This tag is for set theory topics typically studied at the advanced undergraduate or graduate level. These include cofinality, axioms of ZFC, axiom of choice, forcing, set-theoretic independence, large cardinals, models of set theory, ultrafilters, ultrapowers, constructible universe, inner model ...
2
votes
0answers
25 views
Applying the Galvin-Hajnal Theorem?
Working my way through chapter 24 of Jech's Set Theory, but progress is painfully slow. I'm now stuck on question 24.3:
If $2^{\aleph_{\alpha}}\leq\aleph_{\alpha+2}$ holds for all cardinals of ...
4
votes
1answer
58 views
What is the cardinality of $\Bbb{R}^L$?
By $\Bbb{R}^L$, I mean the set that is interpreted as $\Bbb{R}$ in $L$, Godel's constructible universe. For concreteness, and to avoid definitional questions about $\Bbb{R}$, I'm looking at the set ...
0
votes
1answer
55 views
How many Borel conjectures are there
The following may be referred to as Borel conjecture:
Every strong measure zero set of reals is countable.
On the other hand Wikipedia refers to the following as the Borel conjecture:
Let $M$ and ...
-3
votes
0answers
159 views
Is computability theory a joke? [duplicate]
by N.J Wildberger Set Theory: Should You Believe?
I read the book, find some very idea shocking me. The author just destroys everything I had learned from computer science's courses.
Look at last ...
-2
votes
0answers
32 views
Empty Set Axiom + Extensionality Axiom [closed]
I am trying to prove the following two sentences are equivalent:
1) Empty Set Axiom + Extensionality Axiom
2) $(\exists A) (\forall x) \neg(x\in A)$ + Extensionality Axiom
Obviously, as one can see ...
1
vote
2answers
29 views
Finite equivalence class same cardinality
For an equivalence relation $\sim$, if each partition has a finite number of elements, and $X$ is an infinite set, then is it true that $|X/\sim|=|X|$?
I can prove injectivity one way by defining the ...
1
vote
1answer
48 views
Ordered pairs in sets
So I am asked to explain why $(x,y,z)=(x,(y,z))$ is a reasonable definition of an ordered triple. However, I am in a limbo at the moment let alone with trying to explain an ordered pair,(x,y), ...
4
votes
1answer
46 views
$\mathcal{P}(\omega \times \omega)$ contains a copy of every countable ordinal
I am trying to understand this proof of the existence of an uncountable ordinal. I don't see why $\mathcal{P}(\omega \times \omega)$ contains a copy of every countable ordinal as it is said.
For ...
-1
votes
3answers
151 views
What is truth? A puzzle of ZFC and CH [closed]
Given a enough strong formal theory capable to form Continuum hypothesis. from law of excluded middle and Noncontradiction, one and only one of CH and negative CH should be true. but the consistent ...
7
votes
0answers
47 views
Well-orderings and the perfect set property
From a wellordering of an uncountable set of reals, Bernstein constructed a set of reals without the perfect set property. My question is, does an uncountable well-ordering itself violate the perfect ...
4
votes
2answers
43 views
ZF Extensionality axiom
To familiarize myself with axiomatic set theory, I am reading Kenneth Kunen's The Foundations of Mathematics that presents ZF set theory. I haven't gotten really far since I am stuck at the axiom of ...
3
votes
2answers
68 views
Axiom of infinity exceptions?
It is my understanding that the axiom of infinity, or axioms that depend on the axiom of infinity, are the only axioms that are not representable in finite First-Order Predicate Logic. Is this ...
3
votes
2answers
93 views
$V_\omega$ is countable
Is there an easy way to prove this? I found a book that suggests the injection $h:V_\omega\to\omega$ defined by
$$h(\{x_1,x_2,\dots,x_n\})=2^{h(x_1)}+2^{h(x_2)}+\cdots+2^{h(x_n)},$$
but I hit some ...
4
votes
2answers
58 views
$\sum_{\alpha<\omega_2}|\alpha|^{\aleph_0}=\aleph_2\cdot\aleph_1^{\aleph_0}$
I've seen this statement in multiple posts (e.g. here and here), but I can't seem to understand it. I can see why
$$\sum_{\alpha<\omega_2}|\alpha|^{\aleph_0}=\aleph_1^{\aleph_0},$$
by noting that ...
2
votes
1answer
63 views
Is there an element in $^* \Bbb N$ is Dedekind-infinite?
One definition of a finite set is that it can be injected into an initial segment of $ \Bbb N$, thus any $n$ in $\Bbb N$ is finite.
Accordingly, if it's legitmate to define every element in $^* \Bbb ...
3
votes
0answers
45 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 ...
