2
votes
1answer
99 views

What does it mean to say that a forcing “collapses cardinals”?

I hear the following terminology a lot: "So-and-so forcing collapses cardinals." Does this just mean that certain cardinals in the ground model are no longer cardinals in the forcing extension? If ...
6
votes
1answer
85 views

Terminology in forcing

In the context of forcing one reads the relation $p \leq q$ in a poset $P$ as "$p$ extends $q$". A typical example is the poset $P$ of finite partial functions, where one defines $p \leq q$ when $q ...
1
vote
1answer
87 views

$V_\omega$, $\mathcal V^{B}_\omega$, $\mathcal V^{*B}_\omega$ and $\mathcal S^{B}_\omega$: alternative superstructures and properties

I was not able to find a beginner introduction to superstructures and the cumulative hierarchy that makes me able to answer to some of my questions about them so I tried to ask here and I apologize ...
1
vote
0answers
129 views

“Dual cardinality” in the graphs $(V_\alpha,\in_\alpha)$

04-25-2014 Enriched with more details Let define the graphs $(V_\alpha,\in_\alpha)$ in $ZFC$ $V_\alpha$ can be finite too $\in_\alpha \subseteq V_\alpha \times V_\alpha$ and ...
5
votes
1answer
57 views

Supercompact cardinals and being witnessed by a structure of limited rank

Supercompact cardinals have reflection properties. If a cardinal with some property (say a 3-huge cardinal) that is witnessed by a structure of limited rank exists above a supercompact cardinal ...
3
votes
1answer
51 views

How to pronounce the partition relation

The partition relation $$ \kappa \to (\alpha)^m_\lambda $$ says that for any $f:[\kappa]^m \to \lambda$, there is a $X\subseteq \kappa$ such that $f$ is constant on $[X]^m$ and the order type of $X$ ...
2
votes
0answers
64 views

Fields of sets in which, if the l.u.b. of a subset exists at all, it is the union of the subset

I am learning about boolean algebras and how they can be represented as fields of sets. Stone's representation theorem tells us that every boolean algebra is isomorphic to a field of sets. Consider an ...
1
vote
1answer
100 views

What is a finite partial function from $\mathfrak c$ to $D=\{0,1\}$

While I reading a paper, there is a notation is called finite partial function. I searched by google, I cannot find its definition. So I post it here as a question: What is a finite partial ...
4
votes
2answers
110 views

Classes and Sets

Why are equivalence classes called so and not equivalence sets? I am kind of not able to find the difference between a class and a set. What properties that a set have that a class cannot have? It ...
4
votes
0answers
63 views

What is the name of the function that indexes Grothendieck universes?

Assume Tarski-Grothendieck set theory. Then Grothendieck universes form a well-ordered proper class, so we can let $U_\alpha$ denote the $\alpha$'th Grothendieck universe, where $\alpha$ is an ...
3
votes
3answers
108 views

What does “lower density” mean in this problem?

If $\mathscr{U}$ is a ultrafilter on $\omega$, then $\mathscr{U}$ contains a subset $A$ of lower density zero. This is an exercise on page 76 of Problems and Theorems in Classical Set Theory, ...
3
votes
2answers
221 views

What should we call the 'sets' which don't exist under certain set theory axioms?

For example we know that the set of all ordinals does not exist in ZFC, so what should we call it? Set? Collection?
2
votes
2answers
153 views

What is the name of $V_\alpha$?

In the Von Neumann cumulative hierarchy, $V:=\bigcup_\alpha(V_\alpha)$ is called the universe. Is there a name for the individual levels $V_\alpha$? Just as one can say "The closure of $A$ is defined ...
3
votes
3answers
195 views

Class of manifolds is a set?

Is the class of all 2-countable manifolds a set? I think so: each such space is a countable union of sets of cardinality $|\mathbb{R}^n|\!=\!|\mathbb{R}|$, i.e. a manifold has cardinality continuum, ...
1
vote
1answer
84 views

Name for a set which has an order?

As we all know, a set is a collection of elements which have no particular order and no multiplicity. So what do you call a construct which does store its elements in a specific order? What is the ...
2
votes
1answer
115 views

Relative merits, in ZF(C), of definitions of “topological basis”.

Typical Terminology: A basis $\mathcal{B}$ for a topology on a set $X$ is a set of subsets of $X$ such that (i) for all $x\in X$ there is some $U\in\mathcal{B}$ such that $x\in U$, and (ii) if $x\in ...
0
votes
1answer
262 views

Set terminology and symbols in optimization

Coming from engineering background, I'm getting a little lost in terminology and symbols, but I still want to be mathematically precise. In engineering optimization, I often have say two design ...
2
votes
1answer
83 views

Other Names for Sierpinski Reals / Leaning Tower of L'viv

There is a poset constructed by combining in a certain way the usual order on the reals with any well-order on the reals (I can provide details if needed). I've heard it called the "Sierpinski Reals" ...
0
votes
0answers
62 views

Property of a Collection of Subsets Containing the Base Set

The definition of a topology $\mathcal{T}$ on a set $X$ requires several things, one of which is that the set $X$ is an element of $\mathcal{T}$. Similarly, one of the ingredients of the definition of ...
2
votes
2answers
331 views

Confused about Wikipedia definition of NP

I've been checking my understanding of the definitions of NP and NP-complete and I am confused by some of the definitions given on Wikipedia; for example, the article about NP-complete describes NP ...
1
vote
5answers
3k views

What's the difference between open and closed sets?

Especially with relation to topology - rigorous definitions are appreciated, but just as important is the intuition!