Denoting multivariate functions[]
This question isn't related to Rathjen's Psi function, but I thought I would ask it here because the article uses a similar notion. Given the notion "\((a,b)\mapsto f(a,b)\)", is there a way to specify whether it's a binary function accepting arguments \(a\) and \(b\), or if it's a unary function accepting the ordered tuple \((a,b)\) as an argument? I suppose the two notions are distinct, because the graph of the former would be a class containing ordered pairs such as \((((a,b)),f(a,b))\), while the latter would contain ordered pairs such as \(((a,b),f(a,b))\)? C7X (talk) 05:37, 16 November 2020 (UTC)
- The two notions are the same, as the notion of a binary function is defined as an unary function whose domain is a (subset of) direct product. Namely, f(a,b) is just a shorthand of f((a.b)).
- > the graph of the former would be a class containing ordered pairs such as \((((a,b)),f(a,b))\)
- Maybe you made some mistake here. What does ((a,b)) in the first entry mean? (The ordered pair (x,y) is precisely defined as {{x},{x,y}}, but (x) does not make sense unless you specify the intention.)
- p-adic 05:54, 16 November 2020 (UTC)
- I note that {{x}} coincides with {{x},{x,x}} = (x,x), and hence it is not good to use it as a formulation of (x) as it cannot be distinguished from the diagonal ordered pair.
- p-adic 07:26, 16 November 2020 (UTC)
Least weakly Mahlo after least weakly ω-Mahlo[]
A Discord user told me that \(\psi^2_{\textrm{Least }\omega\textrm{-weakly-Mahlo above least }\omega\textrm{-weakly-Mahlo}}(2)\) is equal to this cardinal. Is this correct? C7X (talk) 18:52, 8 March 2021 (UTC)
- What is the difference of the meanings of "after" and "above" in your context? Also, why don't you ask the discord user...? Since it is not so difficult to check the validity of each step to determine a small value of Rathjen's Ψ, the number of steps can be large. Therefore one of the most reasonable way to check is to ask how the one who stated it actually dertermined the value, because he or she has already know each step. In other words, asking others instead of the original one usually wastes time, especially when we want to know an answer whose justification is not quickly given at the first time but can be quickly checked if given.
- p-adic 00:24, 9 March 2021 (UTC)
- OK (in my context "after" and "above" both meant "greater than") C7X (talk) 00:31, 9 March 2021 (UTC)
- > in my context "after" and "above" both meant "greater than"
- Then I think that the statement is wrong. Please tell me if the user told you how he or she verified it.
- p-adic 00:41, 9 March 2021 (UTC)
- Rathjen's Ψ is a little tricky, and hence a guess from "how it works" is usually incorrect. (I think that you know it. I meant that if a user is based on such a guess, you do not have to believe it.)
- p-adic 09:33, 9 March 2021 (UTC)
Extreme strength[]
Bashicu has now stated that "weakly compacts" may even pass the limit of BM4, assuming its well-foundedness. Assuing Bashicu means this OCF (PBot disagrees, but out of a lack of clarification we assume this one), in that case, are there comparisons to Taranovsky's "Degrees of Reflection" for scale? Is it plausible that the limit of this OCF is smaller than C(C(ΩΩ+1,0),0) in DoR, if so, then is it plausible that C(C(ΩΩ+1,0),0) is greater than the limit of BM4? C7X (talk) 01:41, 27 April 2021 (UTC)
Conjunction[]
I agree that ∧ is the standard notation for the conjunction, but π∧α∈X is not a standard use of ∧, is it? The standard use of ∧ is π∈X∧α∈X. If the comma looks unclear, then how about writing (π,α)∈C(α,ρ)^2?
p-adic 14:44, 9 May 2021 (UTC)
Is "π∧α∈X" the reader's perception resulting from replacing a comma with conjunction? In which case, the condition will be \(C(\alpha,\rho)\cap\pi=\rho\color{red}{\land\pi}\land\alpha\in C(\alpha,\rho)\), which isn't well-formed, so the reader can be forced to assume the comma isn't a direct conjunction with π. Unless forcing the reader to assume is considered too unclear to read, then replacing it with (π,α)∈C(α,ρ)^2 is OK, and I think this is the better option.
In general is it a bad idea to force the reader to assume, when one of the ambiguous interpretations isn't well-formed?Tetra was right C7X (talk) 15:17, 9 May 2021 (UTC)
\(\Xi\) function[]
I heard somewhere, I can't remember where, that \(\Xi(1)\) is the least Mahlo cardinal, \(\Xi(2)\) is the least hyper-Mahlo cardinal, \(\Xi(3)\) is the least hyper-hyper-Mahlo cardinal and so on. Is this true? —Preceding unsigned comment added by Binary198 (talk • contribs)
- If I correctly remember, it is false because M^1 is the set of weakly inaccessible cardinals below K. Perhaps you caught a fake, because in my personal experience, more than 80% of statements on Rathjen's Psi from this community and the discord community is false, even if they are regarded as obvious facts.
- p-adic 23:49, 28 January 2022 (UTC)
- Your conclusion includes a false statement. In order to improve your understanding, could you explain why you thought the conclusion? I could not understand your deduction behind "So".
- p-adic 10:06, 29 January 2022 (UTC)
- > I actually don’t remember why I thought this.
- In order to avoid missing reasonings, it is good to add reasons when you state something non-trivial next time.
- > look at remark 4.9 in Proof theory of Reflection
- Actually, I know the remark. That is why I was confident that your conclusion included a false statement.
- > So, Xi(1) = inaccessible, Xi(2) = Mahlo, Xi(3) = Hyper-Mahlo,
- Again, your conclusion includes an error. You need GCH as an additioal axiom to deduce it, i.e. the equalities can be wrong when GCH does not hold. Please add a reasoning when you deduce something non-trivil, instead of omitting it by outting "So".
- p-adic 23:32, 29 January 2022 (UTC)