Activity for Ripheus24
| Type | On... | Excerpt | Status | Date |
|---|---|---|---|---|
| Comment | Post #296181 |
@#117674 in the deduction of a classical-logic conditional $A \rightarrow B$, I saw once that you could assume $A$ and see if you could get $B$ based primarily on that, and if this worked, you'd get a deduction of that conditional. I don't know if this is keyed more to stuff like Gentzen dealt wi... (more) |
— | 4 months ago |
| Comment | Post #296181 |
I should've known that 😅 though by explaining it in terms of $\top, \bot$, you've given me a way to finesse the application of your observation, a refinement based on my (insane!) theory that the imaginary and split-imaginary units (and the $\epsilon$ of the dual numbers) can serve for nonstandar... (more) |
— | 4 months ago |
| Edit | Post #296180 | Initial revision | — | 4 months ago |
| Question | — |
How to find dualized quantifier pairs in substitutionally quantified analysis? Classically, $\exists$ and $\forall$ are dual. These don't seem to be the quantifiers Newton had quite in mind when differentiating. Suppose quantifiers attached separately to the x- and h-terms, "at least zero-many/much x or h." In this context, the h-terms are of a different sortal or typical a... (more) |
— | 4 months ago |
| Comment | Post #296095 |
@#117674 I think that's so, though I've also read something like "a version of CH is derivable using **AD**," or also where $|R|$ is an immediate successor of $\aleph_0$ in a different direction from $\aleph_1$, so it's a delicate variation. And to be sure, most every time I think I "know" more a... (more) |
— | 4 months ago |
| Comment | Post #296095 |
@#118341 I don't accept statements about possibility and impossibility "by definition." The accepted answer to this question showed a way towards adapting wheel theory to alephs, a way that I see no reason to reject. If you can prove internally to that answer's content that it doesn't work, that ... (more) |
— | 4 months ago |
| Comment | Post #296095 |
@#117674 then there is a broader "division error gap" when we come to dividing a singular cardinal by one of the regular cardinals that composes it. For example, 0 and 1 are self-cofinal, but cf(2) = 1, and so on, until $cf(\aleph_0) = \aleph_0$, etc. But in the **AD**-world, $cf(\aleph_3) = \ale... (more) |
— | 4 months ago |
| Comment | Post #296095 |
@#117674 I have to admit I'm not sure I'm familiar with that one. I have sorted through the ones indicated in the SEP entry on continuity and infinitesimals, and on the Wikipedia page for infinitesimals, and I noticed (IIRC) a remark in the entry on signed zeroes that these could be used as "infi... (more) |
— | 4 months ago |
| Comment | Post #296095 |
@#117674 the introduction of the existential quantifier from classical/first-order logic was in part established with reference to the analysis of the ontological argument in theology, i.e. Kant's "existence is not a predicate" was understood/reinterpreted as an existence operator not being in pr... (more) |
— | 4 months ago |
| Comment | Post #296095 |
@#118341 but it's not an unexplained statement, "Aleph subtraction/division are ill-defined," it comes from the properties that make such akin to the 0/0 case, hence my interest in adapting wheel theory to this case. Besides, if I'm approaching this from a realist (Aristotelian if not Platonist) ... (more) |
— | 4 months ago |
| Comment | Post #296095 |
To an extent, I'm using [this MathSE post, primarily including Asaf Karagila's answer](https://math.stackexchange.com/questions/146844/how-to-divide-aleph-numbers), as my point-of-departure. In this case, assimilating cardinal division to division of surreals is not given, so we interpret $\frac{... (more) |
— | 4 months ago |
| Comment | Post #295704 |
@#117674 if I try to read addition off union and multiplication off iterated union, might I adapt something like fuzzy/rough/qua-set/etc. theories to extensions of the concept of union that would allow for "fuzzy addition" and "fuzzy multi-addition" and the like? I suppose it would be painstaking... (more) |
