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While working on an argument over the last few days, I came to a point where I needed to formally render an "even if" sentence. Here and here, I found some good options, but I'm not sure I get the target conclusion using these translations.

So now I wonder: what about "apart from"? I tried a search for "apart from logic symbolic" but that didn't give me any results of the kind I was looking for (one result that looked promising turned out to be a massive .pdf that I couldn't search through without reading the entire document by rote).

Here's the gist of the argument (in "even if" format: please note that this will not be impeccably written out as it stands):

  1. (∃ℵω ∨ ¬∃ℵω) ∧ ∃ℶn (assumption is that "regardless of whether" = "even if").
  2. ??? (missing premise (or premises), something for "apart from" like, "A exists apart from B" and "if A exists apart from B, then..." ???).
  3. ∴ ℶn < ℵω

The goal is to discharge the antecedent in Shelah's result (2ℵn < ℵω) → (2ℵω < ℵω4).

Addendum: this is not an attempt to prove something using normal ZFC logic or axioms. I know that's not to be done as such. One of the interpretations of "even if" that I saw was a modal one. In the logic I'm using, which is supposed to be erotetic in part, I want a principle that goes something like, "If a question falls under an axiom, it has answers under that axiom." Or, then, "If a question X exists apart from an axiom Y, X has answers apart from Y."

Kristian Berry
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    See Apartness relation for an already existing technical concept. – Mauro ALLEGRANZA Apr 13 '21 at 13:26
  • But it seems to me that you are using it as synonym with "independent of". – Mauro ALLEGRANZA Apr 13 '21 at 13:27
  • From #1, it looks like you don't even know whether ℵω exists. If you don't know whether it exists, then you are not going to prove that it is greater than ℶn. – Kevin Apr 13 '21 at 15:15
  • @Kevin, I am having trouble formalizing the idea... I tried out the idea that ℘(K) exists in possible worlds where ℵω doesn't but this seems to commit me to a set-theoretic multiverse, which commitment I'm trying to avoid... – Kristian Berry Apr 13 '21 at 15:36
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    Variations on orthogonality symbol are used to notate "independent of", see e.g. Cook. But what does "exists apart from" really mean? If it means "provable without" then this is a meta-claim, and how is stuffing it into a first order language supposed to work? And "if a question X exists apart from an axiom Y, X has answers apart from Y" is simply false. One can ask if all sets are well-orderable apart from AC, but not answer it. An answer is equivalent to resolving AC. – Conifold Apr 14 '21 at 07:32
  • Is there a difference between a question being asked, and a question's existence? If questions are like propositions in this sense, then might we say that the well-ordering question (if its answer is necessarily true) doesn't exist independent on the AC question (if the answer to the AC question necessarily implies the well-ordering theorem)? Even if we can pose the questions separately? – Kristian Berry Apr 14 '21 at 12:47
  • Or: I have to read up on the evocation relation more, but might we say that the well-ordering question evokes the AC question? – Kristian Berry Apr 14 '21 at 12:51
  • I don't understand your argument even ignoring the missing notation. You are trying to say that (A exists or it doesn't, also B exists) ^ (A is irrelevant to B) => (B is a smaller number than A). How the heck does that work?! – user253751 Apr 14 '21 at 17:38
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    The problem is that what necessarily implies what depends on other axioms. Your "question falls under an axiom" or "question evokes an axiom" bother me because they seem to presuppose a 1-1 relation between individual questions and individual axioms. But there is no such relation. Well-ordering is equivalent to AC under ZF axioms, change something in the ZF background and they may not be equivalent anymore. Sets of representatives "evoke" AC, but Scott's trick allows asking about them without AC. I think your "apart from" must explicitly take backgrounds into account to work. – Conifold Apr 14 '21 at 20:00
  • Technically, I would say that the axioms evoke questions, and there would be no one-to-one correspondence between the set of questions preceded by the axioms and the set of questions that itself preceded the axioms---the former would be the erotetic powerset (if you will) of the latter, instead. The only erotetic set in intrinsic one-to-one correspondence with the set of axioms would be the set of the axioms in interrogative form (same propositions, different "punctuation"). – Kristian Berry Apr 15 '21 at 00:46
  • But aside from all that, yes. For the greater argument, I need something that makes sense of the axiom system as a whole, not just piecemeal. – Kristian Berry Apr 15 '21 at 00:47
  • My next step is to look into epistemic logic... – Kristian Berry Apr 15 '21 at 00:56
  • @user253751 you could say that b is smaller than a when a is a model of ZFC and b is not. Apparently, ℵω is a model of ZFC minus the axiom of replacement (iirc, something along those lines). – Kristian Berry Apr 15 '21 at 02:04

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