The class indexed by real numbers is a set?

• julypraise
In summary, the conversation discusses the concept of set builder notation and its validity in creating a set from a class of sets. It also addresses the convention of indexing by natural numbers and how it can be extended to indexing by any set, including the continuum. The conversation also mentions the possibility of indexing by means of Zermelo's Well Ordering Theorem. Overall, the conversation concludes that indexing by real numbers is a valid form of set builder notation.
julypraise
Let $\mathcal{S} = \{S_{i}:i \in \mathbb{R} \}$ where $S_{i}$ is a set. Then $\mathcal{S}$ is a set? Or, can this notation make sense in some way?

julypraise said:
Let $\mathcal{S} = \{S_{i}:i \in \mathbb{R} \}$ where $S_{i}$ is a set. Then $\mathcal{S}$ is a set? Or, can this notation make sense in some way?

I can't see why you think S couldn't be a set, as long as each $S_i$ is...What did you have in mind?

DonAntonio

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.

julypraise said:
Let $\mathcal{S} = \{S_{i}:i \in \mathbb{R} \}$ where $S_{i}$ is a set. Then $\mathcal{S}$ is a set? Or, can this notation make sense in some way?
Because the $S_i$ are sets, this is valid set builder notation defining a class $\mathcal{S}$. And by the axiom of replacement and the fact $\mathbb{R}$ is a set, the class $\mathcal{S}$ is indeed a set.

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.
Not anything. There isn't, for example, a set of all sets that don't contain themselves!

Hurkyl said:
Not anything. There isn't, for example, a set of all sets that don't contain themselves!

True. Sorry for poor wording. "Almost anything" would've been a better wording.

julypraise said:
Let $\mathcal{S} = \{S_{i}:i \in \mathbb{R} \}$ where $S_{i}$ is a set. Then $\mathcal{S}$ is a set? Or, can this notation make sense in some way?

That's a perfectly valid set. But note that by convention, indexing by $i$ typically indicates indexing over the natural numbers. For clarity, it would be better to write

$\mathcal{S} = \{S_{\alpha}:\alpha \in \mathbb{R} \}$

which provides readers with an indication that we are indexing over a set other than the natural numbers.

DonAntonio said:
I can't see why you think S couldn't be a set, as long as each $S_i$ is...What did you have in mind?

DonAntonio

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.

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.

I see. But, as already noted by others, it is possible to index by means of any set, no matter its cardinality.

In ZFC we can even use Zermelo's Well Ordering Theorem to well order ℝ and then well-order the so indexed sets, if so wanted...

DonAntonio

1. What is a class indexed by real numbers?

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.

2. How is a class indexed by real numbers different from a set?

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.

3. Can a class indexed by real numbers be infinite?

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.

4. How is a class indexed by real numbers used in science?

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.

5. Is the class indexed by real numbers limited to only scientific applications?

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.

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