Talk:Ordinal collapsing function

wall street

I think this is a joke: http://science.slashdot.org/story/10/05/27/0258245/Sudden-Demand-For-Logicians-On-Wall-Street

"In an unexpected development for the depressed market for mathematical logicians, Wall Street has begun quietly and aggressively recruiting proof theorists and recursion theorists for their expertise in applying ordinal notations and ordinal collapsing functions to high-frequency algorithmic trading...."

69.228.170.24 (talk) 06:04, 27 May 2010 (UTC)

Yes, all my sources assure me it's a joke, and nobody has stepped up to provide evidence that infinite ordinals are used in high-frequency trading. John Baez (talk) 21:04, 24 November 2012 (UTC)

Greetings. I am the author. FLeℵgyel (ta|k) —Preceding undated comment added 23:29, 17 November 2013 (UTC)

question on the section: predicative part

It is written

"Now $\psi (\zeta _{0})=\zeta _{0}$ but $\psi (\zeta _{0}+1)$ is no larger, since $\zeta _{0}$ cannot be constructed using finite applications of $\phi _{1}\colon \alpha \mapsto \varepsilon _{\alpha }$ and thus never belongs to a $C(\alpha )$ set for $\alpha \leq \Omega$, and the function $\psi$ remains “stuck” at $\zeta _{0}$"

but should not $\zeta _{0}$ be an element of $C(\zeta _{0}+1)$ since we have $\psi (\zeta _{0})=\zeta _{0}$ and $\zeta _{0}<\zeta _{0}+1$? If this is correct $\psi (\zeta _{0}+1)$ should be larger than $\zeta _{0}$ — Preceding unsigned comment added by 88.131.62.36 (talk) 11:14, 15 June 2013 (UTC)

If you check the definition of C(ζ₀+1), you will see that you would have to show that ζ₀ belongs to it (for some other reason) in addition to ζ₀ < ζ₀+1 before you can conclude that ζ₀ belongs to it on account of being ψ(ζ₀). JRSpriggs (talk) 11:01, 16 June 2013 (UTC)

I was also initially confused about the values of ψ, but I understand it now. "ψ(α) is the smallest ordinal which cannot be expressed from 0, 1, ω and Ω using sums, products, exponentials, and the ψ function itself (to previously constructed ordinals less than α)." The key part is "previously constructed"; I need to be able to create the ordinal number in a finite number of steps from {0, 1, ω, Ω} before I can apply the ψ function to it. Since ζ₀ cannot be constructed in a finite number of steps from {0, 1, ω}, the only way it can be generated is as ψ(Ω); and by definition, ψ(Ω) is not a member of any constructed sets before C(Ω+1). - Mike Rosoft (talk) 05:43, 22 August 2014 (UTC)

Beyond $\Gamma _{0}$ (section: Beyond the Feferman–Schütte ordinal)

There is a statement that: $\psi (\Omega ^{\Omega }+\Omega ^{\alpha })=\phi _{\Gamma _{0}+\alpha }(0)$

...but I guess this would need a reference where someone has worked out the correspondence between the collapsing function and the Veblen functions to this level, because I'm trying to work out the same thing and getting a different answer.

Before you get to $\phi _{\Gamma _{0}+\alpha }(0)$, or even $\phi _{\Gamma _{0}+1}(0)$, you have to take the limit of $0,\ \phi _{\Gamma _{0}}(0)=\Gamma _{0},\ \phi _{\Gamma _{0}}(\Gamma _{0}),\ \phi _{\Gamma _{0}}(\phi _{\Gamma _{0}}(\Gamma _{0})),\ ...$, which should equal $\phi _{\Gamma _{0}+1}(0)$.

Before even that you have to get to $\phi _{\Gamma _{0}}(1)$, which is presumably the supremum of $\phi _{\alpha }(\Gamma _{0}+1)$ for $\alpha <\Gamma _{0}$.

To my reasoning, it should be something like

$\psi (\Omega ^{\Omega }+\Omega ^{\alpha })=\phi _{1+\alpha }(\Gamma _{0}+1)$

and specifically, $\psi (\Omega ^{\Omega }+1)=\epsilon _{\Gamma _{0}+1}$ and $\psi (\Omega ^{\Omega }+\Omega )=\phi _{2}(\Gamma _{0}+1)=\eta _{\Gamma _{0}+1}.$

I think this line of reasoning would also give

$\psi (\Omega ^{\Omega }+\Omega ^{\Gamma _{0}})=\phi _{\Gamma _{0}}(1)$

and for $\alpha \geq 1$, $\psi (\Omega ^{\Omega }+\Omega ^{\Gamma _{0}+\alpha })=\phi _{\Gamma _{0}+\alpha }(0)$.

