**0**

votes

**0**answers

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**

votes

**4**answers

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: ...

**1**

vote

**1**answer

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**

votes

**1**answer

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 ...

**0**

votes

**1**answer

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**

votes

**1**answer

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**

votes

**2**answers

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**

votes

**2**answers

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 ...

**-3**

votes

**1**answer

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**

votes

**0**answers

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**

votes

**3**answers

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**

votes

**2**answers

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**

votes

**1**answer

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

**0**answers

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

**1**answer

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**

votes

**1**answer

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

**2**answers

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**

votes

**0**answers

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

**1**answer

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

**7**answers

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

**1**answer

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

**2**answers

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**

votes

**0**answers

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. ...

**7**

votes

**3**answers

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

**2**answers

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 ...

**-2**

votes

**2**answers

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

**1**answer

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

**2**answers

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

**2**answers

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**

votes

**0**answers

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**

votes

**1**answer

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**

votes

**2**answers

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

**3**answers

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

**1**answer

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

**3**answers

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**

votes

**1**answer

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

**0**answers

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**

votes

**1**answer

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**

votes

**1**answer

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

**1**answer

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

**0**answers

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

**2**answers

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

**1**answer

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**

votes

**1**answer

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**

votes

**1**answer

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

**1**answer

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

**1**answer

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**

votes

**0**answers

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**

votes

**1**answer

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**

votes

**0**answers

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 ...