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 ...
7
votes
2answers
158 views
Is the proper class of all ordinals equivalent to the potential infinity of pre-Cantor times?
My understanding is that the class of all ordinals is, by definition a proper class. This in the end is done to avoid a paradox: the collection of all sets would be paradoxical if you allow it to be a ...
4
votes
2answers
59 views
Are there versions of the axiom of choice that restrict the size of the factors?
One formulation of the axiom of choice is that an arbitrary product of nonempty sets must be nonempty. The axiom of countable choice AC$_\omega$ is known to be strictly weaker than AC, but still ...
3
votes
3answers
176 views
What do these old symbols from set theory mean? (Large E, $\cdot$ and $+$ for sets, and $\ \bar{\!\bar X}\,non\!\geqslant\frak n$)
So, I'm trying to prove the theorems in this paper by Tarski:
On Well-ordered Subsets of any Set, Fundamenta Mathematicae, vol.32 (1939), pp.176-183
but it is from 1939, and I don't recognize a few ...
2
votes
3answers
119 views
Proving the so-called “Well Ordering Principle”
Is there anything wrong with the following proof?
Theorem. Every non-null subset $B \subset\mathbb{N}$ has a least member.
Proof. Assume not. Then, of necessity, we'd have to have $B=\varnothing$, ...
3
votes
0answers
89 views
Does ZFC allow self-reference? [closed]
I heard that, "ZFC theory doesn't allow self-reference."
But I don't know exactly what it means. As we can see in the proof of Godel's incompleteness theorem, we can use method of "Diagonalization" ...
4
votes
2answers
69 views
A reference that forcing with a poset of cardinality less than $\kappa$ preserves supercompact cardinals
I'm looking for a reference for the fact that if $\kappa$ is supercompact and $Q$ is a poset of cardinality less than $\kappa$, then $V^Q\models \check{\kappa} \ \text{is supercompact}$.
Apparently ...
2
votes
2answers
132 views
Beyond Goedel incompleteness and lack of soundness/completeness of higher-order logics
As I understand that there are at least two fundamental limits of the development of the mathematics:
1) Goedel incompeleteness theorems (or more clearly Church thesis) effectively says that there ...
2
votes
2answers
50 views
Is the set of countable successor ordinals countable or uncountable?
I know that $\omega_1$={$\alpha$ : $\alpha$ is a countable ordinal} is uncountable but what about the subset of $\omega_1$ of countable successor ordinals?
5
votes
0answers
72 views
Functions and metafunctions
I didn't get any responses to this question the first time around, so I've tried rewriting parts of it. If there's anything glaringly wrong with the questions I'm asking, please leave a comment!
...
2
votes
2answers
107 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. ...
4
votes
3answers
196 views
Does learning logic and set theory before arithmetic, algebra, and geometry have an advantage?
I'd like to become conversant in a wide variety of serious mathematics, but i'm currently one of those students who did very poorly on mathematical subjects in school, never completing even basic ...
1
vote
1answer
78 views
A proof about $\sigma$-algebras via transfinite induction
This is a proofreading question.
I was trying to help out on this question and in the course of that I encountered the following assertion:
Let $(X, \mathcal A)$ and $(Y, \mathcal B)$ be ...
2
votes
3answers
37 views
Small claim regarding addition of cardinals
I want to show that if $\alpha,\beta$ are cardinals such that $\alpha=\alpha+\beta$ and $0<\beta$ then $ \aleph_{0}\leq\alpha$
It should be fairly simple but for some reason I keep getting stuck.
2
votes
2answers
82 views
How can $\mathbb{N}$ have an upper bound?
"A set A is inductive if every chain in A has an upper bound."
Since $\mathbb{N}$ is a chain, apparently it has an upper bound. But how? I don't understant. How can one find a number greater than ...
3
votes
2answers
91 views
2
votes
0answers
44 views
What are the usual terms for different kinds of well-foundedness?
Smullyan and Fitting say that a class $A$ is well-founded iff every non-empty subclass $B \subseteq A$ has an element $x$ such that $x \cap B = \varnothing$.
In the presence of the axioms of infinity ...
30
votes
9answers
2k views
Infinite sets don't exist!?
Has anyone read this article? Set theory
This accomplished mathematician gives his opinion on why he doesn't think infinite sets exist, and claims that axioms are nonsense. I don't disagree with his ...
4
votes
2answers
82 views
are “all nets in $X$” well defined?
Denote $f:X\to Y$ as a function between topological spaces $X$ and $Y$. One good way for determining whether $f$ is continuous is to check the following statement.
$f$ is continuous iff for every ...
