Questions tagged [set-theory]

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 theory, definability, infinite combinatorics, transfinite hierarchies; etc. More elementary questions should use the "elementary-set-theory" tag instead.

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Skolem's paradox disproof?

Skolem's paradox, derived from the downward Lowenheim-Skolem theorem (dL-S), is that there exists a countable model $M$ of set theory, specifically Zermelo-Fraenkel set theory with the Axiom of Choice ...
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1answer
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On Theorem 17.7 of Jech's Set Theory

Theorem 17.7 of Jech's Set Theory is Kunen's theorem: If $j : V \to M$ is a nontrivial elementary embedding, then $M \neq V$. My main issue lies in one line of the proof, which says: The cardinal $\...
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Skolem Hulls and countable elementary submodels of $H(\theta)$

Let $\theta$ be regular and uncountable. Fix a well-ordering $<$ of $H(\theta)$. Since the structure $(H(\theta),\in,<)$ has a definable well-ordering, every subset $A\subset H(\theta)$ admits a ...
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Assume A is a set and define $\alpha$ to be the set of ordinals dominated by A. Show that (iii) $\alpha$ is the least cardinal greater than card A. [duplicate]

I said: We know that a cardinal is the least ordinal equinumerous to A. We know $\alpha$ is equinumerous to $\bigcup$A, and so $\alpha$ is the least upper bound of A. Thus $\alpha$ is the least ...
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Counterexamples to the Weak Unique Branch Hypothesis and Weak $(\omega_1+1)$-Iteration Hypothesis

This question of mine comes from Woodin's "In Search of Ultimate-L" article, where they define the Weak Unique Branch Hypothesis and Weak $(\omega_1+1)$-Iteration Hypothesis. My confusion ...
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1answer
48 views

Does counting inaccessibles have a fixpoint?

Assume the existence of a proper class of inaccessibles $I$, and let $f : \mathrm{Ord} \to I$ denote the unique strictly increasing class surjection given by Mostowski Collapse. Does $f$ have a ...
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1answer
40 views

Properties of the spaces associated to the uniform ultrafilters of an extender

This question of mine comes from Woodin's "In Search of Ultimate-L" article, where they introduce some variables associated to each extender. I have a question about the remark following ...
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1answer
52 views

Is the leftmost branch function $\text{UB}\rightarrow \omega^\omega$ Borel?

Let's denote by $\text{UB}$ the set of ill-founded trees over $\omega$ having a unique branch. Consider now the function $b: \text{UB} \rightarrow \omega^\omega$, that associates to each tree $T$ in $\...
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1answer
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Bounded subsets of uncoutable totally ordered set

As my username might possibly suggest, set theory and logic is not really an area of mathematics I know much about. But, there is this statement and apparent proof I was able to come up with, both of ...
3
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1answer
72 views

How do you simplify $\cup\{A, \cup A\}$?

I want to simplify $\cup\{A, \cup A\},$ and also $\cup\cup\{A, \cup A\},$ so forth. I thought $\cup\{A, \cup A\}$ would not be simplified more. To say this in plain english, this union is a set that ...
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Ordinal exponentiation — Kunen / Set theory Exercise I.13.39

Exercise I.13.39 in Kunen's Set Theory: If $\kappa$ is an infinite cardinal and $\alpha = \cup_{n < c}X_n$, where $c < \omega$ and each $\textrm{type}(X_n) < \kappa^\omega$, then $\alpha <...
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Can we get something equivalent to $\mathsf{ZFC2}$ by naively allowing the separation and replacement to range over second-order wffs? [duplicate]

I've seen $\mathsf{ZFC2}$ mentioned in a few questions such as this one and I'm curious about ways to axiomatize it that have the fewest differences from plain $\mathsf{ZFC}$ as possible. My question ...
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Symbol for cover

Is there a symbol for the cover operator? Given two family of sets(lets say $\mathcal{A}$ and $\mathcal{B}$) $$ \mathcal{A} = \{A_1, A_2, ..., A_m\}$$ $$ \mathcal{B} = \{B_1, B_2, ..., B_n\}$$ $\...
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1answer
46 views

Can we say that any Dedekind-finite family of sets has a choice function without AC?

Let $(S_i)_{i \in I}$ be a family of non-empty sets where $I$ is a Dedekind-finite set. Can we say without the Axiom of Choice that this family has a choice function? I know that for any finite family ...
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2answers
85 views

If $1+ \alpha = \alpha$, $\alpha$ is an infinite ordinal

I need to prove the following statement: If $1+ \alpha = \alpha$, $\alpha$ is an infinite ordinal. I am trying to use Bernstein's Theorem(CBS) to show if $1+ \alpha \leq \alpha$, i.e., there is an ...
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Is there a name for the family of nested sets generated by $A_{n+1} = \{A_n, f(n+1)\}$ recursion?