1
vote
1answer
45 views
Another version of recursion theorem
Let $h_1:X\times Y\to X$ and $h_2:X\times Y\to Y$ be functions, and $x\in X, y\in Y$.
Then there exist unique functions $f _1:\mathbb{N}\to X$ and $f_2:\mathbb{N}\to Y$ such that:
$f_1(0)=x$
...
10
votes
2answers
108 views
The relationship of ${\frak m+m=m}$ to AC
Two simple questions: (Of course ${\frak m}$ denotes a cardinal in the weak sense in the claims below.)
Can we prove in ZF that $\aleph_0\le{\frak m\Rightarrow m+m=m}$?
If not, what is the ...
3
votes
1answer
87 views
the set of countable sets of Real numbers
I would like to ask some hints towards the proof that The set of countable sets in $\mathbb{R}$ is equinumerous to the set $\mathbb{R}$
1
vote
2answers
97 views
Existence Universal goto-programm (turing machine)
May you can help me out with my problems with source codes. Well first of all we proved that for recursive functions $N:\mathbb N^2\rightarrow \mathbb N$ and $A^k: \mathbb N^k\rightarrow \mathbb N$ ...
3
votes
1answer
66 views
(Non) equivalence of regular cardinal definitions
The usual definition of a regular cardinal is "$\kappa$ is regular if $cf(\kappa) = \kappa$", which, assuming the axiom of choice, is equivalent to this definition: "$\kappa$ is regular iff it cannot ...
4
votes
0answers
53 views
Is $X$ pseudocompact
The following example with a little modified from the handbook of set theoretic topology, Page 574:
Let $\kappa$ be any cardinal for which there exists a family $\{H_\alpha: \alpha < \kappa\}$ ...
5
votes
1answer
54 views
Can't that every standard positive real number is limited be proved?
I'm reading a short section about internal set theory(see here), in which
$x$ is limited in case for some standard $r$ we have $|x| ≤ r$.
while the predicate “standard” is not defined.
I'm ...
-1
votes
2answers
90 views
$\mathbb{R} \setminus \mathbb{Q}$:'a stamping tool' [closed]
What does it mean for the polynomial
$$ a_1x_1+\cdots+a_nx_n=b $$
to have solutions in $\mathbb{R} \setminus \mathbb{Q}$, where $a_i,b\in \mathbb{Q}$?
2
votes
2answers
168 views
how many empty sets are there?
Would I be correct in saying that in the category of sets, the "class of sets that are isomorphic to the empty set is a proper class"?
In other words, there are LOTS of initial objects in the ...
5
votes
1answer
84 views
Inaccessible cardinal and von Neuman Hierarchy
I read a couple of days ago that if $k$ is inaccessible, then $V_k=H(k)$, where $V$ is the von Neumann hierarchy and $H(k)$ the class of sets that are heredititarily of cardinality $< k$.
My ...
0
votes
0answers
41 views
What does commensurable in this context mean?
Suppose that we observe a finite set $T\subseteq U$ where $U$ can be both finite or infinite. Further we assume that a proability measure $P_T$ is defined on all the subsets $S\subseteq T$.
Then by ...
6
votes
2answers
60 views
Zorn's lemma and maximal linearly ordered subsets
Let $T$ be a partially ordered set. We say $T$ is a tree if $\forall t\in T$ $\{r\in T\mid r < t\}$ is linearly ordered (such orders can be considered on connected graphs without cycles, i.e. on ...
3
votes
2answers
27 views
Schema of separation and set of all sets
The schema of separation states (this is probably simplified) states that if $P$ is a property and $X$ is a set, then there exists a set $Y = \{ x \in X : P(x)\}$.
The notes I'm reading say that from ...
2
votes
0answers
62 views
$rk(x\times y)$ and $rk(x^{y})$?
My working ground is $ZF^{-}$, i.e $ZF$ without the Axiom of Foundation.
I define $V_{0}:=0$, $V_{\alpha +1}:=\mathcal P(V_{\alpha})$ and, if $\lambda$ is limit, $V_{\lambda}:=\bigcup_{\beta ...
1
vote
2answers
68 views
maximal tree by Zorn's lemma
In using the Zorn's lemma to show that every connected graph contains a spanning tree, we let $\{T_{\lambda}: \lambda \in \Lambda \}$ be a family of trees contained in $X$ which is totally ordered by ...
-7
votes
1answer
72 views
Sets and Functions prove $B^{A}$ exists [closed]
$B^{A}$ is the set of all functions from A into B.
Prove $B^{A}$ exists.
The hint that is given in textbook is $$B^{A} \subseteq P(A \text{ x } B)$$
The question comes from Hrbacek and Jech's book: ...
