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 ...

learn more… | top users | synonyms

1
vote
1answer
26 views

Least rank of a definable R

The set of real numbers can be defined in many ways, including by Dedekind cuts and Cauchy sequences. The different definitions will give different ranks for the underlying set. There exists a set S, ...
5
votes
1answer
65 views

Defining truth predicates in set theory

In this blog post J.D.Hamkins shows that KM set theory can define truth predicate for first-order set theory, which means, I believe, that there is a second-order definition of such predicate and KM ...
4
votes
4answers
190 views

Does There Exist an Explicit Formula Describing Every Possible Sequence of Numbers?

This thread was previously titled "Does a Set Require an Explicit Formula to Exist?". I'm reading H. Enderton's Elements of Set Theory and working through understanding the Zermelo-Fraenkel axioms. ...
0
votes
1answer
54 views

Absoluteness of $\Sigma_2$ sentences in forcing

Let $M$ be a model of ZFC and $M\models \varphi$ such that $\varphi$ is a $\Sigma_2$ sentence in the language of set theory. Let $M[G]$ some forcing extension of $M$. Is $M[G] \models \varphi$? What ...
1
vote
1answer
47 views

Why adding a club of $\aleph_1$ collapses $\aleph_1$ to $\aleph_0$?

Let $\{S_n \mid n < \omega\}$ be a partition of $\aleph_1$ into countably many disjoint stationary subsets. Why adding a club of $\aleph_1$ to each $\aleph_1 \setminus S_n$ collapses $\aleph_1$ to ...
3
votes
0answers
81 views

What is set theory about? [on hold]

What is the subject set theory about? What knowledge is required? I am thinking about what subjects I will choose, but I am not sure if I should take this one. That is the course content: Brief ...
0
votes
1answer
34 views

What does the notation $H=\{ a | a^2=e \}$ mean? [on hold]

Is it true that the notation $H=\{ a | a^2=e \}$ means $H=\{a,a^2=e\}$?
2
votes
1answer
54 views

Is this description of “sigma-algebra generated by collection of subsets” right?

Disclaimer: sorry for my poor english and edition. Claim: If $M\subseteq \mathcal{P}(X)$, then $\Sigma(M)=M_3$, where: $\Sigma(M)$ is the sigma-algebra generated by $M$ $M_1=\{A\subseteq X:(A\in ...
2
votes
1answer
27 views

Bijection between Natural numbers and Infinite Cartesian product of Natural numbers?

Consider a function $f(n): \mathbb{N} \rightarrow \mathbb{N} \times \mathbb{N} \times ...$ mapping each number $n$ to the set of exponents to raise each prime number $p$ to in order to obtain $n$. For ...
2
votes
2answers
44 views

First order formula defining a predicate which asserts that a set is finite.

Is it possible to define a predicate in the first order language of set theory such that $F(x)$ is true iff $x$ is a finite set? I understand that FOL cannot assert that the domain of discourse is ...
1
vote
2answers
38 views

What are the ramifications of introducing a universal set this way?

What are the ramifications of introducing a universal set using this axiom? $$\exists x : \forall y (y\neq x \rightarrow y\in x)$$
5
votes
1answer
62 views

What is needed to make Euclidean spaces isomorphic as groups?

Consider the abelian groups $G_n=(\mathbb R^n,+)$ for $n\geq1$. Claim: For any $n$ and $m$ the groups $G_n$ and $G_m$ are isomorphic. This claim is true if one assumes the axiom of choice, and I ...
1
vote
2answers
57 views

Definability and the Separation and Replacement Axiom Schemata

I am beginning to study Set Theory, so naturally one might begin with the axioms. When reading the schemata of separation and replacement, my initial thought was, "Why would we need separation if we ...
0
votes
0answers
26 views

Transfinite Induction (Proof Explanation)

The theorem above is extracted from the book 'Introduction to Set Theory' by Hrbacek and Jech. Questions: $1$)I don't understand the successor case. When $\alpha_2=\beta+1$, why suddenly $W_2$ ...
7
votes
1answer
82 views

Indiscernible subsequences

Work in a saturated model $\cal U$ of sufficiently large cardinality. Are there assumptions on $\kappa<|\cal U|$ that guarantee that any sequence $\langle a_i:i<\kappa\rangle$ has an ...
0
votes
0answers
48 views

Models of $\mathbb{N}$ and ZFC

For every model of arithmetic, $\mathbb{N}^M$, in the universe V, does there exist a model of ZFC, $M$, such that $\mathbb{N}^M$ is the standard model of arithmetic as seen in $M$? Clearly, ZFC ...
1
vote
1answer
36 views

Cardinal exponentiation formula

Assume GCH and let $k,m$ be infinite cardinals. I would like to show that $k^m = \max \{ k,2^m \}$. We of course have $k=\beth_a$ and $m=\beth_b$ for ordinals $a,b$. If $a$ is a successor ordinal, ...
2
votes
1answer
110 views

Is the Bourbaki treatment of Set Theory outdated?