In my research, I consider a function $f: \mathbb{N} \rightarrow \mathbb{N} $ and a family of nested sets defined recursively by: $A_{n+1} = \{A_n, f(n+1)\}$ for $n > 1$, and $A_1 = \{ f(1)\}$ as ...
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2answers
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With regards to Zermelo-Fraenkel Set Theory, is the $0$ and $\emptyset$ in $0 =\{ \}= \emptyset$ equal?

I have been studying philosophy for some time now, although my knowledge with respect to mathematics is amateur at best. I recently starting learning set theory on my own, but have ran into an issue (...
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55 views

Some cardinals/ordinals Arithmetic

I am having trouble grasping ideas for the following three problems (and I am unsure whether the third condition even holds). For arbitrary ordinals $\alpha$, $\beta$, (Edit: on the LHS the operations ...
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40 views

(Enderton set theory) Hartog's theorem and cardinal numbers

Assume that A is a set and define (as in Hartogs' theorem) $\alpha$ to be the set of ordinals dominated by A. Show that (i) $\alpha$ is a cardinal number, (ii) card A < $\alpha$ , and (iii) $\...
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Does $\mathsf{ZFC}$ prove that the field of real numbers has one of these compactness properties?

Suppose $\mathcal{A}$ is a $\Sigma$-structure and $\kappa<\lambda$ are infinite cardinals. Say that $\mathcal{A}$-satisfiability is $(\kappa,\lambda)$-compact iff for every theory $T$ in an ...
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1answer
60 views

A proof of Zermelo’s Well-Ordering Theorem using Zorn’s Lemma

The thing is that I have used Zorn’s Lemma to prove that, given a set $A$, it can be well ordered, but I don’t know if I’ve made any wrong assumptions. Let $A$ be a set, and we want to prove that it ...
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1answer
57 views

Is there an ordinal satisfying $\xi$ = $\omega$+$\xi$?

Given that $\omega$ is the $ord(\mathbb{N})$, I need to prove/disprove whether there is an ordinal $\xi$ satisfying the equation $\xi$ = $\omega$+$\xi$. Here, the order given is an anti-...
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1answer
45 views

Show the set of ordinals less than a certain cardinal is uncountable

Sorry for the incomplete title; it would become too long. The question is as follows: for the ordinal $\omega$, consider the equation card $\omega$ $\lt$ card $\xi$ $\leq$ card $2^\omega$. Denote the ...
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3answers
330 views

I am looking for class of math problems which are provable in ZF if and only if they are provable in ZFC

I know that P vs NP and Riemannian hypothesis are of this class but could not find any article on that. I would also appriciate links or books on related theme. My question is: what are some other ...
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27 views

Why can we find certain conditions in a tree forcing $PT_{f,g}$ in the book 'Set theory - on the structure of the real line' by Bartoszynski and Judah

Reading a proof in the book 'Set theory - on the structure of the real line' by Tomek Bartoszynski and Haim Judah, I encountered problems. I have several questions about the proof of Lemma 7.3.7; ...
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1answer
98 views

Decide whether $\{ x | x=x\times x\}$ is a set

This is an exercise my set theory teacher suggested we try and I feel like the answer is yes (because it seems to me that no set other than $\emptyset$ satisfies $x\times x=x$, then the collection ...
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1answer
40 views

Question about ordinal exponentiation.

I am having trouble on how to prove the following identity using ordinal arithmetic. Given a natural number $n > 1$: $$n^{\omega^{\omega}} = \omega^{\omega^{\omega}}$$ I've tried to use the ...
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1answer
79 views

About Construction a one-to-one function from $(a,b)$ onto $[a,b]$

This question asked to construct one-to-one function from $(a,b)$ onto $[a,b]$. I know there is a function but it seems the question to define this function explicitly. How this can be done? Edit I ...
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77 views

What are the current lower bounds on inconsistency of ZFC?

It is unknown whether ZFC (or even PA) is consistent, that is, there might be a proof of a statement of form $P \land \neg P$ in ZFC. So it seems natural to try to find lower bounds for this ...
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1answer
34 views

How to write “the set of all walks of a graph” in formal logic notation?

I have this question on the CS stackexchange to figure out how to model "a walk of a graph" in a custom formal language. I think the problem is that there is not just one walk of a graph but ...
2
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43 views

PCF theory and strongly increasing sequence

I am trying to study PCF theory by reading Don Monk's very detailed and helpful notes and the handbook article by Abraham and Magidor. The definitions of the concepts mentioned below are according to ...
3
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1answer
77 views

Can all structures have “lots of compactness” with respect to expansions by constants?

