Hello. I tried to print "alef null" properly, but the system refuses to permit it. Hence the naught subscript appears in front of the alef, rather than behind it, as normally happens when one places the HTML subscript code after the symbol. I experimented by placing the subscript command in front of the alef character and got two subscripts to the front, as you (maybe) see here: 0א 0. I then tried another command behind the alef and got this absurdity: 0א 00 .
I propose a minor modification of the foundations of transfinite and finite arithmetic.
ω - 1 = א 0. That is, ω is the set containing every natural number, so that it is not one of the naturals on ground that no set can be a member of itself. ω is an ordinal because any member of ω is regarded as less than ω and because there are infinite sets that are "larger" than ω. So א 0 is Cantor's first transfinite cardinal number.
For notation consistency, I write N as a name for ω and N as a name for א 0 . We have N < R, which is a name for the cardinal number of the reals.
It has become a convention to write that 1 + ω = ω , but ω + 1 =/= ω . The justification is that, as an ordinal which has lower cardinality than other infinite sets, there must logically exist such an entity.
My suggestion is that we drop N (aka ω) as the standard reference set for א 0. Instead I recommend N X ∅ such that x = < k, ∅ > ∈ N X ∅ . So one can say that N has the cardinality of N X ∅ , which then defines א 0, which I write as N.
Alternatively, I propose that we define א 0 as N X N so that the ordered pairs are simply two identical integers. This way, using the von Neumann definition of N, every element will be contained by a successor.
We then define the addition of cardinal numbers, finite or infinite, according to their reference sets.
Obviously if N + 1 means N ∪ { ∅ }, then N + 1 = N.
Well, we may however observe that N ∪ 1', where 1' is not an element of N but of, say, N X ∅ , then we may define the addition of a finite element/set (as in the ordering of numbers which are sets of lower numbers) that is in one-one correspondence with the ordinal { ∅ }. That is, 1', being defined as < { ∅ }, ∅ > is in the same ordinal position as 1.
Now 1' could as well represent some irrational, as no irrational is a member of N.
From this, we see that addition of cardinal numbers requires that we assure that X ∩ Y = ∅ .
So N ∪ 1' = N + 1' = 1' + N. Conversely, N + 1 = 1 + N.
This idea is quite minor, as the use of such a ruse to ensure that X ∩ Y = ∅ is standard. But I feel that this tweak is an improvement of the current practice.
So if we wish to express א 0 + א 1 (whatever set that might be), the addition operation requires that
א 0 ∩ א 1 = ∅ .
This idea might (I'm hazy on this) interfere with the ordinal structure of the set universe as spelled out so well in NBG.
Because the continuum hypothesis (is there an "actual infinity" that warrants being tagged with an infinite cardinal number between N and P(N) ?) is independent of axiomatic set theory, I suppose we could risk the above described process.
So N + N' = 2N, where the underline expresses that the last is to be considered a cardinal number. N + R' does not permit multiplication, of course. But N x N', written N2 is bijective with N X N' or N X N. Notionally, it would be better form to write 2'N.
Note that the cardinal numbers follow arithmetical rules, not per se the sets with which they are bijective. That is, we commonly write N2 when we should, to be exact, write N2.
One more point: The null set contains nothing, and so has no cardinality. We don't wish to have ∅ because, say, N ∪ < ∅ , ∅ > is bijective with N + 1', which we may wish to avoid.
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