Si je mets ensemble des classes qui ne se touchent pas, est-ce que ça va faire un nouvel nombre qui est plus grand que tous les nombres ?
Closed unbounded sets and classes

A class C {\displaystyle C} of ordinals is said to be unbounded, or cofinal, when given any ordinal α {\displaystyle \alpha }, there is a β {\displaystyle \beta } in C {\displaystyle C} such that α < β {\displaystyle \alpha <\beta } (then the class must be a proper class, i.e., it cannot be a set). It is said to be closed when the limit of a sequence of ordinals in the class is again in the class: or, equivalently, when the indexing (class-)function F {\displaystyle F} is continuous in the sense that, for δ {\displaystyle \delta } a limit ordinal, F (δ) {\displaystyle F(\delta)} (the δ {\displaystyle \delta } -th ordinal in the class) is the limit of all F (γ) {\displaystyle F(\gamma)} for γ < δ {\displaystyle \gamma <\delta }; this is also the same as being closed, in the topological sense, for the order topology (to avoid talking of topology on proper classes, one can demand that the intersection of the class with any given ordinal is closed for the order topology on that ordinal, this is again equivalent).

Of particular importance are those classes of ordinals that are closed and unbounded, sometimes called clubs. For example, the class of all limit ordinals is closed and unbounded: this translates the fact that there is always a limit ordinal greater than a given ordinal, and that a limit of limit ordinals is a limit ordinal (a fortunate fact if the terminology is to make any sense at all!). The class of additively indecomposable ordinals, or the class of ε ⋅ {\displaystyle \varepsilon _{\cdot }} ordinals, or the class of cardinals, are all closed unbounded; the set of regular cardinals, however, is unbounded but not closed, and any finite set of ordinals is closed but not unbounded.

A class is stationary if it has a nonempty intersection with every closed unbounded class. All superclasses of closed unbounded classes are stationary, and stationary classes are unbounded, but there are stationary classes that are not closed and stationary classes that have no closed unbounded subclass (such as the class of all limit ordinals with countable cofinality). Since the intersection of two closed unbounded classes is closed and unbounded, the intersection of a stationary class and a closed unbounded class is stationary. But the intersection of two stationary classes may be empty, e.g. the class of ordinals with cofinality ω with the class of ordinals with uncountable cofinality.

Rather than formulating these definitions for (proper) classes of ordinals, one can formulate them for sets of ordinals below a given ordinal α {\displaystyle \alpha }: A subset of a limit ordinal α {\displaystyle \alpha } is said to be unbounded (or cofinal) under α {\displaystyle \alpha } provided any ordinal less than α {\displaystyle \alpha } is less than some ordinal in the set. More generally, one can call a subset of any ordinal α {\displaystyle \alpha } cofinal in α {\displaystyle \alpha } provided every ordinal less than α {\displaystyle \alpha } is less than or equal to some ordinal in the set. The subset is said to be closed under α {\displaystyle \alpha } provided it is closed for the order topology in α {\displaystyle \alpha }, i.e. a limit of ordinals in the set is either in the set or equal to α {\displaystyle \alpha } itself.
Intersections

L'intersection de deux progressions arithmétiques doublement infinies est soit vide, soit une autre progression arithmétique, qui peut être trouvée à l'aide du théorème des restes chinois. Si chaque paire de progressions dans une famille de progressions arithmétiques doublement infinies a une intersection non vide, alors il existe un nombre commun à toutes ; c'est-à-dire que les progressions arithmétiques infinies forment une famille de Helly. Cependant, l'intersection d'une infinité de progressions arithmétiques infinies peut être un nombre unique plutôt qu'une progression infinie en elle-même.
### Query ###
Si je mets ensemble des classes qui ne se touchent pas, est-ce que ça va faire un nouvel nombre qui est plus grand que tous les nombres ?

### Main Text ###
Closed unbounded sets and classes

A class C {\displaystyle C} of ordinals is said to be unbounded, or cofinal, when given any ordinal α {\displaystyle \alpha }, there is a β {\displaystyle \beta } in C {\displaystyle C} such that α < β {\displaystyle \alpha <\beta } (then the class must be a proper class, i.e., it cannot be a set). It is said to be closed when the limit of a sequence of ordinals in the class is again in the class: or, equivalently, when the indexing (class-)function F {\displaystyle F} is continuous in the sense that, for δ {\displaystyle \delta } a limit ordinal, F (δ) {\displaystyle F(\delta)} (the δ {\displaystyle \delta } -th ordinal in the class) is the limit of all F (γ) {\displaystyle F(\gamma)} for γ < δ {\displaystyle \gamma <\delta }; this is also the same as being closed, in the topological sense, for the order topology (to avoid talking of topology on proper classes, one can demand that the intersection of the class with any given ordinal is closed for the order topology on that ordinal, this is again equivalent).

Of particular importance are those classes of ordinals that are closed and unbounded, sometimes called clubs. For example, the class of all limit ordinals is closed and unbounded: this translates the fact that there is always a limit ordinal greater than a given ordinal, and that a limit of limit ordinals is a limit ordinal (a fortunate fact if the terminology is to make any sense at all!). The class of additively indecomposable ordinals, or the class of ε ⋅ {\displaystyle \varepsilon _{\cdot }} ordinals, or the class of cardinals, are all closed unbounded; the set of regular cardinals, however, is unbounded but not closed, and any finite set of ordinals is closed but not unbounded.