This is one of those "surely not ..." questions that I embarrassingly can't answer at the moment. Given a structure $\mathcal{A}$ in a language $\Sigma$ and infinite cardinals $\kappa<\...
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48 views

How to prove $|B^A| = |B|^{|A|}$ in sets [closed]

If we show the size of set A like this $|A|$ and $A$ and $B$ are infinite sets how can I prove this: $|B^A| = |B|^{|A|}$
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Is there a structure on $\mathbb{R}$ with a given cardinality of isomorphisms

We have, for example $|\mathrm{Aut} (\mathbb{R})|= |\mathbb{R}^\mathbb{R}|$ $|\mathrm{Aut} (\mathbb{R,+,0})|= |\mathbb{R}^\mathbb{R}|$ $|\mathrm{Aut} (\mathbb{R,<})|= |\mathbb{R}|$ $|\mathrm{Aut} (\...
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55 views

If ZFC is inconsistent, does it follow that Rayo's number is not well defined?

Rayo's number was defined as "The smallest number bigger than any finite number named by an expression in the language of set theory with a googol symbols or less.". It's not clear to me ...
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52 views

Definition of finite in set theory [duplicate]

The usual definition of a finite number $n$ in the ZFC set theory is $n\in\mathbb{N}$, which is equivalent to "$n$ is 0 or a successor ordinal, and so are all its elements". But this is not ...
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1answer
111 views

Partitioning a metric space into Cantor sets

A "Cantor set" is a topological space which is homeomorphic to the standard Cantor set $C$. In my answer to the question Another way for partition of perfect set by user 00GB I pointed out ...
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1answer
79 views

About ZFC and Partition perfect set into perfect sets

I was asking this question Another way for partition of perfect set two days ago and there was a nice discussion over there but now I need to change the question by adding more conditions. Question: ...
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0answers
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Homotopy Type Theory and Foundations of Mathematics [closed]

Is homotopy type theory still being actively investigated in foundations of mathematics? It seems like Harvey Friedman has quite the distaste for it. If HTT were a viable FOM theory, shouldn't there ...
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0answers
11 views

Find the limit point of the set A= { n + 1/(2^m) where n,m belongs to set of natural number} [duplicate]

My answer is coming to as {k; k belongs to set of natural numbers} is my answer correct ?
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1answer
57 views

Infinitely stacked set

suppose we have following sequence: Let $s_0$ be a set. Then let be $s_1 \in s_0$ our next set. In general $s_{n+1} \in s_n$. Assuming ZFC, I was curious about this: Can one prove, that such sequence ...
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0answers
42 views

Should I replace a first-order ZFC set theory each time?

I am studying a formulation of set theory as a first-order theory, which consists of the following languages: Variables $a,b,c,\dots$, called sets, Negation and disjunction $\neg$ and $\vee$, ...
2
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1answer
44 views

Why are there functions in $M[G]$ that are not in $M$?

Could someone help me find the fallacy of my argument : Let V be the universe of all sets, let M be a countable transitive model of ZFC and let $M[G]$ be a generic extension. Let $\alpha, \beta \in M$ ...
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0answers
32 views

Argument for cardinality of all possible strategies for a two-player game

If I understand correctly, a strategy for a two-player game (for either player) is a function from $\omega^{<\omega}$ (i.e. the set of all finite sequences of natural numbers) to $\omega$. Jech ...
2
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1answer
74 views

If $|A| > |B|$, then $|A-B| > |B-A|$?

If $|A| > |B|$, then $|A-B| > |B-A|$? Can it be proven without the Axiom of Choice? I think so. Here’s what I think. Since $|A|>|B|$, there is an injective function from $B$ to $A$ but no ...
4
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1answer
109 views

Does permitting comprehension for all (and only) contingent formulas result in paradoxes?

Does permitting comprehension over all well-formed formulas that are neither contradictions nor tautologies result in paradoxes? I have a hunch that a simple extensional set theory with the "...
2
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0answers
29 views

Why finding an undecidable proposition doesn't imply consistency, if anything is provable in an inconsistent axiomatic system? [duplicate]

I'm reading these independent notes on the Lectures on the Geometric Anatomy of Theoretical Physics, by Dr. Frederic P. Schuller. The first chapter is a brief summary of axiomatic systems and set ...
2
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2answers
142 views

Another way for partition of perfect set

Let $P$ be a perfect $P\subset\Bbb R.$ Then there exists a family $\{P_{\alpha}\subset P\colon \alpha<\mathfrak c\}$ of pairwise disjoint perfect subsets such that $$P=\bigcup_{\alpha<\mathfrak ...
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1answer
56 views

Is ZF equivalent to every subhierarchy coming from below being a set, over Zermelo + Ranks?

Define hierarchy as a family of sets well ordered by inclusion such that each successor is the powerset of its immediate predecessor, and the limit sets are the unions of all their predecessors. A ...
3
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1answer
42 views

What does the parameter $p$ refer to in this context?

I'm reading about the axioms of set theory (from Jech), and I'm having some confusion. There are various parts where it talks about well-formed formulas that include a parameter $p$. This appears here:...

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