What is X? Ordered Proper-Class-Sized Sequence

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In summary, the conversation discusses a proper-class-sized sequence, denoted as a, that consists of elements from 0 to 9. These elements are ordered lexicographically or antilexicographically to form a class called X. However, it is noted that X cannot technically be a proper class due to its inclusion of other proper classes. This is resolved by considering X as a "second order class". The conversation also touches on the idea that X cannot contain every linearly ordered set.
  • #1
Dragonfall
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This may be a little hand-wavy:

Let [tex]a[/tex] be an ordered, proper-class-sized sequence [tex]a=(a_0,a_1,...,a_{\omega},...,a_{\omega_2},...,a_{\omega_{\omega}},...)[/tex] where [tex]a_i, i\in\mathbb{O}rd[/tex] are, say, 0,...,9. So that if we look only at those [tex]a[/tex] whose expansion on [tex]a_{\omega}[/tex] onwards are 0, we'd get something like the real numbers.

We order these things lexicographically (or antilexicographically, whichever it is that the reals are ordered by, I can never remember). So let X be the class of these things. What is X?
 
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  • #2
I just realized that X can't technically be a proper class since X contains proper classes. Formally we'll have to invoke some sort of type theory argument and call X a "second order class" or something. But that's just a technicality.
 
  • #3
I think what I mean is that "there is no class X such that every linearly ordered set is isomorphic to a subset of X".
 

What is the definition of a proper-class-sized sequence?

A proper-class-sized sequence is a sequence of objects that is infinitely long and contains more objects than can be counted by any finite number. It is called a proper class because it cannot be treated as a set within the framework of set theory.

What is the difference between a proper-class-sized sequence and an infinite sequence?

The main difference between a proper-class-sized sequence and an infinite sequence is that a proper-class-sized sequence contains more objects than can be counted by any finite number, while an infinite sequence contains an uncountable number of objects but it is still finite in size.

How is a proper-class-sized sequence represented mathematically?

A proper-class-sized sequence is usually represented using set-builder notation, where the objects in the sequence are defined by a property or rule. For example, the proper-class-sized sequence of all positive integers can be represented as {x | x > 0}.

What is the significance of proper-class-sized sequences in mathematics?

Proper-class-sized sequences play an important role in mathematics, particularly in the study of set theory and logic. They help to define and understand the concept of infinity and provide a way to reason about objects that are too large to be contained within a set.

Can a proper-class-sized sequence be visualized or imagined?

No, it is not possible to visualize or imagine a proper-class-sized sequence because it contains an infinite number of objects that are beyond our ability to comprehend or visualize. It can only be represented and understood mathematically.

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