Tagged Questions
2
votes
4answers
162 views
What's the definition of $\omega$?
This is a follow up on a comment to one of my previous questions. What's the definition of $\omega$?
Are the following equivalent definition of $\omega$:
$\omega$ is the initial ordinal of ...
4
votes
1answer
81 views
What is the definition of the well-founded part of a model of set theory?
I've been trying to understand John Steel's various notes on inner model theory, but the one thing that trips me up is what he calls the well-founded part of a model of set theory. What exactly is the ...
2
votes
1answer
76 views
Discussion: Differing definitions for the rank of a set
I've just identified that the definition we used for the rank of a set in my set-theory class (1.) is different than the one I commonly find on the web (2.).
$\text{rank}(A)=\min\{\alpha\mid ...
3
votes
2answers
135 views
Which algebraic structure captures the ordinal arithmetic?
Consider the set class $\mathrm{Ord}$ of all (finite and infinite) ordinal numbers, equipped with ordinal arithmetic operations: addition, multiplication, and exponentiation. It is closed under these ...
17
votes
3answers
474 views
Conflicting definitions of “continuity” of ordinal-valued functions on the ordinals
I've encountered the following definition in Kunen, Levy, and other places: A function $\mathbf{F}:\mathbf{ON}\to\mathbf{ON}$ is continuous iff for every limit ordinal $\lambda$, we have ...
4
votes
1answer
272 views
Equivalent definitions of ordinals?
The first definition of an ordinal number I found was that an ordinal number is the $\in$-image of a well-ordered set $(A,\lt)$. From this definition it was derived that an ordinal is just the set of ...
