If you have facts that contradict me then fire away, maybe I'll learn something.
You're going to need better sources.
Philosophy Explorer wrote: ↑Sun Mar 25, 2018 6:06 pm
Aleph 0 or Aleph null is the infinity of the natural numbers.
Tru dat.
No, that's the content of the
Continuum hypothesis (CH). We know that CH is independent of the axioms of ZFC. And what's worse, it's independent of every sensible additional axiom anyone has ever thought of. No matter what naturalistic or reasonable axioms you try to add to set theory, there's a model where CH is true and another model where CH is false.
Nobody knows what the cardinality of the reals is. The best we can do is prove (easily, exercise for the reader) that the cardinality of the reals is 2^(Aleph-0). But nobody knows how big that might be in terms of the Alephs. That is, 2^(Aleph-0) might be Aleph-1, or it might be Aleph-47, or it might be Aleph-(Aleph-47) for all we know.
Interestingly both Gödel, who proved that CH is consistent with ZFC, and Cohen, who proved that CH is independent of ZFC, did not believe that 2^(Aleph-0) = Aleph-1.
There is a plausibility argument (though of course not a proof). For all finite n, 2^n is a lot larger than n. Why should this not be true in the transfinite realm as well? Putting it another way, the successor operation is far far weaker than exponentiation. You have to do a lot of successors to get from 2^4 to 2^5 for example. By analogy, exponentiation is a far more powerful operation than succession in the transfinite realm. That's actually Cohen's heuristic argument for the falsity of CH.
Again no, for the same reason that it depends on CH. The cardinality of all curves in the plane (or in n-space, makes no difference) is 2^2^(Aleph-0). See
https://math.stackexchange.com/question ... ty-aleph-2.
It's likely that the persistent misunderstanding that this value is Aleph-2 came originally from a mistake in George Gamow's book One, Two, Three, Infinity as outlined in the Stackexchange link.
Philosophy Explorer wrote: ↑Sun Mar 25, 2018 6:06 pm
It's unknown or unproven that there's an infinity in-between Aleph 0 and Aleph 1.
It's easily proven that there is no such cardinal. Aleph-1 is the very next transfinite cardinal after Aleph-0. There is no cardinal between them. Again, this is a common misunderstanding.
What you are thinking of is CH, the question of whether there is a cardinal strictly between Aleph-0 and 2^(Aleph-0). That's unknown.
But what this means is that for all we know, 2^(Aleph-0) might be some humongous Aleph such as Aleph-4838374873874. That would leave lots of Alephs between those two.
But between Aleph-0 and Aleph-1, there are no cardinals, and that follows directly from the definition of the Alephs.
Philosophy Explorer wrote: ↑Sun Mar 25, 2018 6:06 pm
Two infinite sets are the same size if one-to-one correspondence exists between the two sets (bijection).
"Same size" is misleading, because bijection is a very week metric for equality of size. For example the interval of real numbers [0,1] has the same cardinality as [0,2] and the same cardinality as the entire set of real numbers. But their lengths are very different: 1, 2, and infinity, respectively.
Whatever that means. +infinity and -infinity are numbers in the
extended real numbers. We use the extended real numbers to simplify the expression of various statements in calculus and measure theory.
Also, there's an extensive theory of transfinite cardinal and ordinal numbers. Aleph-0 is a number, as is Aleph-47.
So when you say, "infinity is not a number," that claim needs to be put into context.
Finally, I note that
I explained these points to you over three years ago in these two threads:
https://forum.philosophynow.org/viewtop ... 26&t=15446
https://forum.philosophynow.org/viewtop ... 26&t=13711