On page 5 of the following write up, the author asks why the Bourbaki did not notice that their system of Zermelo set theory with AC was inadequate for existing mathematics. Throughout the rest of the ...
0
votes
1answer
42 views

Existence of non measurable set and ZF theory?

Does statement: 'Existence of non measurable set' consistent with ZF theory. or if I throw Axiom of choice from ZFC theory. Can I prove or disprove existence of a countably additive measure function ...
6
votes
2answers
126 views

In the surreal numbers, is it fair to say 0.9 repeating is not equal to 1?

I find the surreal numbers very interesting. I have tried my best to work through John Conway's "On Numbers and Games" and self-teach myself from some excellent online resources. I have prepared a ...
0
votes
1answer
29 views

How to check which axioms hold for models in set theory?

I started a class in set theory. The professor drew a few diagrams, all of them having big circles on the outside. Inside there are two small circles marked $a$ and $b$. And they have arrows between ...
2
votes
2answers
84 views

“Partitioning” an uncountable set “equally”

I just observed a simple fact about the real numbers, that if U is an uncountable subset of the set of real numbers, then there exists a real number r, such that both the set V of all elements of U ...
1
vote
1answer
29 views

Forcing $M[G] \models CH$

I have seen the proof of transitioning from a model in which CH holds to a model in which CH fails. However, how do we force the other direction? More explicitly, suppose that $M$ is a countable, ...
0
votes
0answers
35 views

A lattice generated by two particular sublattices of the lattice of binary relations

Let $U$ be some set. Let $\Gamma$ be the set of all finite unions $\bigcup_{X\in S}(X\times Y_X)$ where $S$ is a finite partition of $U$ and $Y_X\in\mathscr{P}U$ for every $X\in S$ (that is $Y$ is a ...
0
votes
0answers
70 views

Another conjecture about filters and cartesian products

Now I present a similar but different (weaker) question (the difference is a different definition of the set $\Gamma$): Let $U$ be some set. Let $\Gamma$ be the set of all finite unions ...
2
votes
2answers
98 views

How does one prove $ZF\vDash MC\Rightarrow AC$?

This is somewhat adressed to Andreas Blass, whose papers I have read, in particular I make reference to an old paper of his »Existence of Basis implies the Axiom of Choice« (84). Anyone who happens to ...
-1
votes
0answers
106 views

Is the axiom of choice constructive in the constructible universe?

Even ZF has some non-constructive elements mostly due to contradiction proofs. For example one may be able to construct a sequence of objects some of which have a given property without being able to ...
1
vote
1answer
53 views

Proof that ZF set theory implies Weak König's Lemma

In some of my other questions and in several references one finds the statement that ZF axioms imply WKL I have searched for the proof of this, but I so far cannot find a proof. I am looking ...
0
votes
0answers
85 views

Existence of a Banach space with algebraic dimension $\alpha \neq \aleph_0$

For any given cardinal number $\alpha \neq \aleph_0$, is there a Banach space $X$ with algebraic dimension $\alpha$?
3
votes
1answer
32 views

Is there a cardinal $\kappa$ with uncountable cofinality such that $\kappa<\kappa^{\aleph_0}$?

Is there a cardinal $\kappa$ with uncountable cofinality such that $\kappa<\kappa^{\aleph_0}$? The question is motivated by the observation that $\kappa< \kappa^{{\rm cf}\kappa}$ for any ...
2
votes
0answers
68 views

If models of set theory can be construed as categories, can notions of forcing be construed as functors?

Consider the following passage from Blass and Scedrov's paper "Complete topoi representing models of set theory"(Annals of Pure and Applied Logic, vol. 57 (1992),PP. 1-26) where 'set theory' in this ...
2
votes
1answer
49 views

A question about cardinals with countable cofinality

This question bothers me as I am not sure if we could drop the assumption of uncountable cofinality: Let $\kappa$ be an uncountable cardinal and let $x_\alpha, y_\alpha$ be collections of real ...
4
votes
0answers
66 views

Has anyone proposed an axiom for ZFC that implies that many different cardinal characteristics are distinct?

I think it is more interesting when cardinal numbers are distinct, rather than equal. Has anyone proposed an axiom for ZFC that, in one fell swoop, implies that many different cardinal characteristics ...
2
votes
2answers
59 views

A question about the cardinality of a set of functions with finite support where the domain of each function has cardinality aleph-null.