It would eventually agree with the rest of the article at $\psi (\Omega ^{\Omega }.(1+\alpha ))=\Gamma _{\alpha }$.

--Stephen J. Brooks (talk) 16:10, 30 July 2021 (UTC)

Merge discussion

About Arai psi function. The article is currently marked as single source , so I think it was a bit too early to create the stand-alone article. Editors who participated in this discussion may be interested in improving references to other articles called X-psi functions.--SilverMatsu (talk) 09:31, 29 October 2021 (UTC)

@SilverMatsu: I agree with you. Do merge Arai psi function to here. --Justanothersgwikieditor (talk) 09:06, 25 November 2021 (UTC)

@Justanothersgwikieditor: Thank you your reply. Done. --SilverMatsu (talk) 09:52, 25 November 2021 (UTC)

Another, simpler, ordinal collapsing function

Please see User:JRSpriggs/Ordinal notation. JRSpriggs (talk) 18:41, 2 April 2022 (UTC)

Possible error in one of the example ψ-functions?

Before the simultaneous recursion is introduced, there is a circular reference via ψ₁. ψ₀(α) is defined using a set C₀(α), which includes the ordinal ψ₀(α+2). However, ψ₁(α+2) is defined using a set C₁(α+2), which contains ψ₀(α0+1), which means ψ is defined by a non-terminating inductive clause. So I think this version of ψ is not just hard to extend as mentioned in the article, but not a well-defined system of functions. I'm not sure how it should be fixed while keeping its current purpose, an intermediate strengthening of ψ which is meant to lead into the system with arbitrarily many ψ-functions. C7XWiki (talk) 22:13, 25 September 2022 (UTC)

Other similar OCFs > Arai's ψ

In the article, it is said that ψΩ(ε{Ι+1}) is the TFBO in the section of Arai's ψ function, where I denotes the least recursively inaccessible. This is wrong. The TFBO is the proof theoretic ordinal of ID_ω and the recursive collapse of ε{Ω_ω+1} with respect to an appropriate ordinal collapsing function, Ω_ω representing either the least limit of infinite cardinals or least limit of admissibles depending on if recursive or non-recursive analogues are used. ψΩ(ε{I+1}) is the proof theoretic ordinal of KPi (sometimes denoted with capital I) and is much larger than the TFBO.

The confusion between the TFBO and the proof theoretic ordinal of KPi probably comes from the fact that KPi = KPI and KPl have similar names, while one (KPi) has a proof theoretic ordinal of ψΩ(ε{I+1}) while the other (KPl) has the TFBO as proof theoretic ordinal. One has probably confounded these two theories and has mistakenly said that ψΩ(ε{I+1}) is the TFBO.

Should this be changed? 213.93.13.10 (talk) 13:54, 3 March 2024 (UTC)

Also, to add on to my previous comment:

In the section ‘collapsing large cardinal’, the following is stated about the PTO of KPi:

"Roughly speaking, this collapse can be obtained by adding the α ↦ Ω_α function itself to the list of constructions to which the C(·) collapsing system applies."

This is wrong in two different ways. Firstly, adding α ↦ Ω_α to the construction does not increase the strength of the ordinal notation that much as you'd have gaps between ε{Ω_α+1} and Ω_{α+1} for each positive α, thus one should also make a collapsing function for each Ω_α as done in ‘Going beyond the BHO’. Secondly, and more importantly, this approach would only reach the extended Buchholz ordinal (collapse of Φ₁(0)) when done right, which is way below the proof theoretic ordinal of KPi. To get to the proof theoretic ordinal of KPi one must use a large cardinal I (the least weakly inaccessible) used in a collapsing function diagonalizing over Ω_α in the same way that the main OCF on the article diagonalizes over ε_α.

It is also stated that the PTO of KPi is the collapse of the least weakly inaccessible cardinal. In the article, this is stated confusingly and I think it should be clarified what is meant exactly. The collapse of a weakly inaccessible cardinal is not the PTO of KPi, the collapse of the least ε-number after the least weakly inaccessible is. Same for KPM and Π₃-ref.

I'll probably make edits to the article after a few days if no-one comments to disagree that changes to the article with respect to what is listed above should be made. 213.93.13.10 (talk) 20:05, 3 March 2024 (UTC)