3
votes
1answer
58 views
Exercise in Just/Weese (amoeba forcing) (1/2)
I solved the following exercise (18.3):
Can you tell me if I got it right? Thanks:
18.3.(a) The inclusion "$\supseteq$" follows immediately from the definition. For "$\subseteq$" let $U \in ...
5
votes
1answer
93 views
About cardinalities of almost disjoint systems of functions
Let us call a system $\{f_i; i\in I\}$ of functions $f_i\colon\omega\to\omega$ almost disjoint if the set $\{n\in\omega; f_i(n)=f_j(n)\}$ is finite whenever $i\ne j$. (I am not entirely sure whether ...
3
votes
1answer
45 views
“big” Hausdorff space with dense subspace of given cardinality
In a topology course we proved the following proposition:
Let $A$ be an infinite set. Then there exists a Hausdorff space $X$ of cardinality $|\mathfrak{P}(\mathfrak{P}(A))|$ which contains a ...
2
votes
1answer
26 views
Well-founded part of a graph
Let (A,R) be a graph. Define by transfinite recursion:
$
W_{0}=\emptyset
\\
W_{\alpha+1}=\{a \in A : ext_{R}(a) \subseteq W_{\alpha +1}\}
\\
W_{\alpha}=\cup_{\beta < \alpha} W_{\beta} \text{if ...
8
votes
1answer
138 views
Stone's Representation Theorem and The Compactness Theorem
If you're working on $ZF$ and you assume the compactness theorem for propositional logic, then you have the prime ideal theorem, and thus you can show that the dual of the category of Boolean algebras ...
3
votes
3answers
41 views
Limit Ordinal Question
how to prove this result.
$\alpha$ is limit ordinal if and only if $\beta<\alpha$ implies $\beta+1<\alpha$ for any $\beta$.
I am getting confused because of the definition of limit ...
3
votes
1answer
60 views
well-ordering principle
I'm confused by the well-ordering principle . The proof is clear but I don't have any idea why is it true . It says that every non-empty set can be well-ordered but $C^{1}$ is a non-empty set but ...
1
vote
3answers
65 views
a finite algorithm mapping from $\omega \times \omega$ to $\omega$ possible?
We know that $\omega \times \omega$ is isomorphic to $\omega$, but I am not sure if there would exist a finite algorithm mapping from $\omega \times \omega$ to $\omega$. An algorithm would of course ...
1
vote
2answers
44 views
Existence of a singleton set (AC for a singleton set)
If $x$ is a nonempty set, is it in general possible to prove from ZF that there exists a singleton set the only element of which is in $x$ ?
This would be the result of applying the axiom of choice ...
6
votes
3answers
101 views
Proof of $CFE \implies BPI$
(CFE): Every filter of closed sets can be extended to a maximal one.
(BPI): Every Boolean algebra contains a prime ideal.
I am reading Herrlich's and Stepran's paper "Maximal filters, continuity and ...
2
votes
2answers
32 views
length of a normal not cofinal sequence
Let $\kappa$ an infinite cardinal. Does every normal sequence $\langle\alpha_\xi\rangle \subseteq \kappa$ with $\sup \alpha_\xi<\kappa$ is of length $<cf\kappa$ ?
I think yes but I have some ...
4
votes
1answer
119 views
Is ZFC without Axiom of Infinity consistent?
The incompleteness theorem states that one cannot prove whether ZF or ZFC is consistent, but what about ZFC withouth Axiom of infinity? (Assuming the empty set exists)
Furthermore, let $M$ be a ...
1
vote
1answer
54 views
Free algebra (Boolean algebra)
Could someone give me a simple explanation of Free Algebra on $\kappa$. How to construct free($\omega$).
here is it says
http://en.wikipedia.org/wiki/Free_Boolean_algebra
free($\omega$) is equal to ...
3
votes
1answer
58 views
GCH implies that $2^{<\kappa}=\kappa$
If GCH holds, then
$ 2^{<\kappa}=\kappa$ for all $\kappa$
It is true that $2^{<\kappa}=\sup_{\delta <\kappa}(2^\delta)$
some explain this for me. Thanks in advance
1
vote
1answer
56 views
The class of cardinal numbers is well ordered
I'm looking for a proof that the class of cardinal numbers is well ordered under the order relation $|A|\leq |B| \Leftrightarrow$ exists an injection $f:A \to B$.
In fact, I've found a very beautiful ...
1
vote
2answers
85 views
Question about proof in Jech's The axiom of choice ($AC_\omega$)
I'm looking at the following in Jech's The Axiom of Choice on page 20:
2.4.1. Example: The Countable Axiom of Choice implies that every infinite set has a countable subset.
Proof. Let $S$ be ...
