Tagged Questions
2
votes
2answers
53 views
Weakly inaccessible cardinals and Discovering Modern Set Theory
So I've been trying to teach myself some set theory and I've come across some exercises in Just and Weese's Discovering Modern Set Theory. To whit:
Pg. 180
Definition 20: A cardinal $\kappa$ is ...
3
votes
0answers
48 views
Can any large cardinal axiom be reached by a recursive sequence of unbounded sequences and fixed points?
The lowest large cardinals can be seen as a simple recursive rule that defines an unbounded sequence, then defines a fixed point of the sequence, then an unbounded sequence of fixed points, a fixed ...
3
votes
1answer
94 views
Is the expressive power of infinitary logic language $L(\infty,\infty)$ larger, the same or smaller than that of ZFC+large cardinal axioms?
In a previous question I learned that the power of statements of the form $\Pi_m^n$ or $\Sigma_m^n$ for arbitrary positive $m$ and $n$, is smaller than that of ZFC. For instance, the GCH cannot be ...
2
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2answers
111 views
Is the class of all ordinals independent of set theory?
I am little confused with this. In any model of set theory (or is only in ZFC?), we start with an initial set, and then by transfinite recursion we build the universe of sets, which is a proper class. ...
3
votes
1answer
111 views
How many weak/strong limit cardinals exist under different assumptions?
I am trying to hastily answer the question in the title because I need it for another problem. I have been consulting multiple sources and am confused about the standard meanings of the following ...
3
votes
1answer
99 views
Measurable cardinal existence condition
We know that a cardinal $k > \omega$ is measurable if there is a measure function $\mu :2^k\mapsto \{0, 1\}$ that satisfies the following 3 conditions:
1.$<k$ -additive: for every set I of ...
5
votes
2answers
253 views
mahlo and hyper-inaccessible cardinals
Wikipedia states that a Mahlo cardinal is hyper-inaccessible, hyper-hyper-inaccessible, etc. Is this a characterisation of Mahlo? If not what about "alpha = hyper^alpha-inaccessible" with the obvious ...
