forcing, large cardinals, descriptive set theory, infinite combinatorics, cardinal characteristics, forcing axioms, ultrapowers, measures, reflection, pcf theory, models of set theory, axioms of set theory, independence, axiom of choice, continuum hypothesis, determinacy, Borel equivalence ...

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50 views

About the limitation by size

This could be a big post, so I'll try to summarize my thoughts and divide them into several questions. When working in category theory, I used to choose the following definition. A category $C$ is ...
6
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4answers
360 views

Direct axiomatization of ordinal and cardinal numbers

Again, this question is related (**) to a previous one: in standard books on basic set theory, after stating the axioms of ZFC, ordinal numbers are introduced early on. Afterwards cardinals appear: ...
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1answer
85 views

Absoluteness and Tree Representations

Suppose $T$ is a tree on $\omega \times \omega \times \delta$ for some ordinal $\delta$ is a homogeneous tree (with some coherent set of measures witnessing homoegeneity). ($T$ can have additional ...
2
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1answer
122 views

Some very weak statements on choice

This is a follow-up question to Does $|(X\times\{0\}) \cup (X\times\{1\})| \leq |X|$ for $X$ infinite imply ${\sf AC}$? Consider the statements $(\text{S}1)$ For any infinite set $X$ there ...
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1answer
119 views

Frankl's union-closed sets conjecture for infinite families

This question is motivated by Frankl's union-closet sets conjecture. Let $X$ be a non-empty set. We say that a family ${\cal A} \subseteq {\cal P}(X)$ is union-closed if $\emptyset\notin{\cal A}$ and ...
2
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1answer
113 views

Does $|(X\times\{0\}) \cup (X\times\{1\})| \leq |X|$ for $X$ infinite imply ${\sf AC}$?

Consider the statement For any infinite set $X$ there is an injection $\varphi$ from $(X\times\{0\}) \cup (X\times\{1\})$ into $X$. Does this imply the ${\sf AC}$?
15
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2answers
278 views

A nice subcategory of the category of measurable spaces

Is there some notion of "nice" measurable spaces and "nice" maps between them which satisfies the following properties? The real line equipped with the Lebesgue $\sigma$-algebra is nice. Any ...
12
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2answers
365 views

tree properties on $\omega_1$ and $\omega_2$

Are the following mutually consistent (relative to large cardinals)? (1) There are no $\omega_2$-Aronszajn trees. (2) There is an $\omega_1$-Kurepa tree. In the models I know of the tree property ...
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1answer
137 views

Axiom of countable choice need for the cantor-bernstein theorem [closed]

Is the axiom of countable choice need for proving the cantor-bernstein theorem?
2
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0answers
76 views

Is there a transcendental definable function between algebras of elementary embeddings?

Let $\lambda$ be a cardinal. Let $\mathcal{E}_{\lambda}$ be the set of all elementary embeddings from $V_{\lambda}$ to $V_{\lambda}$. If $f:V_{\lambda}\rightarrow V_{\lambda}$ is a function and ...
13
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3answers
1k views

How would calculus be possible in a finitist axiom system?

I am interested in learning a little more about finitism, currently about which I only know a few encyclopedic paragraphs. I know that during some time, some mathematicians like Kronecker thought ...
4
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2answers
188 views

A weak kind of fixed point

Let $X$ be a set and let $\cal A$ be a non-empty subset of $P(X)$ with the property that whenever $A_1 \subseteq A_2 \subseteq \cdots $ is an increasing chain of elements of $\cal A$ then $\cup_i A_i ...
16
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1answer
1k views

Analogues of Primitive Recursive Functions

Let $\mathbf{A}$ be an admissible set (possibly with urelements). I am wondering if there is some good notion of "primitive recursive arithmetic" relative to $\mathbf{A}$. More precisely, I would like ...
1
vote
0answers
153 views

Reference request: Models of isomorphic languages result into isomorphic categories

This is basically a reference request by someone who has not been educated as a logician and would like to be rigorous about certain preliminary aspects of model theory. Fix an uncountable universe ...
6
votes
1answer
364 views

Taller models of ZFC

This question is somewhat related to a previous one, where I asked for new forms of infinite beyond the cardinal hierarchy. Using forcing techniques, at least the ones I know of, one starts from a ...
9
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1answer
310 views

Every measure on a set $X$ extends to the power set of $X$: Consistent or not with ZF?

Question. Is it consistent with ZF that every (countably additive, non-negative) measure $\mu: \Sigma \to \bf R$, where $\Sigma$ is a sigma-algebra on a given set $X$, extends to a (countably ...
7
votes
2answers
213 views

How can any theory prove well-foundedness of ordinals above $\omega_1^{\text{CK}}$?