Suppose that $M$ is a well ordered set with at least two elements and that $L$ is a well ordered set with the same cardinality as the natural numbers. Let $[L, s.f, M]$ be the set of all functions ...
4
votes
2answers
160 views

Is there a minimal axiomatization of ZFC?

Working in ZFC, does there exist a set $\Sigma$ of sentences which axiomatizes ZFC (i.e. every sentence in $\Sigma$ is provable from your favorite axiomatization of ZFC, and vice versa) and is minimal ...
4
votes
1answer
118 views

Proof of “If $ZFC$ proves there is an inaccessible cardinal, then $ZFC$ is inconsistent”.

Let $I$ be the statement "there is an inaccessible cardinal". I'm aware of two proofs of "If $ZFC\vdash I$ then $ZFC$ is inconsistent". One proof uses the Second Incompleteness Theorem, which I ...
14
votes
1answer
330 views

What axioms does ZF have, exactly?

While trying to find the list of axioms of ZF on the Web and in literature I noticed that the lists I had found varied quite a bit. Some included the axiom of empty set, while others didn't. That is ...
1
vote
1answer
44 views

If $\beth_1$ is weakly inaccessible, are any of the cardinal characteristics of continuum provably strictly less than $\beth_1$?

Assume ZFC+"$\beth_1$ is weakly inaccessible." Are there any cardinal characteristics of the continuum mentioned at wikipedia that can thereby be proved to have cardinality strictly less than ...
2
votes
1answer
59 views

Cardinal characteristics

Assuming Continuum hypothesis is not true, How many cardinals $k$ exist which are $\aleph_1 < k < \mathfrak c$? Can I assume that there is a finite number of these cardinals or is there an ...
3
votes
1answer
74 views

Is intuitionistic naive set theory consistent?

I'm asking because the usual argument that a set either belongs in itself or not doesn't apply. I did a quick search and it appears that the logic is also required to be contraction free. If it's ...
1
vote
3answers
59 views

Are dimensions redundant?

I am fairly new to this so apologies for informal terminology. After I discovered what space filling curves are, I came to the conclusion that any point in any number of dimensions can be represented ...
4
votes
2answers
79 views

Is it meaningless to say $M\prec N$ for two proper class models?

Kunen in page 88 of his "Set Theory" book says: ... For a specific given $\varphi$, the notion $M\prec_{\varphi}N$ (i.e. $\forall \overline{a}\in M~~~M\models \varphi ...
2
votes
1answer
65 views

Does this argument rely on countable choice?

Consider the following Theorem: Any algebraic field extension $K|F$ of infinite degree contains finite subextensions of arbitrarily high degree. Proof: We'll prove that, for any n, there's a ...
6
votes
4answers
361 views

Are there counter intuitive interpretations of ZF or ZFC?

Are there interpretations of ZF or ZFC that are non standard in the sense that $\epsilon$ is interpreted in a counter intuitive way that intuitively has nothing to do with "belongs to" or "is part of" ...
4
votes
3answers
156 views

Models of set theory

How can one talk of a semantics or model of set theory (lets say ZF or ZFC) when the definition of a structure (and potential model) needs a carrier set in the first place (by its definition)?
3
votes
1answer
58 views

Why are the hypotheses of Zorn's lemma met in this proof about decomposing a Hilbert space into invariant subspaces?

Let $H$ be a separable complex Hilbert space and let $\mathcal{A} \subset B(H)$ be an algebra of bounded linear operators on $H$ which is closed under adjoints. I've just read a very short proof that ...
1
vote
2answers
44 views

$\kappa\cdot\kappa= \kappa,$ for infinite cardinals

I am trying to understand the proof that uses a maximal-lexicographic ordering. For an infinite ordinal $\kappa,$ the canonical well-ordering of $\kappa \times \kappa,$ denoted by $<_{cw}$ is ...
2
votes
0answers
63 views

A couple of questions on ordinal numbers

While going over von Neumann's definition of ordinal number I made a couple of conjectures whose veracity I have not been to able to decide yet. I share them here in order to pick up hints ...
1
vote
3answers
60 views

How to prove that a set is infinite iff it is Dedekind infinite?

I need to prove the following: A set $X$ is infinite if and only if it is equipotent to a proper subset of itself Here, $X$ is defined to be infinite if $|X|$ is not a non-negative integer or ...
3
votes
2answers
87 views

Is there any infinite quantity small enough to be affected by finite changes?

Hilbert's paradox of the Grand Hotel shows us, among other useful things, that the cardinality of any infinite set is a quantity equal to n more than itself for any finite n. I am interested in ...