**2**

votes

**2**answers

189 views

### What do we call functions that behave like predicate symbols?

Assume a metatheory that supports lambda-abstraction, and an object language that is merely first-order. Now let $\varphi$ denote a formula in the object language with one free variable $x$. Then we ...

**2**

votes

**1**answer

212 views

### Partial interpretation of an iteration

Suppose that $\langle\mathbb{P_\alpha,\dot Q_\beta}\mid \beta<\delta,\alpha\leq\delta\rangle$ is a system of iterated forcing.
Let $\dot a$ be a name in $\mathbb P_\delta$, and let $G_\alpha$ be a ...

**1**

vote

**1**answer

119 views

### Free monoids and full transformation monoids

I asked this question on math stack exchange, but I didn't get any responses. So, now I am motivated to ask it here. Is the class of free monoids first order axiomatizable? And what about the class of ...

**4**

votes

**1**answer

281 views

### Essential incompleteness via diophantine formulas?

Work in the first order language of number theory, consisting of the symbols $\mathbf{0}$, $\mathbf{S}$, $\boldsymbol{+}$, and $\boldsymbol{\cdot}$, and let $Q$ denote Robinson's arithmetic.
By a ...

**2**

votes

**3**answers

751 views

### An established proof in Wang Tile which I doubt

When I was reading the paper:
Wang, Hao. "Notes on a class of tiling problems." Fundamenta Mathematicae 82.4 (1975): 295-305.
from http://matwbn.icm.edu.pl/ksiazki/fm/fm82/fm82119.pdf
I could not ...

**6**

votes

**1**answer

226 views

### Characterization of intermediate submodels of generic extensions

Question 1: Suppose that $V[G]$ is a set generic extension of $V$ by some forcing notion $P\in V,$ and suppose that $W$ is a model of $ZFC, V \subset W \subset V[G].$ Can we find a forcing notion ...

**2**

votes

**0**answers

321 views

### Topological proof of a result in Logic

I proved the result below using logic. My questions:
Can this theorem be proved by purely topological means?
Do you know any theorems that either can be used to prove the same result, or which give ...

**9**

votes

**2**answers

835 views

### Is there any research on set theory without extensionality axiom?

In practice (say, in computer science), collections with many "labels" ("identities"), or collections which occur in many copies, are more frequently used than sets. Such collections do not satisfy ...

**2**

votes

**1**answer

442 views

### Is this system incomplete?

Let $\mathbf{SBM}$ be the normal modal logic system defined as $\mathbf{T}$ plus the following two axioms:
$$\mathrm{SB}: \Box(\Diamond p \rightarrow p)\rightarrow (p \rightarrow \Box p)$$
...

**4**

votes

**1**answer

85 views

### Counterexample for closedness under union of $\prec_{\infty,\kappa}$ chains

Assume $\kappa$ is uncountable and $\phi$ is an $L_{\infty,\kappa}$ sentence. Let $K$ be the collection of models of $\phi$ partially ordered by $\prec_{\infty,\kappa}$. It is well-known that $K$ is ...

**5**

votes

**2**answers

839 views

### Why can't mathematics be formalised in terms of classes rather than sets? [closed]

I've always been curious about the seeming compulsion to found mathematics upon sets, be it ZF(C) or some other system. Of course, there are other approaches these days like category theory and type ...

**21**

votes

**1**answer

602 views

### Is there a model of ZF set theory with a set that does not inject into the cardinals?

Question. Is there is a model of ZF set theory with a set $X$ that does not inject into the cardinals?
I use the term "cardinal" here in the ZF sense, so they are not necessarily well-orderable.
To ...

**7**

votes

**1**answer

295 views

### Is there a name for infinite words containing every finite words?

Apparently, the closest thing I've found would be normal number http://mathworld.wolfram.com/NormalNumber.html
But requiring that every finite words occurs is weaker than this property. So I'm ...

**4**

votes

**1**answer

191 views

### Necessity of omega-models in second order arithmetic

Are there examples of independence results over subsystems of true second order arithmetic that cannot be established using omega-models? To rule out trivial examples, let us assume that the base ...

**7**

votes

**2**answers

309 views

### Proof complexity of two directions of equivalency?

This question is not precise, but I believe has a precise formulation.
Consider a mathematical theorem which gives an equivalency between two conditions. As an extreme example:
\begin{theorem}
A ...

**7**

votes

**2**answers

345 views

### Are there “non-constructive” sets in second-order arithmetic?

Consider $\phi(A)$ a formula of second-order arithmetic with one free variable $A$ of type "set". Suppose $\exists A : \phi(A)$ is a true sentence. Does it follow (not in second order arithmetic ...

**10**

votes

**4**answers

1k views

### Does formalizing math require search and creativity, or is it near-mechanical?

I remember reading somewhere that it takes about a week to convert a page of math into something a proof-assistant like Isabelle or HOL Light would accept.
Is this type of conversion something that ...