3
votes
2answers
57 views
Indecomposable limit ordinals
A limit ordinal $\gamma>0$ is said to be indecomposable iff $\nexists\alpha,\beta<\gamma$ such that $\alpha+\beta=\gamma$. In view of this definition, I’m trying to prove the equivalence of the ...
1
vote
1answer
29 views
Possible typo in Just/Weese II
I'm just skimming a few pages about Martin's axiom and I cannot seem to make sense of the following exercise:
It seems to me that the exercise is trivial hence I deduce that I am missing something ...
2
votes
3answers
90 views
Subsets as non-mathematical objects?
I think of mathematical objects as individual things that exist by their own (either abstractly or concretely) and can be represented mathematically.
When thinking of subsets, I'm in doubt if ...
2
votes
2answers
68 views
intersection of the empty set and vacuous truth
Let $\mathbb S = \varnothing$.
Then from the definition: $ \bigcap \mathbb S = \left\{{x: \forall X \in \mathbb S: x \in X}\right\}$
Consider any $x \in \mathbb U$.
Then as $\mathbb ...
6
votes
1answer
88 views
How to prove that a regular cardinal cannot be expressed as a union of sets with less cardinality?
My question is about the following:
Using the Axiom of Choice show that:
If $\kappa\ge\omega$ is a regular cardinal, $\gamma\le\kappa$, and $\langle A_\alpha\mid\alpha\lt\gamma\rangle$ is a ...
0
votes
0answers
89 views
$S^2-D$ being equidecomposable with $S^2$ and its details
In the Wikipedia proof of Banach-Tarski paradox (http://en.wikipedia.org/wiki/Banach-Tarski_paradox#Some_details.2C_fleshed_out), there is a stuff about $S^2-D$ being equidecomposable with $S^2$ as ...
2
votes
1answer
71 views
when is $\mathbb P_{\mathcal {A}}$ separative?
This is exercise $II.16$ of Kunen's set theory.
Some background:
$\Bbb P_{\mathcal A}=\{\langle s,F\rangle:s\subseteq\omega, |s|<\omega, F\subseteq \mathcal A,|F|<\omega\}$, $\langle ...
16
votes
5answers
404 views
Why hasn't GCH become a standard axiom of ZFC?
I've never seen a text that includes GCH in the ZFC axioms. I presume this means that GCH has not achieved widespread acceptance. This seems surprising to me, given that:
The cardinal numbers ...
1
vote
1answer
106 views
understanding provability
I am still confused about provability.
.
.
Let a statement P is, sort-of-says like this.
P: ( "X is provable" ∧ "P is provable" )
If ( X is provable ∧ P is provable ) is provable → (P is ...
2
votes
2answers
54 views
How to find a condition that “decides” the value of a name for an ordinal
In a paper I'm reading involving forcing, the following is used without proof, and it seems to be stated as if it is "obvious," but I don't see why. It's probably either easy or known, but I might ...
0
votes
1answer
62 views
Operating on a set of sequences - such as adding sequences and so on possible even when sequence is coded as number?
Suppose that there is a way to code some set of sequences into number. Then one is given one number. Suppose that we want to multiply all numbers in each sequence by that number and get the coded ...
9
votes
1answer
81 views
Proof of a basic $AC_\omega$ equivalence
On Wikipedia it is mentioned that "... in order to prove that every accumulation point $x$ of a set $S\subseteq \mathbf R$ is the limit of some sequence of elements of $S\setminus \{x\}$, one uses (a ...
5
votes
0answers
133 views
Weakly Compact Cardinals and the Extension Property
From $\textit{The Higher Infinite}$ by Kanamori, in his proof (page 39 - 41) that $\kappa$ is weakly compact if and only if $\kappa$ satisfies the Extension Property, the proof does something that I ...
4
votes
1answer
47 views
Is there an AD-family $\mathcal Q$ of size $\aleph_1$ which is not a Luzin gap?
This is exercise $II.10.$ of Kunen's set theory:
Show (in ZFC) that there exist almost disjoint families $\mathcal A,\mathcal B\subseteq P(\omega)$ such that $|\mathcal A|=|\mathcal B-\mathcal ...
1
vote
1answer
63 views
Consistency proof in $\mathrm{ZFA}$
Suppose we work inside a model $M$ of $\mathrm{ZF}$ and we prove that there exists a model of $\mathrm{ZFA}$ that satisfies $\mathrm{Con}(T)$, where $T$ is any theory stronger than $\mathrm{PA}$. ...
12
votes
3answers
156 views
Axiom of Choice and Determinacy
In my set theory course we have been talking about the axiom of determinacy. One of the first things we showed was that $AD$ and $AC$ are incompatible. We later showed that $ZF+AD$ implies the ...
