What is the biggest ordinal that exists metamathematically?

  • Context: Graduate 
  • Thread starter Thread starter Garrulo
  • Start date Start date
  • Tags Tags
    Ordinal
Click For Summary

Discussion Overview

The discussion revolves around the concept of the largest ordinal that exists metamathematically, exploring the nature of ordinals, cardinals, and the implications of infinite sets. Participants engage with both theoretical and conceptual aspects of ordinals and their relationships to larger infinities.

Discussion Character

  • Exploratory
  • Debate/contested
  • Conceptual clarification

Main Points Raised

  • One participant questions the existence of the biggest ordinal, suggesting that it can always be increased by adding one.
  • Another participant raises the issue of the lack of a maximum cardinal in metamathematics and inquires about the collection of mathematical sets that exist in this context.
  • Several participants mention Graham's number as a large mathematical construct, but one clarifies that it is finite and not relevant to the discussion of transfinite ordinals.
  • A participant argues that for any set, there exists a larger set, leading to the conclusion that there is no largest set, which complicates the notion of a set of everything.
  • Another participant introduces the concept of a proper class, which cannot be contained within another class, as a way to address the limitations of sets.

Areas of Agreement / Disagreement

Participants express differing views on the existence of the largest ordinal and the implications of infinite sets. There is no consensus on the nature of ordinals versus cardinals, and the discussion remains unresolved regarding the largest ordinal and the concept of proper classes.

Contextual Notes

Some statements depend on specific definitions of ordinals and cardinals, and the discussion touches on unresolved mathematical concepts related to infinite sets and proper classes.

Garrulo
Messages
61
Reaction score
0
What is the biggest ordinal that exists metamathematically??
 
Physics news on Phys.org
Whatever the biggest ordinal is, I can name it plus 1. Discuss.
 
Well, if there is not a maximum cardinal metamatemathically existing, ¿which is the colection of this matematical sets than exist metamathematically
 
But the Graham´s number is mathematically finite, isn´t it? I talk about ordinal numbers transfinite
 
Given any set - infinite or not - there is another set that is larger. If the set is infinite, then the other set is a "larger infinity" which means that it is so big that there is no way to ever match it up with the first.

It follows that there is no largest set and the idea of a set of everything makes no sense.
 
Not a set, but it exists the concept of propper class. A class that can´t be in another class
 

Similar threads

Undergrad When can Ordinal Variables be treated as Interval Variables?
  • · Replies 8 ·
Replies
8
Views
3K
Undergrad Limits of Two Ordinal Sequences
  • · Replies 14 ·
Replies
14
Views
2K
Undergrad Ordinal Arithmetic: Proving X is Countably Compact
  • · Replies 1 ·
Replies
1
Views
2K
Undergrad Enumerating a Large Ordinal: Can We Find a Limit to the Continuum?
  • · Replies 27 ·
Replies
27
Views
4K
Graduate Measures of Center/Spread in Categorical/Ordinal
  • · Replies 1 ·
Replies
1
Views
2K
MHB Properties of the Ordinals ....
  • · Replies 1 ·
Replies
1
Views
1K
Graduate How can we construct ordinals after large Veblen?
  • · Replies 18 ·
Replies
18
Views
5K
Undergrad A Sequence T based on the Rule of Three
  • · Replies 15 ·
Replies
15
Views
2K
Undergrad What is before the first transfinite ordinal, omega?
  • · Replies 12 ·
Replies
12
Views
4K
Graduate Fundamental sequences for the Veblen hierarchy of ordinals
  • · Replies 6 ·
Replies
6
Views
3K