It means, by iterating a U {a} finitely many times starting with a being the empty set,
that is, the whole set is generated from the empty set
It obviously cannot be infinite unless we somehow find an extremely weird way to do so, which is the aim of this whole interfinite project to make countable sets singular
Hmm... Interestingly enough, remember the paper 'notes on finites sets' you read a little while ago? Their definition is strong enough to disprove the theorem(I think)
I recall some time earlier when I was experimenting with that, I can somehow define sets that are less than finite (as the union of them will be finite),but their power sets will become very strange such as having two subsets with some shared elements but their intersection will be empty
however I never get to the point to test them with that theorem which showed their finite sets is equivalent to the natural numbers thus I think bijections will make them ceased to exist
for example imagine the power set of {a,b,c,d,e} is only the subsets {a,b,c}, {c}, {c,d,e}, then it seems at first glance one can made these weird sets
The current experiment is to have some weird finite sets X that bijects with some natural number n, yet have powersets that are of smaller cardinality than P(n). Then one can have weird things like "2+2+2=3"
this will basically mean the powersets of these sets X will be prescribed as an axiom and leading to weird things that some partitions of X can be empty
This is why I read that notes on finite set, because it gave a notion of finite that is elementary enough and make no explicit reference to the naturals
thus allow us to investigate what really means to be finite in order to create weird sets from that concept
In summary, there are three kinds of weird sets I want to create:
1. Subfinite sets: A set X which does things like "1+1+1+1+1=3" etc.
2. Interfinite sets: Some not infinite sets such that finite union of them is infinite. Alternately, find some set theory that retain as much ZF as possible that make aleph0 singular
3. (I cannot think of a name) set: arbitrary union of finite set is finite. Tobias and allesandro have mentioned $V_{\omega}$ is one such model and it models ZFC-infinity
I think it is still subfinite since even a disjoint union of a finite and subfinite set will have a set that has less subsets than the powerset of a set formed by collecting all its individual elements
(I have wrote a proof of this in an earlier experiment but it is not quite fleshed out)
It should be noticed that proof might be wrong because I used the misunderstood version of an inductive system, so subfinite sets may be unable to exist
I might need to revise the proof later. Anyway, to make things clear, we can start with defining what a subfinite set is:
It will be interesting to see where exactly they fail, the more axiom we find to be problematic the closer we are to find some model of interesting finite sets
Then, after creating it we will start developing analysis using it and define primes(although I highly doubt that primes will be definable) and move all math to this theory
A set $T$ is subfinite if $T \subset A$ where $A$ is finite, $|T|=n$ for some $n\in \Bbb{N}$ and $|\mathcal{P}(T)|<|\mathcal{P}(n)|<|\mathcal{P}(A)|$
"Basically, T are sets where the powerset operation gone screwed and unable to sprew all the possible subsets that is to be expect of finite set with that many elements"
The powerset operator is defined to be all the subsets of a given set. This left behind a convenient loophole where it could be that all the subsets of said set just happened to be less than we expected
What I am suspecting should I push along this route is that associativity of Union may be broken, but I am not sure yet
i think when I work with these weird things long enough I might start to have the following conjecture:
> Almost all weird system breaks associativity of the logical operator "or"
to me that does not seemed far fetched as amorphous sets are defined to be sets which always partition into a finite and an infinite subset
So subfinite set in essence took this one step further by having most partitions to be empty
