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julypraise
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Let [itex]\mathcal{S} = \{S_{i}:i \in \mathbb{R} \} [/itex] where [itex]S_{i}[/itex] is a set. Then [itex]\mathcal{S}[/itex] is a set? Or, can this notation make sense in some way?
julypraise said:Let [itex]\mathcal{S} = \{S_{i}:i \in \mathbb{R} \} [/itex] where [itex]S_{i}[/itex] is a set. Then [itex]\mathcal{S}[/itex] is a set? Or, can this notation make sense in some way?
Because the [itex]S_i[/itex] are sets, this is valid set builder notation defining a class [itex]\mathcal{S}[/itex]. And by the axiom of replacement and the fact [itex]\mathbb{R}[/itex] is a set, the class [itex]\mathcal{S}[/itex] is indeed a set.julypraise said:Let [itex]\mathcal{S} = \{S_{i}:i \in \mathbb{R} \} [/itex] where [itex]S_{i}[/itex] is a set. Then [itex]\mathcal{S}[/itex] is a set? Or, can this notation make sense in some way?
Not anything. There isn't, for example, a set of all sets that don't contain themselves!Whovian said:A set can be of anything. Even {Lincoln, Charizard, {Fish Fingers, Custard}} is a set. Your set is, therefore, valid. Note that its cardinality is one of the alephs, I don't know which one.
Hurkyl said:Not anything. There isn't, for example, a set of all sets that don't contain themselves!
julypraise said:Let [itex]\mathcal{S} = \{S_{i}:i \in \mathbb{R} \} [/itex] where [itex]S_{i}[/itex] is a set. Then [itex]\mathcal{S}[/itex] is a set? Or, can this notation make sense in some way?
DonAntonio said:I can't see why you think S couldn't be a set, as long as each [itex] S_i[/itex] is...What did you have in mind?
DonAntonio
julypraise said:You know, the concept of indexing in my mind (in my intuition) is kind of a countable process. But then now the index set is a continuum. So I thought it might not be possible; I mean this kind of indexing might not be possible by ZFC.
A class indexed by real numbers refers to a set of objects or elements that can be assigned a unique numerical value according to the real number system. This means that every element in the class can be identified by a real number, making it a well-defined and organized collection.
While both a class indexed by real numbers and a set are collections of elements, the main difference is that a set can only contain elements with distinct and well-defined properties, while a class can contain elements with more complex or abstract properties that cannot be easily defined. Additionally, a set is a well-defined collection, while a class is a collection that is defined by a property or rule.
Yes, a class indexed by real numbers can be infinite. Since real numbers have infinite values, a class indexed by real numbers can also have an infinite number of elements. However, it is important to note that not all classes indexed by real numbers are infinite, as it depends on the specific properties and rules of the class.
A class indexed by real numbers is a useful tool in science for organizing and categorizing data. It allows for a more precise and systematic way of analyzing and interpreting data, as well as making predictions and drawing conclusions. Additionally, real numbers are often used to represent physical quantities in scientific calculations and experiments.
No, a class indexed by real numbers can be used in various fields and applications, not just limited to science. It can be used in mathematics, economics, computer science, and many other disciplines that require the organization and categorization of data. Real numbers are a fundamental concept in many areas of study, making a class indexed by real numbers a valuable tool in any field.