I am studying set theory and I am confused in the following:
Are limit ordinals the same as infinite ordinals? I would say yes since the least non-zero limit ordinal is $\omega$.
Infinite limit ordinals are the same as limit ordinals? Yes, by definition of limit ordinals.
What are infinite cardinals?
Thanks.
On
No: $\omega+1=\omega\cup\{\omega\}$ is infinite, but it is not a limit ordinal, because it has a largest element, namely, $\omega$. In fact $\omega+n$ is not a limit ordinal for any positive integer $n$; the next limit ordinal after $\omega$ is $\omega+\omega$, the limit of the successor ordinals $\omega+n$.
Any ordinal that can be written in the form $\alpha+1$ for some ordinal $\alpha$ is a successor ordinal, i.e., not a limit ordinal.
A cardinal is simply an ordinal $\alpha$ that does not admit a bijection with any smaller ordinal; if in addition $\alpha$ is infinite, then it’s an infinite cardinal.
On
No. All limit ordinals are infinite,1 but not all infinite ordinals are limit ordinals. To point out, note that every ordinal has a successor, which is a strictly larger set (inclusion-wise, not cardinality-wise). So every limit ordinal has a successor ordinal which is also infinite.
Cardinals are ordinals which have the property that no smaller ordinal has the same cardinality. In other words, there is no initial injection from the ordinal into a smaller ordinal. Note that if such injection does exist, then it necessarily does not preserve the ordering.
We can show that if $\alpha$ is an infinite ordinal which is also a cardinal, then it is a limit ordinal. But not all limit ordinals are cardinals. For example if $\alpha$ is an ordinal, then $\alpha+\omega$ is a limit ordinal, but there is always a bijection between $\alpha$ and $\alpha+\omega$.
Footnotes.
Limit ordinals are not the same as infinite ordinals. $\omega+1$ is an infinite successor ordinal.