0
votes
3answers
42 views

What is the usual definition of “ordinal number”?

My definition for the ordinal number is "von-Neumann ordinal". I thought this is the only definition for the ordinal, but i found some other definitions in wikipedia. What is the usual definition of ...
2
votes
1answer
51 views

Distinguishing characteristic and spurious properties

When working in set theory with definitions of somehow primitive notions like ordered pairs (Kuratowski) or ordinal numbers (von Neumann) one is supposed to use only the characteristic properties of ...
2
votes
4answers
266 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
106 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
150 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 ...
4
votes
2answers
173 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 ...
19
votes
3answers
579 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
307 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 ...