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Results tagged with set-theory
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user 5513
forcing, large cardinals, descriptive set theory, infinite combinatorics, cardinal characteristics, forcing axioms, ultrapowers, measures, reflection, pcf theory, models of set theory, axioms of set theory, independence, axiom of choice, continuum hypothesis, determinacy, Borel equivalence relations, Boolean-valued models, embeddings, orders, relations, transfinite recursion, set theory as a foundation of mathematics, the philosophy of set theory.
12
votes
3
answers
1k
views
Drawing conclusions by NOT using AC.
The existence of non-measurable subsets and functions on $\mathbb{R}$ require the use of the axiom of choice. That is, there exist models of ZF in which all subsets of (and hence all functions defined …
- 4,790
6
votes
Accepted
Cardinality of the set of all paths in the infinite complete infinitary tree
Assuming your path has countable length, the set of all paths in a $k$-ary tree will have cardinality $k^{\aleph_0}$. Indeed, at each step you have $k$ choices, and there are $\aleph_0$ steps (think o …
- 4,790
3
votes
Accepted
Ultrafilters containing the image of a filter
Note that the image of a filter on $X$ will be $G$ if and only if it contains the filter consisting of all preimages of ''big'' sets in $Y$. Note also that we can combine two filters $F$ and $F'$ (tha …
- 4,790