$\newcommand{\omegaoneck}{\omega_1^{\text{CK}}}$ Pardon the extremely basic question - this isn't quite my area - but I'm confused about the definition of proof theoretic ordinals. The proof ...
13
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0answers
370 views

Is $\mathbb{Z}^{\omega}$ ever the union of a chain of proper subgroups each isomorphic to $\mathbb{Z}^{\omega}$?

Recall that the covering number $cov(B)$ is the least cardinal $\kappa$ such that $\kappa$ meagre sets cover the real line. Andreas Blass and John Irwin http://www.math.lsa.umich.edu/~ablass/bb.pdf ...
4
votes
1answer
177 views

A Question on HOD, V and GCH

The theorem 1.1 of the following paper: Mohammad Golshani, V, HOD, and the GCH, Journal of Symbolic Logic. states that: Theorem: Assume $V\models ZFC+GCH+~\text{There exists ...
43
votes
7answers
5k views

Why hasn't mereology succeeded as an alternative to set theory?

I have recently run into this wikipedia article on mereology. I was surprised I had never heard of it before and indeed it seems to be seldom mentioned in the mathematical literature. Unlike set ...
17
votes
1answer
533 views

Dual Borel conjecture in Laver's model

A set $X\subseteq 2^\omega$ of reals is of strong measure zero (smz) if $X+M\not=2^\omega$ for every meager set $M$. (This is a theorem of Galvin-Mycielski-Solovay, but for the question I am going to ...
2
votes
2answers
508 views

Can we compute every definable number with knowledge of the halting problem?

Suppose we knew the answer to the halting problem, and the halting problem for this new system with the old halting problem solved. And so on. Would this allow us to compute every definable number?
4
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0answers
135 views

Set theory and forcing from the point of view of a formal system $G^+$ of Gentzen type

There are four papers by Vladimir Alfeevich Kuznetsov, which discuss the above titled topic: (1) Some problems in set theory from the standpoint of a formal system G+ of Gentzen type. (Russian) Akad. ...
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3answers
1k views

Existence of prime ideals and Axiom of Choice.

One of the must obvious equivalences of Axiom of Choice is the converse of Krull Theorem. Bernhard Banaschewski in the Article titled by A New Proof that “Krull implies Zorn” showed a very simple ...
4
votes
2answers
559 views

Is platonism regarding arithmetic consistent with the multiverse view in set theory?

A "truth" platonist for arithmetic believes, given a statement in the language of arithmetic, that the problem whether the statement is true has a definite answer. Prof. Hamkins has argued for a ...
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votes
2answers
214 views

Selection problem in a collection of non-empty sets

Is there a set $X\neq\emptyset$ and a collection ${\cal F}\subseteq {\cal P}(X)\setminus\{\emptyset\}$ of non-empty subsets of $X$ with the following properties? $a\in {\cal F} \implies |a|\geq 2$, ...
5
votes
1answer
248 views

Statements that Could be Forced by Ultrapowers

Ultrapower of a structure is a very flexible mathematical creature in comparison with the ground structure and its ordinary products. Depending on the nature of ground structure and the good ...
2
votes
2answers
282 views

Non-Formal Applications: Higman and Kruskal

After looking through many papers, I noticed that most of the discussions and proofs for Higman's Lemma and Kruskal's Tree Theorem only have formal applications in set theory, logic, and type theory. ...
2
votes
2answers
423 views

Is there a countable pseudocharacter Hausdorff space,such that…?

Let X be a Hausdorff space and Difine the Property A as following: if $\mathscr{U}$ is a collection of open sets of X that witnesses Hausdorff property of X (= $\forall x,y \in X$, there exist two ...
11
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0answers
175 views

Is there a locally compact, $\omega_1$-compact, not $\sigma$-countably compact space of size $\aleph_1$?

There are old ZFC examples due to Eric van Douwen that satisfy all the properties in the title, except that they are of cardinality $2^{\aleph_0}$, so the answer to the title question is YES if the ...
6
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1answer
321 views

all subsets borel

Assume Martin's axiom plus $\neg CH$. It is well known, via almost disjoint forcing, that every set of reals of size less than continuum is an example of a metric space whose subsets are all ...
3
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2answers
262 views

Is there an uncountable Borel almost disjoint family?

Here we are considering subsets $\mathcal{F}$ of $2^\omega$, which are in correspondence with families of subsets of $\omega$ (sets of "reals"). Such a family is Borel if it is a Borel subset of ...
7
votes
3answers
880 views

About the axiom of choice, the fundamental theorem of algebra, and real numbers

About fundamental theorem of algebra, there is a large collection different demonstrations. I ask: is there some proof that avoids AC (choice axiom)? In a general topos (with natural number object) ...
14
votes
1answer
229 views

Is the Martin's axiom number $\mathfrak m$ regular

The Martin's axiom number $\mathfrak m$ is the least cardinal $\kappa$ for which $\text{MA}_\kappa(\text{ccc})$ is false, i.e. the least cardinal such that there exists a ccc poset $P$ and a family ...
5
votes
3answers
605 views

What does the axiom of replacement mean and why should I believe it?

