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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1answer
43 views

Cardinality of the set of all monoids with countably many elements

How can I prove assuming the continuum hypothesis, that the cardinality of the set of all monoids with countably many elements has cardinality the same as that of the power set of the real numbers. ...
2
votes
1answer
59 views

Doubt about the proof of $V_\omega\models\mathsf{Separation}$

In Kunen's 'set theory', he introduce following theorem: Suppose that for each formula $\phi(x,z,\vec{w})$ with no variable besides the displayed ones free,$$\forall z, \vec{w}\in M:\{x\in z:\phi ...
3
votes
3answers
159 views

The Free Set Lemma

The statement of the lemma is as follows: if $$f: \omega_1 \rightarrow \{x\ :\ x\ \textrm{is finite}\}$$ then there is an uncountable $S \subseteq \omega_1$ such that for all distinct $\alpha,\ \beta ...
6
votes
1answer
80 views

Regarding functions on $\omega_1$

I've been trying to prove a property that apparently all functions $g: \omega_1 \rightarrow \omega_1$ have, where $\omega_1$ is the least uncountable ordinal. For $\alpha \in \omega_1$, define ...
1
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1answer
61 views

A Question Regarding the Origin of the Axiom of Symmetry

It is my understanding that Chris Freiling's "Axiom of Symmetry" is based on a counterexample to CH given by Sierpinski in his book "Hypothese de continu". Since I neither read nor speak French, I ...
2
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0answers
81 views

Formula Complexity of $\models_n$

I want to show $\models_0$ is $\Sigma_1$, and $\forall n \geq 1, \models_n$ is $\Sigma_n$. So for the base case, $\models_0 \ulcorner \phi \urcorner$ is true iff $\ulcorner \phi \urcorner \in ...
3
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0answers
46 views

Is there any filter space characterization for strong cardinals?

The theory of filter spaces is introduced by Apter, Diprisco, Henle & Zwicker, in their joint paper: Arthur Apter, Carlos Di Prisco, James Henle, and William Zwicker, Filter spaces: towards a ...
2
votes
1answer
113 views

How to define mathematical objects from scratch?

Are there any "Zermelo-type theories" for some other type of mathematical objects except for sets? I.e., axiomatic systems for mathematical objects formalized for first-order logic using a language ...
17
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2answers
543 views

Was there anybody before Cantor who conjectured existence of infinities of different sizes?

Georg Cantor is formally known as the first one who discovered existence of infinities of different sizes. But the history of thinking about the concept of "infinity" in maths and philosophy goes back ...
1
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1answer
65 views

A question from Kunen's book: chapter VII (H9), about diamond principle

Assume $(\mathbb{P}$ is c.c.c.$)^M$ and $\Diamond$ holds in $M[G]$. Show that $\Diamond$ holds in $M$. Hint: It is sufficient to verify $\Diamond^-$ in $M$. Should I try to create a ...
1
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1answer
23 views

Initial segments of well-ordered sets are isomorphic

I want to prove that if $(X,\prec)$ and $(Y,<)$ are well-ordered sets that X must be isomorphic to an initial segment of $Y$ or vice versa. I am trying to do this by defining the function: ...
0
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1answer
34 views

Disjoint union of sets $A_i$ indexed by a set $I$ is still a set?

I have a question in set theory considering a disjoint union of sets. My question is: if we form the following disjoint union $$\bigsqcup_{i \in I} A_i$$ where the $A_i$ are are sets and $I$ is a set ...
7
votes
3answers
191 views

Why is the Power Set Operation Inherently Vague?

It is a somewhat common view among mathematicians/philosophers (who have an opinion on the subject) that the power set operation is inherently vague. They go on to say that its inherent vagueness is ...
7
votes
1answer
96 views

Consistency strength of the “club ultrafilter”

What are the consistency strengths of $$ZF+``\text{The club filter on $\omega_1$ is an ultrafilter}"$$ and $$ZF + DC + ``\text{The club filter on $\omega_1$ is an ultrafilter}"?$$ I know that the ...
4
votes
2answers
66 views

Does every set have a derangement? [duplicate]

A derangement of a set $A$ is a bijection from $A$ to itself with no fixed points. Is it the case that every infinite set has a derangement.
5
votes
1answer
133 views

What goes wrong in the following argument that our conception of “set” is inconsistent?

This question might sound facetious, but it is a genuine question which I am very much interested in. I apologize in advance if it is too conceptual or philosophical, but I'm optimistic that I might ...
4
votes
1answer
90 views

Does it make sense to form a set whose elements are proper classes?