3
votes
1answer
104 views
Model theory question with finiteness
It should be pretty elementary, but I can't really see it.. Cheers to anyone who can help me
Let $\Psi$ be a transitive set that is a model of $ZF$. Then, $a$ is finite if and only if $\Psi \models ...
3
votes
2answers
75 views
Why does every countable limit ordinal have cofinality $\omega$?
According to Wikipedia, if $\alpha$ is a countable limit ordinal, then $\mathrm{cf}(\alpha)=\omega$. It is intuitively clear to me that it should be so. Certainly the cofinality of such an ordinal ...
3
votes
4answers
94 views
What are the prerequisites to Jech's Set theory text?
I'm looking for a book to self-study axiomatic set theory, and heard this was a classic. What are the main prerequisites for this text? My knowledge of set theory isn't too great. Probably the only ...
0
votes
1answer
28 views
How to understand $f$ is one-to-one in Jech, 'Set theory' Lemma 3.10.?
I saw T. Jech, 'Set theory' Lemma 3.10:
An infinite cardinal $\kappa$ is singular iff there exists a cardinal $\lambda<\kappa$ and a family $\{S_\xi :\xi<\lambda\}$ of subsets of $\kappa$ ...
0
votes
2answers
68 views
Closed, unbounded subset of a cardinal.
I missed two lectures in my set theory course, and now I don't understand the homework problems.
One is this: let $\kappa$ be a regular uncountable cardinal. Show that the following sets are closed ...
2
votes
1answer
33 views
Antichain and predense set
I am trying to prove
$A$ is a maximal antichain i ff $A$ is predense.
It is problem 16 from link here. If $A$ is maximal antichain then is easy to prove that $A$ is predense. I stuck on proof ...
4
votes
3answers
128 views
A basic doubt on axiom of foundation of Zermelo-Fraenkel set theory
I have read in a book that the "axiom of foundation prevents anomalies such as a
set being an element of itself".
Now, axiom of foundation says that there exist an element in every set which is ...
4
votes
0answers
69 views
Half of a Cohen real
I recently heard from a friend that Zapletal gave a talk at Toronoto where he constructed a proper forcing which adds a real which infinitely often equals every ground model real but doesn't add a ...
5
votes
1answer
53 views
$ZFC^- + AFA$ and infinite cardinals
$ZFC^-+AFA$ is a non-well-founded set theory, where $ZFC^-=ZFC-FA$ is $ZFC$ without the axiom of foundation, and $AFA$ is an anti-foundational axiom
With the axiom of foundation we have that every ...
3
votes
1answer
54 views
Size Of Proper Classes
There is a well-known hierarchy of infinite cardinalities for sets. I've heard it said that proper classes are from a certain point of view "too large" to be sets.
Are some proper classes larger ...
1
vote
1answer
57 views
Jech: Set Theory exercise 3.13, how do I avoid Choice?
In an effort to finally learn set theory rigourously, I've decided to start plowing through Jech's Set theory, making sure to do each of the exercises.
Here is Jech's problem 3.13 (pg. 34 in the ...
2
votes
2answers
73 views
Does well-ordering of the proper class of cardinal numbers imply choice?
It is well-known (forgive the pun) that the axiom of choice (which states that the product of every non-empty family of non-empty sets is non-empty) implies that the proper class of all cardinal ...
1
vote
1answer
62 views
Zorn's Lemma and Posets
If $A$ is a poset in which every chain has an upperbound in $A$ ($a$ be any element in $A$). There exists at least one maximal element $m$ in $A$ such that $m \geq a$.
Whats the difference between ...
7
votes
2answers
122 views
Cardinality of the set of open functions?
I was wondering what is the cardinality of the set $ \{ f : \mathbb{R} \to \mathbb{R} \mid f \text{ is open } \} $ (i.e., $f(U) \subseteq \mathbb{R}$ is open for all open $U$).
There are at least $c = ...
2
votes
1answer
64 views
Can we write every uncountable set $U$ as $V∪W$, where $V$ and $W$ are disjoint uncountable subsets of $U$? [duplicate]
Is it true that for every uncountable set $U$, we can write $U=V∪W$, where $V$ and $W$ are disjoint uncountable subsets of $U$ ?
1
vote
3answers
86 views
Dedekind Finite set contains Dedekind Finite subsets
I would be grateful for some help in proving:
If a set is Dedekind Finite then every subset of it must be Dedekind finite too. I tried a reductio ad absurdum way of thinking but I can't seem to find ...
3
votes
2answers
79 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 ...
1
vote
0answers
59 views
Is a “model” only a proper model if everything in it's definition is also explicitly constructed?
Say you have some collection of axioms and you find a set $X$ fulfilling them. And the definition of the set $X$ involves another concept, e.g. the real number, but only in a way which refers to the ...
