All Questions
Tagged with ordinals definition
27
questions
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How is transifnite recursion applied?
I've been struggling to understand how ordinal addition, multiplication, and exponentiation, along with the Aleph function $\aleph$, are defined using Transfinite Recursion in Jech's Set Theory or ...
- 5,166
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1
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166
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What ways are there to define $\aleph$?
I've seen some posts on this website that consist in providing and comparing different proofs of a theorem (e.g. for Taylor's Theorem or trigonometric identities). Currently I'm reading Holz's ...
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How to define ordinal addition
From Jech's Set Theory:
We shall now define addition, multiplication and exponentiation of ordinal numbers, using Transfinite Recursion.
Definition 2.18 (Addition). For all ordinal numbers $\alpha$
$...
- 5,166
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What is a sequence of ordinals?
What is a sequence of ordinals?
The concept of a sequence of ordinals shows up here and here, and in the definition of cofinality in Jech, third edition, page 31.
$\text{cf}(\alpha) =$ the least ...
- 11.9k
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68
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Confusion in the definition of Ordinal Arithmetic
I'm trying to fill in some gaps regarding the definition of ordinal arithmetic. In particular, we define
$$ \alpha +_{ON} \beta =
\begin{cases}
\alpha & \beta = 0, \\
S(\alpha +_{ON} \...
- 115
2
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1
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141
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About the definition of hereditary cardinality
I've seen these two definitions of sets that are hereditarily of cardinality $< \kappa$.
$x$ is hereditarily of cardinality $< \kappa$ iff $|trcl(x)| < \kappa$
$x$ is hereditarily of ...
- 12k
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34
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$B$ is complete $\iff$ every member of $B$ is a subset of $B$; is there another term for 'complete'?
I'm reading Suppes' Axiomatic Set Theory, and this definition is on page 130 when we begin building the ordinal numbers. I can't seem to find any reference on either MathSE or google about complete ...
- 59
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How can I tell if some set is a von Neumann stage $V_\alpha$ for some ordinal $\alpha$?
For some set $X$, is there some statement that's equivalent to "there exists an ordinal $\alpha$ such that $X$ is the von Neumann stage $V_\alpha$ in the von Neumann hierarchy of sets", but ...
- 1,181
1
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1
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Defining Long Line like $\mathbb R \times [0,1)$ (cartesian product of real line with unit interval) instead of using ordinals?
I was trying to understand the long line, and came across this reddit thread: https://www.reddit.com/r/math/comments/apfzi/in_topology_the_long_line_or_alexandroff_line_is/, in which there's a comment ...
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2
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286
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Intuition behind recursive ordinals and their relationship to ordinal functions
From what I understand, an ordinal $\alpha$ is recursive if it is the order type of a subset of $\mathbb{N}$ that is well-ordered by a recursive relation $\prec$ (meaning, $\mathbb{1}_\prec:\mathbb{N}\...
- 509
1
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1
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198
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Ordinals: is transitivity not implied by well-ordering?
It is my understanding that ordinals can be defined in a variety of ways, the one I am looking at (this is in a context where ZF has not been formally introduced yet) is the following:
An ordinal is ...
- 509
1
vote
1
answer
576
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Difficulty in understanding cantor normal form
Cantors normal form of x is defined as the following
$x = \omega^{a_1} n_1 + \dots + \omega^{a_k} n_k$, Where $x$ is an ordinal and where $\langle a_i \rangle$ is a strictly decreasing finite sequence ...
- 347
4
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2
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847
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Difference Between Cantor's Ordinals and Von Neumann Ordinals?
Cantor's Naive Set Theory allows the construction of the set of all ordinals, which contains itself, which triggers the Burali-Forti Paradox. ZFC both disallows a set of the size of all ordinals and ...
- 317
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2
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Section on Rank in Enderton's Text.
I am confused with a statement Enderton made in his text, Elements of Set Theory on page 202, Chapter 7. There were two Lemmas,
Lemma 7Q: For any ordinal number $\delta$ there is a function
$F_{\...
- 9,618
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votes
1
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Definition of a recursive ordinal
I'm having trouble understanding the definition of reclusive or computable ordinal - Wikipedia defines it as follows:
"...an ordinal $\alpha$ is said to be recursive if there is a recursive well-...
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