In chapter 4 of Handbook of Categorical Algebra, vol 1, the author defines a "subobject of $A$" as "an equivalence class of monomorphisms with codomain $A$" (for a suitable notion of equivalence). He ...
2
votes
1answer
70 views

Can arithmetic truths fix the truth value of the Continuum Hypothesis?

Many logicians and philosophers believe that all sentences expressible in the language of Peano Arithmetic have determinate truth-values, even though no nice formal system can capture all of these ...
3
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0answers
59 views

Equivalent forms of Jensen's diamond principle [duplicate]

I try to prove that these four statements are equivalent: $\Diamond$ There are $A_\alpha\subseteq \alpha\times\alpha$ for $\alpha<\omega_1$ s.t. for all $A\subset \omega_1\times\omega_1$, ...
2
votes
1answer
52 views

Axiom of Choice Equivalent

I'm trying to prove the following statement is equivalent to the Axiom of Choice: "For any set $A$, there exists a function $F$ with dom $F = ⋃A$ and for each $x ∈ ⋃A$, $x ∈ F(x) ∈ A$." (1) The ...
4
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0answers
55 views

$\mu$-clubs and stationary sets consisting of elements with cofinality $\mu$

Let $\mu < \kappa$ be infinite cardinals. A set $C$ is called a $\mu$-club in $\kappa$, if it is unbounded in $\kappa$ and contains all its limit points of cofinality $\mu$. Now let $T \subset S ...
2
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0answers
39 views

Show that if A is a finite set of non empty sets, then A has a choice function.

I'm trying to prove that if A is a finite set of non empty sets, then A has a choice function. My approach is to construct such function. Since it is a finite set we can say that $A=\left \{ ...
3
votes
1answer
77 views

Is the first-order incompleteness of a theory (like arithmetics, set theory or logic itself) avoidable in a second or higher-order axiomatizations?

Can we avoid the first-order incompleteness of a theory (like arithmetics or set theory) in a second-order theory which contains the previous? How does it depend on the chosen semantics or models? If ...
6
votes
1answer
114 views

Does iterating the consistency of ZFC answer any natural questions?

The following is a natural question that occurred to me, but I'm not sure if it's even well-defined since I haven't read the literature on iterating consistency statements. Let $Con_0(ZFC)=Con(ZFC)$ ...
4
votes
1answer
90 views

What goes wrong when you try to reflect infinitely many formulas?

The reflection principle in ZFC shows that you can construct a set that reflects finitely many formulas. Suppose we wanted to reflect {$\phi_n$} and we construct a set $M_n$ to reflect $\phi_1, ... , ...
4
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0answers
68 views

Strange Consequences of Large Cardinals in Probability

Large cardinal axioms are very strong hypothesizes and as any other strong hypothesis they have many strange consequences in mathematics. On the other hand we know that if we bring even the least ...
10
votes
2answers
901 views

How to find the shortest proof of a provable theorem?

Roughly speaking, there are some fundamental theorems in mathematics which have several proofs (e.g. Fundamental Theorem of Algebra), some short and some long. It is always an interesting question ...
0
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0answers
47 views

Books and articles on model theory for set theory

I'm interested in books and/or articles which explore a little more in depth the model theory of set theory. I'm aware that most books on set theory have a section or two on models (e.g. Jech, Kunen), ...
2
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0answers
54 views

Monomorphisms in a concrete category

Let $\mathcal{C}$ be a concrete category, i.e., a category which admits a faithful functor $C:\mathcal{C}\rightarrow \mathsf{Set}$. It is certainly not the case that $f$ a mono in $\mathcal{C}$ ...
0
votes
1answer
70 views

Induction in a first order system with ZF

Suppose I have some characterization of the natural numbers $N$ in a first-order system under ZF. To be precise, I have $N = \lbrace n: \forall w:( w\space is\space inductive) \rightarrow n \in w ...
3
votes
1answer
56 views

If $2^{\aleph_{\beta}}\geq \aleph_{\alpha}$, then $\aleph_{\alpha}^{\aleph_{\beta}}=2^{\aleph_{\beta}}$.

If $2^{\aleph_{\beta}}\geq \aleph_{\alpha}$, then $\aleph_{\alpha}^{\aleph_{\beta}}=2^{\aleph_{\beta}}$. Proof: Note that if $\beta \geq \alpha$, then we have ...
0
votes
1answer
126 views

Is Bell's Notion of “Abstract Set” Flawed?

Consider the following definition of "abstract set" given by John L. Bell (who wrote the book "Set Theory: Boolean-valued Models and Independence Proofs") from his preprint "Abstract and Variable ...
11
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0answers
152 views

How much set theory does the category of sets remember?

