This question is from Introduction to Mathematical Logic
If the set of symbols of a consistent generalized theory $\mathbf K$ has cardinality $\aleph_\alpha$:, then $\mathbf K$ has a model of cardinality $\aleph_\alpha$·
As a high-school student , I have never encountered even $\aleph_0$ in my life.much less I know about $\aleph_\alpha$. The only thing I know is that they are cardinal numbers.
So, I think I have to back of a bit and learn a thing or two about cardinals.Can someone give a reference about this cardinal and ordinal numbers so I can do this kind of proofs involving cardinal numbers?
Best Answer
To take this off the unanswered list, I would like to elaborate Paul Sinclair's comment. Outside of higher set theory or logic, there is almost never a need to deal with actual cardinals. This is because every statement of the form "$S$ has cardinality $k$" is equivalent to "$S$ bijects with $k$". And in ordinary mathematics, we almost always have no need to know what kind of set this $k$ is. In particular, the statement you quoted is equivalent to:
This can be proven without any notion of cardinality (or similar) at all. The main reason to invoke the notion of cardinals is if we want to have a canonical representative for each collection of sets that bijects with one another, just like we have each $k∈ℕ$ as a canonical representative for the collection of all sets of size $k$. Even then, for the purpose here (i.e. countable sets) we could very well use $ℕ$ as our natural representative...