**1**

vote

**0**answers

51 views

### Is it possible to classify all the inequivalent classes of first order sentence with quantifier rank fixed

It is known that for symbols with finite many relations, the number of inequivalent class of first order sentence with quantifier rank $m$ is finite. But is it possible to list (classify) them? At ...

**2**

votes

**1**answer

227 views

### Absolutness of $\Pi_1^1$ statements

Shoenfield absoluteness is well known for $\Pi_2^1$-statements, but it does not hold between a countable transitive model of ZFC and the universe.
But it is also known that $\Pi_1^1$ statements are ...

**12**

votes

**3**answers

469 views

### Formal/rigorous treatment of (im)predicativity/predicativism

There are several places on the web where one may find quite intuitively understandable accounts of (im)predicativity; here on MO I found two questions with very good detailed answers (Predicative ...

**5**

votes

**0**answers

383 views

### Last Status of Feferman's Conjecture on Indefinite Value of Continuum

The "true" value of $2^{\aleph_0}$ is one of the most fundamental open questions of mathematics and its philosophy. Hundreds of set theoretic results during the last century don't say anything more ...

**4**

votes

**2**answers

312 views

### Overspill in models of arithmetic

Assume that $M$ is a non-standard model of complete arithmetic, i.e. of the theory $Th(\mathbb{N})$. Suppose that $R$ and $S$ are proper cuts of $M$. (With a cut, I mean a subset of the universe of ...

**19**

votes

**1**answer

654 views

### Why isn't this a computable description of the ordinal of ZF?

In a previous MO question, I was told by several commenters that
(a) it's known that there exists a computable ordinal $\alpha_{ZF}$ that "encodes the strength of ZF set theory" (i.e., a least ...

**2**

votes

**1**answer

113 views

### Question on deriving $\alpha \rightarrow \Box \alpha$ in modal logic KTU

Let K and T be the usual modal logical principles $\Box (\alpha \rightarrow \beta) \rightarrow (\Box \alpha \rightarrow \Box \beta)$ and $\Box \alpha \rightarrow \alpha$. Let U be the modal logical ...

**2**

votes

**0**answers

121 views

### characterization of all periodic tiling of a simple set of Wang Tile

Consider a set of Wang Tile such that all the edges are either 1 or 0.... there are 16 elements in such a set.
Now, I wish to characterize all the periodic tilings of this set (better if they are ...

**0**

votes

**1**answer

82 views

### Question about the consistency of assuming (via axiom) that $\kappa < \nu$ for certain pairs of cardinal numbers provably satisfying $\kappa \leq \nu$

Call an ordered pair of formulae $\langle P(\kappa,\tilde{a}), Q(\nu,\tilde{a})\rangle$ in the language of $\{\in\}$ unproblematic iff
ZFC proves that for all $\tilde{a}$ and all cardinal numbers ...

**48**

votes

**3**answers

2k views

### How do I verify the Coq proof of Feit-Thompson?

I probably don't have the appropriate background to even ask this question. I know next to nothing about formal or computer-aided proof, and very little even about group theory. And this question is ...

**10**

votes

**1**answer

624 views

### Where can I find Gonthier's Coq code proving the four color theorem?

In a 2008 article in the Notices, Georges Gonthier announced a computer-checked proof of the four color theorem using Coq:
Gonthier, Georges. Formal proof—the four-color theorem.
Notices Amer. ...

**14**

votes

**0**answers

251 views

### Computability of Brauer groups

A friend of mine and I were talking about computable algebra, and this question came up. The answer may already be known, but I couldn't find it with Google:
Suppose I have a countable field, $k$. ...

**0**

votes

**0**answers

126 views

### Basis of periodic tiling of Wang tile

Given a set of Wang tile,
Given 3 periodic tiling: A, B, C
We define 3 vector F[A], F[B], F[C]
each vector correspond to the appearing frequency of each type of tiles in the tiling.
Now, we ...

**1**

vote

**1**answer

90 views

### simple cycle analog in 2D (with application in tiling)

We know that any closed cycle of a graph could be decomposed into sum of simple cycles. To translate this theorem into tiling of 1D (Wang tile). We know that any 1D periodic tiling could be ...

**11**

votes

**1**answer

352 views

### Is it possible for a theorem to be constructive only in a non-constructive metatheory?

There are several theorems in category-theoretic logic which say something like, "any proposition in X logic that is provable in topos logic assuming (the law of excluded middle and) the axiom of ...

**6**

votes

**2**answers

494 views

### Are all functions in Bishop's constructive mathematics continuous?

When I started to write this question I was really confused, but now I think I am starting to get it. Nonetheless I'd like some confirmation that my understanding is correct.
I have read, heard ...

**1**

vote

**1**answer

171 views

### relationship between corner tile and edge tile of wang tile

It is clear that any corner type of Wang Tile could be converted to edge type of Wang Tile by defining the edge color according to the corner color.
However, could we convert edge type of Wang Tile ...

