring in large debt!
I'm not sure what your point is. Can you clarify?
Well, they are not really "free". The government still has to pay for them. At first glance, that is not the individual's problem, but sooner or later, it will still be.
The government naturally collects money from people from whom it is relatively easy to do that: the wage slaves in Brazil. In theory, they would try to mostly collect it from who has the highest income, but those are exactly the people who will most easily adjust to whatever is needed to avoid paying. For example, they avoid wage slavery. They do not receive too much income through that kind of contracts.
The poor, on the other hand, do not have much income anyway. It does not even matter how they receive it. So, it is not the poor who can keep that system financially afloat.
So, a strategy of large public expenditure depends on having a large middle class of easily taxable wage slaves, which is exactly the demographic that tends to disappear: smart enough to make more money than the poor in society but not smart enough to prevent the government from taxing their income away.
Quoting Nagase
In that case, it is the government which incurs the debt.
They will try to get it back from the middle class, but only if spending on university really leads to increasing the size of the middle class of wage slaves. If the students rather graduate into working part time jobs at Starbucks, this strategy will still fail.
The biggest problem is that university does not increase the graduate's productivity. In fact, it is bad business to waste 4+ years of your life memorizing phone books replete with unimportant trivia.
If the government's plan fails financially, there is a real risk that their debt will still somehow end up being your problem. The funny thing is that the more money you have, the less it will affect you, because in that case, you already have lots of workarounds for the problem of the government trying to collect money from you.
I mentioned (putting it in these terms now) that for any theory T and any cardinality C, if there is a model M of T, then there is a model M* of T such that the universe for M* has a member of cardinality C. (Though originally I forgot to mention the obvious qualification that this pertains when there is a model M of T.)
You asked me what I had in mind. So in my previous post, I gave the proof.
Let's recap the discussion. I mentioned that the universes of ZF-Inf are all infinite, and remarked that this easily followed from the Power Set axiom. You then replied that "for any theory, and for any cardinality, there is a model of the theory such that the universe of the model has a member of that cardinality." I was puzzled by this and offered two interpretations: (i) on one interpretation, you were mentioning the fact that every theory with an infinite model has a model in any cardinality or (ii) you were mentioning the fact that, for any model M, it is possible to construct another model M* whose domain is composed of whatever you want. Now, (ii) was irrelevant to my remark, because I was talking about the size of the model, not the size of its members. And (i) forgets the fact that some theories have only finite models (say T implies "there is a y such that for every x, x=y").
Quoting Nagase
When you say "its sets", I take it you mean the universes's sets, i.e. the sets that are in the universe. Of course, for any particular universe, there is a bound on the sizes of the sets. So I took it that you meant that for any set size there are universes for the theory that have members of that set size. And I agree. My point is that this holds for any theory whatsover, regardless of its axioms or, in particular the power set axiom.
Quoting Nagase
If by 'members' you mean members of the universe of the model, then the above is incorrect. Trvially, let the universe of some model be {2}. The universe has cardinality n=1 but it has a member of cardinality n+1. And beyond the trivial, for infinite universes, it is not precluded that a universe is of cardinality C but the universe has a member of cardinality C+1, where C+1 is the successor cardinal of C.
In the remarks you quote, I had in mind transitive models. My bad for not making that clearer.
That works. Thanks.