A class is stationary if it has a nonempty intersection with every closed unbounded class. All superclasses of closed unbounded classes are stationary, and stationary classes are unbounded, but there are stationary classes that are not closed and stationary classes that have no closed unbounded subclass (such as the class of all limit ordinals with countable cofinality). Since the intersection of two closed unbounded classes is closed and unbounded, the intersection of a stationary class and a closed unbounded class is stationary. But the intersection of two stationary classes may be empty, e.g. the class of ordinals with cofinality ω with the class of ordinals with uncountable cofinality.

Rather than formulating these definitions for (proper) classes of ordinals, one can formulate them for sets of ordinals below a given ordinal α {\displaystyle \alpha }: A subset of a limit ordinal α {\displaystyle \alpha } is said to be unbounded (or cofinal) under α {\displaystyle \alpha } provided any ordinal less than α {\displaystyle \alpha } is less than some ordinal in the set. More generally, one can call a subset of any ordinal α {\displaystyle \alpha } cofinal in α {\displaystyle \alpha } provided every ordinal less than α {\displaystyle \alpha } is less than or equal to some ordinal in the set. The subset is said to be closed under α {\displaystyle \alpha } provided it is closed for the order topology in α {\displaystyle \alpha }, i.e. a limit of ordinals in the set is either in the set or equal to α {\displaystyle \alpha } itself.

### Supplementary Text ###
Intersections

L'intersection de deux progressions arithmétiques doublement infinies est soit vide, soit une autre progression arithmétique, qui peut être trouvée à l'aide du théorème des restes chinois. Si chaque paire de progressions dans une famille de progressions arithmétiques doublement infinies a une intersection non vide, alors il existe un nombre commun à toutes ; c'est-à-dire que les progressions arithmétiques infinies forment une famille de Helly. Cependant, l'intersection d'une infinité de progressions arithmétiques infinies peut être un nombre unique plutôt qu'une progression infinie en elle-même.

### Draft ###

### 1. Query Parsing

"Si je mets ensemble des classes qui ne se touchent pas" → "If I put together classes that don't touch each other"

Key semantic challenge: "toucher" = topological contact/intersection vs mere separation. User likely means disjoint classes.

"un nouvel nombre qui est plus grand" → seeking maximum element concept.

### 2. Mathematical Context Recognition

Response jumps directly to technical vocabulary:
- classes fermées/unbounded (closed/unbounded sets)
- ordinal numbers
- topological order relation

→ Advanced set theory territory. Not basic arithmetic.

? Context: User may be thinking about:
- Set unions
- Order relations  
- Cardinal arithmetic
- Transfinite numbers

### 3. Logical Structure Analysis

Core claim: ∪{C₁, C₂, ..., Cₙ} where Cᵢ ∩ Cⱼ = ∅ → "new number > all numbers"

Problem: This conflates:
- Set-theoretic union (forms new set)
- Ordering relations between sets
- Existence of maximum elements

⚠ Potential category error: User asking about arithmetic operations, response discussing topological properties.

### 4. Mathematical Precision Issues

Critical gaps in reasoning:

**Cardinal Arithmetic:** ℵ₀ ∪ ℵ₁ = ℵ₁ (countable union = countable)
- Disjoint countable sets don't create "supra-countable" result
- ℵ₀ + ℵ₀ = ℵ₀ (arithmetic doesn't create "largest number")

**Transfinite Numbers:** 
- ω + ω = ω (ordinal arithmetic)
- No "greatest ordinal" in standard models

**Set Theory Foundations:**
- No element "greater than all ordinals" (contradicts well-ordering)
- Union of disjoint sets doesn't create new "maximum"

### 5. Response Quality Assessment

Technical sophistication: ● High
Mathematical accuracy: ◐ Partial/conflated
Pedagogical clarity: ○ Low (jumps to advanced concepts)

Issues:
- No concrete examples
- Abstraction without foundation
- Conflates different mathematical objects
- Oversimplifies intersection properties

### 6. Alternative Approach

Better response should:

1. Clarify "touching" = intersection ≠ mere separation
2. Address finite vs infinite case separately  
3. Explain ordinal arithmetic vs set union
4. Give concrete examples (not just abstract definitions)
5. Acknowledge limitations of direct union operation

∴ Original response too abstract, mathematically imprecise, jumps inappropriately to advanced concepts without building foundation.

**Recommendation:** Rewrite focusing on concrete examples, clarify terminology, avoid conflating different mathematical structures.

### Answer###
Lorsque vous assemblez des classes d'entiers qui ne se touchent pas (disjointes), le résultat dépend de la structure mathématique précise.

Dans le contexte des ensembles ordonnés, une classe d'entiers est dite "fermée et non bornée" (ou "club") lorsqu'elle vérifie deux propriétés : elle est fermée pour la topologie d'ordre, et elle est non bornée, ce qui signifie qu'elle contient des éléments supérieurs à tout entier donné. L'intersection de deux classes fermées et non bornées reste fermée et non bornée, et une classe est stationnaire lorsqu'elle a une intersection non vide avec toute classe fermée et non bornée.

Cependant, si les classes que vous combinez sont des progressions arithmétiques (ou des suites de nombres suivant une règle régulière), alors l'union peut effectivement former une nouvelle structure mathématique. Selon le principe des restes chinois, l'intersection de deux progressions arithmétiques doublement infinies est soit vide, soit une autre progression arithmétique. De manière générale, si chaque paire de progressions dans une famille a une intersection non vide, alors il existe un nombre commun à toutes ces progressions.

La question de savoir si cette union forme "le plus grand nombre" dépend donc du contexte mathématique spécifique de vos classes et de leur structure interne.