All Questions
Tagged with ordinals foundations
22
questions
3
votes
1
answer
204
views
Ascertaining whether "absolute" standardness of $\omega$ is actually possible in ZFC
Books on set theory seem to at least strongly imply that there can exist some $\mathsf{ZFC}$ universe whose $\omega$ is "standard" in an "absolute" sense (order-isomorphic to the ...
- 1,024
2
votes
1
answer
166
views
Alpha recursion - Constructible universe and Analytical hierarchy
Alpha recursion and Constructible universe are very intertwined, because the first is based on the concept of admissible ordinal $\alpha$ which is defined as an ordinal such that $L_\alpha$ - a set ...
- 443
2
votes
2
answers
145
views
What is the meaning of "induction up to a given ordinal"?
Given an ordinal $\alpha$, what does it mean: "induction up to $\alpha$"? When $\alpha=\omega$, is this is ordinary mathematical induction? Also, Goodstein's Theorem is equivalent to "...
- 4,432
2
votes
1
answer
285
views
What is an example of a statement equivalent to $\omega^{\omega}$-induction?
If $\alpha$ is a countable ordinal and $A$ is the set of natural numbers having well-ordering of type $\alpha$, does this mean that $\alpha$-induction (transfinite induction up to $\alpha$) is ...
- 4,432
3
votes
1
answer
263
views
What is the proof-theoretic ordinal of true arithmetic?
The proof theoretic ordinal of $PA$ is $\epsilon_0$. My question is, what is the proof-theoretic ordinal of true arithmetic, i.e. $Th(\mathbb{N})$?
I’m assuming you get something bigger than $\...
- 10.3k
0
votes
0
answers
89
views
How is Zorn's lemma proven in ZF(C), actually?
I want to prove some results like that every vector space has a base without the use of Zorn's lemma becuase I want to practice set theory and become confident with things like ordinals. I have some ...
- 1,697
0
votes
0
answers
233
views
Higher order arithmetic, hierarchies and proof theoretic ordinals
I would like to consider a generalization of the notation $\Pi$ and $\Sigma$ used for the arithmetical hierarchy $(\Pi^0_n$, $\Sigma^0_n)$ and the analytical hierarchy $(\Pi^1_n$, $\Sigma^1_n)$ to ...
- 443
1
vote
1
answer
243
views
Is there a sequence of extensions of ZFC where the corresponding sequence of proof theoretic ordinals has $\omega_1^{CK}$ as least upper-bound
I was reading this question on MO where they define an infinite sequence of extensions of ZF by creating iteratively a new theory which includes the consistency of the previous ones. The definition ...
- 443
0
votes
1
answer
186
views
Is this a proper recursive ordinal notation for ordinals < $\omega^2$?
After making another question about ordinal notation I want to clear some confusion I have about the topic.
Let consider ordinals less than $\omega^2$ (or in $\omega^2$) , any of such ordinals can be ...
- 443
0
votes
1
answer
94
views
Is there a set in ZFC that can not be obtained from Ordinals
Is there a set in ZFC that can not be obtained from ordinals (defined according the Von Neumann definition) via set operations (union, intersection, set difference) and power set?
- 443
1
vote
0
answers
96
views
What is the Turing degree of the set of True formula of Arithmetic whose order is an infinite ordinal
This question was originally posted as a part of this other question, but I was suggested to make a new question for this part.
In the first question I asked about the Turing degree of the set of ...
- 443
4
votes
1
answer
321
views
What is the Turing degree of the set of true formula of Second Order Arithmetic?
The set of true formula of First Order Arithmetic is not arithmetical (by Tarski's undefinability theorem) and it has Turing degree $\emptyset^{(\omega)}$.
What about the set of true formula of ...
- 443
4
votes
1
answer
127
views
Characterization of V-stages in the absence of foundation or replacement
In the absence of foundation and replacement, we can say that $x$ is an ordinal iff $x$ is hereditarily well-founded (with respect to the relation $\in$) and hereditarily transitive, where
$$ \text{$...
- 6,037
6
votes
2
answers
476
views
The Definition of Ordinals and the Axiom of Regularity
In the first edition of his set theory textbook, Pinter defines an ordinal number to be a set which is transitive and (strictly) well-ordered by the membership relation $\in$. I believe this is the ...
- 1,107
4
votes
1
answer
537
views
Justification of ZFC without using Con(ZFC)?
I saw this post by Carl Mummert, which describes how one can 'see' the reasonableness of ZFC, via iterating the power-set operation starting from the empty set. However, this iterations proceed in ...
- 59.2k