All Questions
Tagged with foundations ordinal-numbers
8 questions
6
votes
2
answers
500
views
Replacement axiom and the von Neumann hierarchy
Within ZFC, the von Neumann hierarchy consists of sets $V_\alpha$ indexed by ordinals, subject to the following conditions:
$V_0=\varnothing$.
$V_{\alpha+1}=\mathcal P(V_\alpha)$.
$V_\lambda=\bigcup_{...
- 447
5
votes
0
answers
261
views
Higher order arithmetic, hierarchies and proof theoretic ordinals
I asked this question on MSE some days ago but I have not received any answer so I have decided to post it here.
I would like to consider a generalization of the notation $\Pi$ and $\Sigma$ used for ...
- 161
6
votes
1
answer
935
views
Smallest ordinal modelling $\aleph_1$?
Let $X_1$ be the class of all ordinals $\alpha$ such that there exists a transitive model $M$ of ZF(C) such that $M$ thinks that $\alpha$ is $\aleph_1$.
Every class of ordinals has a minimum element (...
- 395
10
votes
4
answers
1k
views
Direct axiomatization of ordinal and cardinal numbers
Again, this question is related (**) to a previous one:
in standard books on basic set theory, after stating the axioms of ZFC, ordinal numbers are introduced early on. Afterwards cardinals appear: ...
- 7,853
13
votes
3
answers
2k
views
Consistency of Analysis (second order arithmetic)
Is there a proof of the consistency of Analysis (second order arithmetic), which is similar to Gentzen's proof of the consistency of arithmetic?
Update:
Which (different) methods can be used to ...
- 32.1k
12
votes
2
answers
747
views
Ways to define "definability"
The notion of a definable set is not expressible in the language of set theory: there is no formula $\delta(x)$ that is equivalent with there being a formula $\phi(y)$ such that $x = \lbrace y : \phi(...
- 9,720
0
votes
1
answer
308
views
Good set theory in which to study ordinal-indexed sequences?
I'd like to "model" the absolute complement of a set $X$ as the ordinal-indexed sequence $\alpha \mapsto V_\alpha \setminus X$ where $V_\alpha$ is the $\alpha$ stage of the cumulative hierarchy. My ...
- 3,793
8
votes
1
answer
3k
views
Foundations: Existence of uncountable ordinals.
This isn't really a research question, but at least it's research-level mathematics. I'm talking with some other people about the first uncountable ordinal, and I want some facts to inform this ...
- 2,754