**1**

vote

**0**answers

108 views

### Graph theoretical representation of Wang Tile

We note that for one dimensional tiling problem of Wang Tile could be represented by a graph. Each cycle on the graph represents a periodic solution.
However, is there a well established counter-part ...

**10**

votes

**1**answer

332 views

### Does small forcing preserve CH?

Suppose CH holds and $\mathbb{P}$ is a poset of size $\omega_1$, such that forcing with $\mathbb{P}$ preserves $\omega_1$. Does forcing with $\mathbb{P}$ preserve CH? If $\mathbb{P}$ is proper then ...

**12**

votes

**0**answers

174 views

### Can Gentzen-style proofs give omega-consistency and beyond?

In 1936, Gentzen famously showed that Primitive Recursive Arithmetic, plus the assumption that the ordinal $\epsilon_0$ is well-founded, is able to prove Con(PA). But of course, Con(PA) doesn't yet ...

**14**

votes

**6**answers

2k views

### What “forces” us to accept large cardinal axioms?

Large cardinal axioms are not provable using usual mathematical tools (developed in $\text{ZFC}$).
Their non-existence is consistent with axioms of usual mathematics.
It is provable that some of ...

**3**

votes

**2**answers

218 views

### Sufficient Condition for Defining $\in$

Consider the first order language $\mathcal{L}=\{\in,\in'\}$ with two binary relational symbols $\in , \in'$ and $ZFC$ as a $\{\in\}$-theory. If we define $\in'$ using $\{\in\}$-formula $\varphi(x,y)$ ...

**1**

vote

**1**answer

223 views

### A Special Pair of Formulas

Consider the first order language $\mathcal{L}=\{\in,\subseteq\}$ and $\{\in\}$-theory $\text{ZFC}$. Is there a formula $\psi (x,y) \in \{\subseteq\}-Form$ with the following ...

**27**

votes

**1**answer

895 views

### Is there a computable ordinal encoding the proof strength of ZF? Is it knowable?

In comments on Quora (see, for example, here, here, here), Ron Maimon has repeatedly expressed the strong opinion that Hilbert's program was not killed by Gödel's results in the way typically ...

**10**

votes

**2**answers

604 views

### Questions about Prikry forcing and Cohen forcing

I have some questions.
The first one is about the product of Prikry's forcing.
Let $\kappa$ be a measurable cardinal, $U_1, U_2$ be normal measures on $\kappa$ and let $\mathbb{P}_{U_1}, ...

**6**

votes

**1**answer

218 views

### Sets of cardinalities of bases without choice

For a vector space $V$, let $BS(V)$ be the set of cardinalities (not necessarily $\aleph$s) of bases of $V$. Of course, in ZFC each $BS(V)$ is a singleton, but supposing the axiom of choice fails, it ...

**9**

votes

**1**answer

303 views

### Inner model in which every uncountable cardinal is large

The following is known:
$(*)$ If $0^\sharp$ exists, then any uncountable cardinal is is an inaccessible cardinal (and even more) in $L$.
My question is that:
Are there any large cardinal ...

**3**

votes

**1**answer

166 views

### Proof of existence of recursively inaccessible and Mahlo ordinals

As in title - I'm looking for a proof of the existence of a countable recursively inaccessible or recursively Mahlo ordinals, especially the first one. When looking for it in all the papers I stumbled ...

**4**

votes

**0**answers

189 views

### $\infty$-Borel Determinacy?

An $\infty$-Borel set is a set $X\subseteq\mathbb{R}$ which has an $\infty$-Borel code - a set $r$ coding the construction of $X$ via open sets, complementation, and well-ordered unions (see ...

**10**

votes

**0**answers

253 views

### cardinals below the critical point of a generic embedding

This may be an easy question. Are the cardinals below the critical point of a precipitous ideal embedding always absolute between the generic ultrapower and the generic extension?
To focus on the ...

**1**

vote

**2**answers

175 views

### Is there an intuitionistic generalized boolean algebra (of Stone)?

A "boolean algebra without the greatest element" was called by Stone "generalized boolean algebra" and he axiomatized it. Is there any publication about "preudo-boolean algebras without the greatest ...

**10**

votes

**1**answer

343 views

### Applications of SCH outside of set theory

Recall that the Singular Cardinals Hypothesis (SCH) says that if $\kappa$ is a singular cardinal and $2^{cf(\kappa)}<\kappa,$ then $\kappa^{cf(\kappa)}=\kappa^+.$ Clearly it has many applications ...

**12**

votes

**0**answers

566 views

### Does every Aronszajn tree has a Suslin or a Special subtree?

Question: Does every $\omega_1$-Aronszajn tree contains a Suslin sub-tree or a special Aronszajn sub-tree?
Recall that Suslin trees are $\omega_1$-trees (trees of height $\omega_1$, and countable ...