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**7**

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219 views

### How can any theory prove well-foundedness of ordinals above $\omega_1^{\text{CK}}$?

$\newcommand{\omegaoneck}{\omega_1^{\text{CK}}}$
Pardon the extremely basic question - this isn't quite my area - but I'm confused about the definition of proof theoretic ordinals.
The proof ...

**9**

votes

**2**answers

481 views

### Why is there a need for ordinal analysis?

Consider the Peano axioms. There exists a model for them (namely, the natural numbers with a ordering relation $<$, binary function $+$, and constant term $0$). Therefore, by the model existence ...

**6**

votes

**1**answer

262 views

### Which ordinals are proof-theoretic ordinals?

Few months ago I have posted this question on MO, but I must admit that at the time, admittedly, I had no idea on how technical proof-theoretic considerations can be. I have decided to revise this ...

**6**

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**2**answers

219 views

### Models of PRA/EFA with induction on $X$ but not $\omega^X$

As I currently understand it, induction on formulas containing $N+1$ first-order quantifiers is required to prove the well-ordering of the ordinal $(\omega \uparrow\uparrow N) < \epsilon_0$, that ...

**22**

votes

**1**answer

792 views

### Why isn't this a computable description of the ordinal of ZF?

In a previous MO question, I was told by several commenters that
(a) it's known that there exists a computable ordinal $\alpha_{ZF}$ that "encodes the strength of ZF set theory" (i.e., a least ...

**16**

votes

**0**answers

240 views

### Can Gentzen-style proofs give omega-consistency and beyond?

In 1936, Gentzen famously showed that Primitive Recursive Arithmetic, plus the assumption that the ordinal $\epsilon_0$ is well-founded, is able to prove Con(PA). But of course, Con(PA) doesn't yet ...

**29**

votes

**1**answer

1k views

### Is there a computable ordinal encoding the proof strength of ZF? Is it knowable?

In comments on Quora (see, for example, here, here, here), Ron Maimon has repeatedly expressed the strong opinion that Hilbert's program was not killed by Gödel's results in the way typically ...

**10**

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**1**answer

355 views

### What is the proof-theoretic ordinal of PA + Con(PA), PA + Con(PA + Con(PA)) etc., and why?

I seem to remember having read that the proof-theoretic ordinal (sup of ordinals the theory can prove well-ordered) of $\mathsf{PA} + \mathsf{Con}(\mathsf{PA})$ is the same as that of $\mathsf{PA}$, ...

**2**

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**0**answers

513 views

### What is the role of the (formalized) omega rule in Ramified Analysis?

In the 1960's, Feferman and Schutte did groundbreaking proof-theoretic work to find out the strength of predicative systems of second-order arithmetic. They used the ramified theory of types, a ...

**3**

votes

**2**answers

312 views

### What is the proof-theoretic ordinal of Hyperarithmetical Comprehension?

As I discuss in my answer here, it seems to me that Solomon Feferman shows, on pages 10-11 of his seminal 1964 paper "Systems of Predicative Analysis", that if you consider predicative second-order ...

**4**

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**1**answer

317 views

### Can the Burgess-Hazen analysis of Predicative Arithmetic be extended to Transfinite Types?

Around page 300 of his book "Mathematical Thought and its Objects", Charles Parsons discusses the work of Edward Nelson, who believes that mathematical induction is impredicative, because it can be ...

**2**

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**0**answers

408 views

### Is there a notion of “predicative given the real numbers”?

A definition is called impredicative if it involves quantification over a domain that contains the thing being defined. For instance, if you define hereditary property to be a property which applies ...

**31**

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**3**answers

2k views

### Why do stacked quantifiers in PA correspond to ordinals up to $\epsilon_0$?

I am trying to understand why induction up to exactly $\epsilon_0$ is necessary to prove the cut-elimination theorem for first-order Peano Arithmetic; or, as I understand, equivalently, why the length ...

**4**

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**1**answer

305 views

### ERA, PRA, PA, transfinite induction and equivalences

I'm quite sure I don't understand very well the links between proof theoretical ordinals of theories, the axioms of transfinite induction and the objects a theory can prove to exist.
For instance I'm ...

**8**

votes

**2**answers

631 views

### Ordinal Analysis of Peano Arithmetic with Restricted Induction

If we take Peano Arithmetic and restrict induction to formulas over various fragments of the arithmetic hierarchy, say to the $\Sigma^0_n$ formulas for various $n$ or some other interesting fragments, ...