Can potential x actual set membership evade set theoretic paradoxes?

If we informally view a set as a container, and view set membership as an invitation for entry into a container. If that invitation is fulfilled, i.e. there is an instance of entry into the container, then that kind of membership is actual, if never fulfilled then it is potential.

Lets denote set membership by $ \in$ ,actual membership by $ \in^!$ , and potential membership by $ \in^*$ , the latter is definable in terms of the former ones as a set membership that is not actual.

We can sketch that in spatial closed figures, so membership would be invitation for entry of objects to the compartment of space inside the closed figure. Obviously the figure itself even if it invites itself for entry (thus it would be a member of itself) cannot actuate this invitation, so in this circumstance it would be merely a potential member of itself.

Now the rules about set memberships (in $ \sf FOL(=,\in, \in^!)$ ) would be:

Membership: $ \forall x \forall y \ (y \in^! x \to y \in x)$

Extensionality: Two sets are equal only if they have the same members

$ $ \forall x \forall y \ [\forall z (z \in x \leftrightarrow z \in y ) \to x=y]$ $

Comprehension: If $ \phi^!$ is a formula that only use predicates of equality $ =$ and actual membership $ \in^!$ , in which the symbol $ “x”$ doesn’t occur free, then all closures of:$ $ \exists x \forall y \ (y \in x \leftrightarrow \phi^!)$ $ ; are axioms.

Define: $ A(x) \iff \forall y \ (y \in x \to y \in^! x)$

Where $ A$ means Actual set; so a set is actual if and only if its membership is its actual membership, for example the empty set $ \emptyset$ .

Actuality: $ \forall x [A(x) \leftrightarrow \forall y \in x (A(y))]$

Actualization: if $ \phi$ is a formula in which $ x$ is not free, that only use predicates $ “=”$ and $ “\in”$ , then all closures of: $ $ \forall y (\phi \to A(y)) \to \exists x \forall y (y \in x \leftrightarrow \phi)$ $ ; are axioms.

Induction: if $ \mathcal Q$ is a definable predicate, then all closures of: $ $ \mathcal Q(\emptyset) \land \forall \text{natural } n \ [\mathcal Q(n) \to \mathcal Q (n \cup \{n\})] \ \to \forall \text{natural } n \ (\mathcal Q(n))$ $ ; are axioms.

Where $ natural$ is defined as finite von Neumann ordinal in the customary manner [using membership relation $ \in$ ].

This theory would interpret Zermelo set theory over the realm of Actual sets, and it can actually prove its consistency over that realm. I’m not sure if it can even interpret ZFC? However this theory has a universal set!

Having a universal set, this theory definitly looks suspecious, so the chances of an inconsistency are high.

Question 1: is there a clear inconsistency with this theory?

Question 2: had there been prior known endeavors along lines of actual and potential membership that is similar to the above account?

Is Waterdeep faction membership mutually exclusive?

Waterdeep: Dragon Heist has multiple factions which PCs can belong to, or join. These include Lord’s Alliance, Emerald Enclave, Bregan D’aerthe, etc. If a PC matches the requisites for more than one of these factions, can he join multiple ones?

For example, could a drow druid join both Emerald Enclave and Bregan D’aerthe? Or is membership exclusive, and PCs have to pick a single faction to work with?

Yesterday, I told my players that they could belong to multiple factions as long as their missions and actions didn’t cross each other. But it was a DM’s call in the moment, and I wanted to confirm my ruling.

Suggestions for membership level names

I wanted you guys opinion on something. I'm launching a membership product that has 3 different levels that people can join. Originally I was going to use "Bronze" "Silver" and "Gold" membership levels, but I would really like something more creative than those descriptions. Any ideas? This is for a real estate investing membership site if that helps.

Thanks in advance!


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Any chance to survive after “Apple Developer Program membership will be terminated” message?

I’ve got “your Apple Developer Program account has been flagged for removal message on the 3rd May 2019. On 15th May 2019, I’ve got the “your Apple Developer Program membership will be terminated message. Apple does not share specific reasons for that decision, as usual.

However, as of 20th May the account is still active: I can access it and make changes.

Since the 15th of May I’ve removed from sale all apps that could have caused Apple’s anger. I’ve disabled all corresponding In-App purchases. I’ve deleted most of the apps. I can’t delete one of them because there’s an In-App purchase with “in Review” status.

Is there still a chance for canceling the termination of my membership? How long does it usually take to terminate a membership?

Waiting for death is sort of worse than the death itself.

Hide items in list based on group membership

I have a list of items in Sharepoint. I want to allow group members to submit items to the list, but I only want owners to be able to view the items.

Microsoft has an example of hiding fields in rows based on the row data:

"style": {   "display": {     "operator": "?",     "operands": [       {         "operator": "==",         "operands": [           "@me",           "[$]"         ]       },       "",       "none"     ]   } } 

But I want to a) hide entire rows and b) base it on group membership instead, and I can’t find anything in their sample catalog that applies.

Magento 2.3.x Membership / Recurring Billing Solution

I’m looking for a way to setup a membership based store using magento 2.3.x. I need to be able to allow the customer to subscribe to a membership on a monthly or yearly billing basis ( ARB) that also changes the user to a new usergroup so that I can display custom pricing etc.

I have found a membership extension (mageplaza) that changes the customer to a new group, but it doesn’t natively support automatic rebilling for the membership. Anyone have any suggestions or know if this is possible? Payment gateway being used is

Drupal Commerce membership for family members

I’m attempting to setup Drupal 7 to sell memberships. The basics are all setup, and selling roles seems to be working fine. However, I’m getting stuck on additional memberships for the user’s family.

I have full membership, and junior membership. I’d like the head of household to be able to purchase their membership, add one for their spouse, and a junior membership for anyone under 18. All with unique member IDs. I cannot require separate accounts for each person as a family of 5 would need to create 5 accounts and process 5 transactions for membership.

Ideally, I want something like the following:

  • Full Member – John Smith – $ 10
  • Full Member – Jane Smith – $ 10
  • Junior Member – Joe Smith – $ 5
  • Junior Member – Sarah Smith – $ 5

Total $ 30

Each gets a member ID, and is listed in the view/report as a member-member type.

Membership Entity gets me closer, but still a far way away.

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