— | 6 months ago |
| Comment | Post #295704 |
@#53036 unfortunately, it seemed like most of the "settled" information about sesquation I could find focused on how it would still conform to the normal "2 by 2 is 4" parameter. The rest was (to my memory) all relatively old (e.g. a 2006 report) guesstimation/graph work. Should I look less for d... (more) |
— | 6 months ago |
| Edit | Post #295704 | Initial revision | — | 6 months ago |
| Question | — |
Would sesquation have identity element 0, 1, or something else? If we allow a schematic of hyperoperations $\uparrow^a$ to take values for $a \not\in \mathbb{N}$, then setting $a = 0 := Succ(n)$, sesquation occurs for $\uparrow^{\frac{3}{2}}$. This is intended to be, as literally as can be, intermediary between $+$ and $\times$. Now what would its identit... (more) |
— | 6 months ago |
| Comment | Post #295526 |
We'd also have the following inequation/equation string:
$0/0 \neq 0/1 = 0/2 = ... = 0/n = \frac{0}{n+1} = ... \neq \frac{0}{\aleph_0} \neq \frac{0}{\aleph_1} \neq ...$
I don't know if that has much value, and whether all its value is aesthetic and subjective for me. But I thought it was an... (more) |
— | 7 months ago |
| Comment | Post #295526 |
@#53410 not necessarily helpful as an aid in theorizing in terms of infinitesimal analysis, at least not in a well-explored direction, but something occurred to me in that if $0/\aleph_0$ is not equal to 0, it's not equal to 0/1, 0/2, 0/3, etc., but that might allow the abstract introduction of a... (more) |
— | 7 months ago |
| Comment | Post #295526 |
@#53410 I saw at the end of Carlström's 2001 paper that wheel theory "has $\sqrt[n]{m}^{p/q} = \sqrt[q \times n]{m^p}$. In particular, $x^{1/0}$ and $x^{0/0}$ is defined in this way." Now does this eventually allow us to use e.g. $\aleph_0/0$ as an exponent, etc.? I would assume so, and then that... (more) |
— | 7 months ago |
| Edit | Post #295545 | Initial revision | — | 7 months ago |
| Question | — |
Is it possible to force $2^{\mathfrak{X}} = 2^{\aleph_0}$ for some non-aleph $\mathfrak{X}$? Easton's theorem (Wikipedia) per ZFC allows a vast range of formulas for infinite powersets; a relevant "base case" (so to speak) is the allowance for forcing $2^{\aleph0} = 2^{\aleph1}$, a proposition also known as Luzin's hypothesis (Wikipedia). Try as I might, I've not made as much progress... (more) |
— | 7 months ago |
| Comment | Post #295526 |
So I came up with a bunch of alternative moments here. For example, we could write the formula like $//\aleph_0(2x\/\aleph_0 + /\aleph_0^2)$ (I think, not 100% sure though). Also, assume that we start with $x = \frac{0}{0}$ itself but with $\epsilon = 0$, then at the "end" we have $2\frac{0}{0}$,... (more) |
— | 7 months ago |
| Comment | Post #295526 |
@#53410 now my next step is trying to see about inserting $/\aleph_0$ in the basic slope formula $\frac{2x\epsilon + \epsilon^2}{\epsilon}$ for infinitesimal $\epsilon$. I'm looking to see if this justifies the classical "practice of neglect" for consolidating the formula as 2*x*. Let's see... ok... (more) |
— | 7 months ago |
| Comment | Post #295527 |
@#117674e OK. I'm mostly in the dark about this beyond what I knew for the basis of my question, and some obsession with proving Euler right about the usefulness of 0/0 :p (more) |
— | 7 months ago |
| Comment | Post #295527 |
Good resources to look into :D (more) |
— | 7 months ago |
| Comment | Post #295526 |
This is what I was hoping for, and more. Thank you so much! (more) |
— | 7 months ago |
| Edit | Post #295503 | Initial revision | — | 7 months ago |
| Question | — |
Is there a stable/consistent extension of wheel theory to alephs? In wheel theory (MathSE) (see also Wikipedia here and nLab here), $0/0$ is more tractable than usual. There is a unary version of division, $/a$, which is involutive: $//a = a$, and $0/0$ is used as an absorbing element generally. So the theory goes on to allow for working with division by zero t... (more) |
— | 7 months ago |