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 ...

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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 ...
5
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1answer
48 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 ...
5
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0answers
38 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 ...
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5answers
759 views

Is there a statement whose undecidability is undecidable?

We know there are statements that are undecidable/independent of ZFC. Can there be a statement S, such that (ZFC $\not\vdash$ S and ZFC $\not\vdash$ ~S) is undecidable?
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0answers
25 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 $W_2$ ...
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0answers
45 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 ...
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1answer
29 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, ...
3
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5answers
863 views

cardinality of all real sequences

I was wondering what the cardinality of the set of all real sequences is. A random search through this site says that it is equal to the cardinality of the real numbers. This is very surprising to me, ...
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0answers
68 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 ...
3
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1answer
118 views

A conjecture about filters and finite unions of cartesian products

Let $U$ be some set. Let $\Gamma$ be the set of all finite unions of cartesian products ($X_1 \times Y_1 \cup \dots \cup X_n \times Y_n$) of sets on $U$. Obviously, $\Gamma$ is a a distributive ...
2
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1answer
104 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 ...
2
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2answers
81 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 ...
2
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2answers
83 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 ...
0
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1answer
40 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
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2answers
121 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
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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
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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 ...
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1answer
27 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, ...
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0answers
101 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 ...
2
votes
1answer
155 views

Why does the existence of independent statements not prove consistency?

I've read before that, by the Principle of Explosion, if a theory is inconsistent, then absolutely any statement can be proven within it. Obviously, there are statements which are independent of ZFC ...
0
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0answers
81 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 $\alpha$?
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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 ...
3
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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 ...
69
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15answers
6k views

Why did mathematicians take Russell's paradox seriously?

Though I've understood the logic behind's Russell's paradox for long enough, I have to admit I've never really understood why mathematicians and mathematical historians thought it so important. Most ...
2
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0answers
64 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
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2answers
58 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
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1answer
111 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 ...
2
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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 ...
14
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1answer
326 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 ...
4
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0answers
63 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 ...
4
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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 ...
1
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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
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1answer
57 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
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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 ...
6
votes
4answers
360 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
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2answers
319 views

Order-isomorphic with a subset iff order-isomorphic with an initial segment

Let $(X, \prec)$ and $(Y, <)$ be two well-ordered sets. I want to show that if $X$ is isomorphic to a subset of Y then $X$ is isomorphic with an initial segment of $Y$. (The other direction is of ...
3
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1answer
57 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 ...
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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 ...
36
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11answers
9k views

Why is “the set of all sets” a paradox?

I've heard of some other paradoxes involving sets (ie, "the set of all sets that do not contain themselves") and I understand how paradoxes arise from them. But this one I do not understand. Why is ...
4
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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
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1answer
64 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 ...
4
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3answers
153 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)?
4
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2answers
174 views

How much set theory is necessary for serious logic?

I'm currently studying logic at my university and I have been trying to squeeze in as much set theory on the side as possible. Considering that I am spending quite a lot of time studying set theory ...
1
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2answers
43 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
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0answers
62 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 ...
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3answers
56 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 ...
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2answers
156 views

Is Infinity Needed in Maths? Does Infinity Actually Exist? [closed]

I'm asking this question as I have been having an on going online debate with a friend of mine. I claimed that Infinity does in fact exist in Maths and in Reality, as there's a whole plethora of ...
3
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2answers
86 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 ...
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0answers
19 views

question regarding limit of the a sequence $[x]$ along the a ultrafilter $U$

I've some question regarding limit of the a sequence $[x]$ along the a ultrafilter $U$ Its written let $U$ be an ultrafilter on $N$ where $N=\{1,2,....\}$. Now let $[x]=(x_i \mid i\in N)$ be a ...
3
votes
1answer
46 views

Absolute coequalizers in $\mathbf {Set} $

Let $ A $ be a set and let $ R\subseteq A\times A $ be an equivalence relation on $ A $. Denote by $ p, q $ the projections $ R\longrightarrow A $ on the first and second factor, respectively. The ...