Question. Let $M$ be a model of enough set theory. Then we can form a category $\mathbf{Set}_M$ whose objects are the elements of $M$ and whose morphisms are the functions in $M$. To what extent is ...
5
votes
3answers
348 views

Good Sources for Lecture Movies in Set Theory, Logic and Philosophy of Maths

Of course as any other researcher I'm not able to attend any scientific event in my research area. But it is always interesting and useful to watch the lecture movies of these events. I will ...
0
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1answer
50 views

Construction of natural numbers in ZF/ZFC without class predicates.

I'm told by some websites that it's possible to formally construct the naturals in ZF without recourse to either predicates involving classes ("all inductive sets", yuck) or some sort of "intuitive ...
5
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0answers
46 views

Type-definable Forcing or forcing in a non-first order setting

Roughly speaking, in set forcing the forcing notion is a set from ground model's perspective and in class forcing its a definable subset of the ground model given by solutions of some formula with ...
2
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0answers
34 views

Is it true that each large cardinal which is not first order expressible has no extender characterization?

It is well-known that Reinhardt cardinal (i.e. The critical point of a non-trivial self-elementary embedding of the universe in $ZF$) is not first order expressible. Does this imply that Reinhardt ...
4
votes
2answers
219 views

Is there any category theoretic proof for independence of Continuum Hypothesis?

Both of set theory and category theory could be a foundation for mathematics. Many set theoretic arguments could be translated to a category theoretic argument and vice versa. Question: Is there ...
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1answer
69 views

ineffable, weakly compact, and ? cardinal

In the online book, page 312, http://projecteuclid.org/download/pdf_1/euclid.pl/1235419485 what cardinal notion do we get, by requireing that X in the bottom of the page, is not only stationary ...
0
votes
0answers
37 views

Does this version of Schröder-Bernstein-Cantor imply choice? [duplicate]

Consider the following statement: $(*)$ For all sets $A$,$B$ and surjections $f\colon A \rightarrow B$, $g\colon B \rightarrow A$ there is a bijection $h\colon A \rightarrow B$ Given choice, this ...
1
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1answer
28 views

Generic in Boolean-Valued-Models

Let $M$ be a transitive $\in$-interpretation of a extension $T$ of $ZF$ in $ZF$,and let $B$ such that $$T\vdash B\in M\wedge B\text{ is a complete Boolean algebra}$$ Then, using the fact that any set ...
2
votes
1answer
60 views

Equivalent condition for CH

I would like to know why the following condition $\otimes$ is equivalent with the Continuum Hypotheses. $\otimes$ There exists a sequence $<A_{\alpha} | \alpha < \omega_1>$, such that for ...
1
vote
2answers
31 views

Are the hyperreal numbers densely ordered?

Using the construction mentioned in this post are the hyperreals densely ordered? If not, is there a construction in which they are? This should be a rather straightforward question, but my brain ...
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4answers
164 views

Partition of $\mathbb{R}^+$ into two semigroups

Can the semigroup $(\mathbb R^+ ,+)$ be partitioned into two semigroups? I have been trying but haven't found anything, please help.
2
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0answers
61 views

Kanamori's proof that Vopenka's principle implies the existence of extendibles.

In The Higher Infinite, Kanamori gives a proof that Vopenka's principle implies the existence of (many) extendibles. The relevant theorem is Proposition 24.15 on p.337. Working in a $V_\kappa$ which ...
0
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1answer
52 views

What is the formal way to define “class” in ZFC?

Unlike axiomatic systems deal with classes such as NBG, the term "class" is not a word in ZFC. How do I formally treat classes? Here is an example of what I'm exactly talking about. Example ...
1
vote
1answer
99 views

There are any known example of independence proofs about independence result?

(This question is inspired by deleted question, and the questioner who write the deleted question wrote new question.) It is well-known that consistency of "ZF + DC + every set of reals are Lebesgue ...
2
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2answers
140 views

Different models of ZF disagree on equality of explicit recursively enumerable sets

Assuming that ZF is consistent, are there two recursively enumerable sets defined by explicit enumerators that are the same in one model of ZF+Con(ZF) but different in another model of ZF+Con(ZF)? If ...
0
votes
1answer
37 views

Normal tree with $\aleph_1$ nodes and no branch of cardinality $\aleph_1$

I am trying to understand how a normal tree with $\aleph_1$ node can fail to have a branch of cardinality $\aleph_1$. Consider the tree of height $\omega_1$ whose nodes are $\mathbb{Q}$-valued ...
5
votes
1answer
154 views

Second reading on set theory? Any recommendations?

I have in past six-ish months studied through the Herbert Enderton's Elements of set theory book. Up to the point the book is great,I loved most parts of it and learned almost everything up to the ...