Here Professor Blass describes the following cumulative hierarchy of sets: Begin with some non-set entities called atoms ("some" could be "none" if you want a world consisting exclusively of ...
8
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1answer
343 views

Edge chromatic number of hypergraphs

This is question Selection problem in a collection of non-empty sets with a simplification in criterion 3. Is there a set $X\neq\emptyset$ and a collection ${\cal F}\subseteq {\cal ...
5
votes
0answers
137 views

The Hales-Jewett Theorem for an infinite alphabet

Recall the Hales-Jewett Theorem: HJT: Given a finite alphabet $A$ and some $r \in \mathbb{N}$, there is some $H \in \mathbb{N}$ such that whenever $A^H$, the set of all length-$H$ words from $A$, ...
6
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1answer
187 views

Does “$|{\cal P}_2(X)| = |X|$ for $X$ infinite” imply ${\sf (AC)}$?

This comes from a comment made by user bof in this thread. Let $X$ be a set, define ${\cal P}_2(X) = \big\{\{a, b\}: a\neq b\in X\big\}$. Consider the statement ${\sf (S)}$ If $X$ is an ...
12
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1answer
764 views

Reverse-engineer forcing: am I reinventing the wheel?

In the course of a project I’m working on, I’ve started playing around with a sort of “reverse-engineering” forcing. It seems interesting, but I have a sinking feeling I’m reinventing the wheel; does ...
4
votes
1answer
173 views

Are there first order theories of interest to an algebraist or at least a model theorist of large cardinal consistency strength?

I am wondering if there are some first order theories of algebraic structures or structures of interest to model theorists of large cardinal consistency strength or at least unexpectedly high ...
3
votes
0answers
324 views

“Nicely” strong measure zero sets

This question is essentially an expanded version of the unanswered half of Two strengthenings of "strong measure zero". A set $X$ of reals is strong measure zero if, for any $f: ...
24
votes
2answers
1k views

A set that can be covered by arbitrarily small intervals

Let $X$ be a subset of the real line and $S=\{s_i\}$ an infinite sequence of positive numbers. Let me say that $X$ is $S$-small if there is a collection $\{I_i\}$ of intervals such that the length of ...
16
votes
1answer
472 views

Two strengthenings of “strong measure zero”

A set $X\subseteq\mathbb{R}$ is strong measure zero if, for every sequence $(\epsilon_i)_{i\in\mathbb{N}}$ of positive reals, there is a sequence $(I_i)_{i\in\mathbb{N}}$ of open intervals covering ...
15
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1answer
349 views

A problem of Keisler and Tarski

The following question dates back to Keisler and Tarski: From accessible to inaccessible cardinals, Fund. Math. 53, 1964 and also perhaps Mazur: On continuous mappings of Cartesian products, Fund. ...
16
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1answer
486 views

What sort of cardinal number is the Löwenheim-Skolem number for second-order logic?

In their paper "On Löwenheim-Skolem-Tarski numbers for extensions of first order logic", Magidor and Väänänen make the following statement: "For second order logic, $LS(L^{2})$ ...
5
votes
1answer
253 views

How much choice does a linear or well-order on cardinals imply?

It is well-known that if the natural (partial) order on the class of cardinal numbers is a linear order, then it is in fact a well-order and the axiom of choice holds. I was, however, interested in ...
3
votes
1answer
128 views

Invariants of category in Polish spaces

Consider the invariants appearing in the Cichoń's diagram: $add(\mathcal I)$, $cov(\mathcal I)$, $non(\mathcal I)$, $cof(\mathcal I)$, where $\mathcal I$ is either the ideal of null sets for the ...
7
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0answers
160 views

Can you define a probability measure on the set of countable transitive models of ZFC?

It is well known that the set of hereditarily countable sets $H(\omega_1)$ —or, if you prefer, $H_{\omega_1}$— has cardinality $2^{\aleph_0}$, and I understand that every countable ...
10
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1answer
335 views

What sort of large cardinal can continuum be?

I have stumbled across a related question asking which large cardinal properties can hold for $\aleph_1$. This question is probably also related, asking in what ways $\aleph_0$ is a "large" cardinal. ...
10
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0answers
245 views

Is the universality of the surreal number line a weak global choice principle?

I'd like to consider the principle asserting that the surreal number line is universal for all class linear orders, or in other words, that every linear order (including proper-class-sized) linear ...