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12h
answered Infinite set always has a countably infinite subset
14h
comment Which axioms of ZFC are required to prove the existence of $\aleph _ 0$?
@AsafKaragila I took that to be your meaning. Silly me.
1d
comment Which axioms of ZFC are required to prove the existence of $\aleph _ 0$?
@AsafKaragila OK, so ZFC is insufficient. Interesting.
1d
comment Why is $1+2+3+\cdots = 0 $?
This is a good illustration that anything can follow from a falsehood. A simpler example: If $0=1$ then I am the king of France. Alas, $0\neq 1$.
1d
comment Which axioms of ZFC are required to prove the existence of $\aleph _ 0$?
The two lists of axioms given so far differed considerably. Which is correct? My guess is neither, but the authors both obviously understood what I was after. As far as me saying what $\aleph_0$ actually is, I posted definitions and links to articles at Wiki. Do you take exception to these definitions? That would be interesting. Anyway, this discussion is dragging on. Unless you can list the axioms I requested, do not expect a reply from from me.
2d
comment Which axioms of ZFC are required to prove the existence of $\aleph _ 0$?
@AsafKaragila I am not putting forth any theories of my own here. I am only asking for a list of axioms. If it is impossible, too time-consuming or too difficult to compile such a list, just say so.
2d
comment Which axioms of ZFC are required to prove the existence of $\aleph _ 0$?
@AsafKaragila I can find the definition at Wiki (see links above). I want to know what axioms are required to construct or prove the existence of $\aleph_0$. Is that too provocative a question?
2d
comment Which axioms of ZFC are required to prove the existence of $\aleph _ 0$?
@hardmath You need the axiom of infinity to construct the set of natural numbers $N$. Then there is some object called $\aleph_0$ that somehow denotes the size of $N$. Can it be be shown to exist using only ZFC? Or are other axioms required? Is that such a hard question?
2d
comment Which axioms of ZFC are required to prove the existence of $\aleph _ 0$?
What "logic" have I presented? I have only asked questions. Are they too provocative? If so, I will consider withdrawing my question.
2d
revised Which axioms of ZFC are required to prove the existence of $\aleph _ 0$?
added 502 characters in body
2d
revised Which axioms of ZFC are required to prove the existence of $\aleph _ 0$?
added 171 characters in body
2d
comment Which axioms of ZFC are required to prove the existence of $\aleph _ 0$?
By this logic, would not the set of even numbers be identical to the entire set of natural numbers since they both have the same size?
2d
comment Which axioms of ZFC are required to prove the existence of $\aleph _ 0$?
The size of the set of natural numbers is identical to the set itself? Really????
2d
comment Which axioms of ZFC are required to prove the existence of $\aleph _ 0$?
I mean an object used to denote the size of the set of natural numbers? Is that not the usual meaning?
2d
comment Which axioms of ZFC are required to prove the existence of $\aleph _ 0$?
According to the commentary, that proves "The existence of omega (the class of natural numbers)." Is that really the same as proving the existence of the cardinal number $\aleph_0$?
2d
comment Which axioms of ZFC are required to prove the existence of $\aleph _ 0$?
Can there not exist an inductive set $Y$ and $y\in Y$ such that $y\notin X$?
2d
comment Which axioms of ZFC are required to prove the existence of $\aleph _ 0$?
Sorry, I missed the edit.
2d
comment Which axioms of ZFC are required to prove the existence of $\aleph _ 0$?
But $X$ is just one inductive set.
2d
comment Which axioms of ZFC are required to prove the existence of $\aleph _ 0$?
You wrote: "Write $\mathcal{I}$ for the collection of all inductive subsets of $X$, which exists by separation." Separation on what set?
2d
asked Which axioms of ZFC are required to prove the existence of $\aleph _ 0$?