# lo.logic's questions - English 1answer

2.994 lo.logic questions.

### 10 Circular, or missing, definition in set theory?

7 answers, 1.919 views set-theory lo.logic
The extensionality axiom states that sets are the same if they contain the same elements, or conversely, sets are different if there is an element of one that is not an element of the other. This ...

### 10 History of the notation for substitution

One of the very common notations for syntactic substitution is $[\ /\ ]$. However, there seems to be an inconsistency in the literature about its usage. Many write $[t/x]$ for "substitute $t$ for $x$...

### 5 Determinacy of (infinite, possibly loopy) combinatorial games

I am looking for references and hopefully enlightening proofs of the following statement(s) concerning the determinacy of not-necessarily-well-founded (i.e., possibly infinite, possibly loopy) ...

### 8 Are all generalized Scott sets realized as generalized standard systems?

Below, I've focused on PA when lots of other theories would do. If replacing PA with a different theory leads to a more answerable question, feel free to do so. The standard system of a nonstandard ...

### 3 Singular compactness for stationary reflection?

Let $\lambda\geq \omega_2$ be a regular cardinal. The weak reflection principle for $[\lambda]^\omega$ ($WRP([\lambda]^\omega)$) asserts that for any stationary $S\subset [\lambda]^\omega$ there ...

### 113 Solutions to the Continuum Hypothesis

Related MO questions: What is the general opinion on the Generalized Continuum Hypothesis? ; Completion of ZFC ; Complete resolutions of GCH; How far wrong could the Continuum Hypothesis be?; When was ...

### -2 Are there paradoxes in ZF + (the Axiom of Choice for finite sets)? [on hold]

1 answers, 269 views set-theory lo.logic axiom-of-choice
The good intuitive reasons that the Axiom of Choice (AC) for arbitrary sets-- that a function f with f(i) in S(i) for each i in {i}, for any non-empty sets {i} and all S(i), exists-- doesn't follow ...

### 9 Infinite games: are they well defined?

It is just my curiosity about this question where we have an infinite game and (according to the answers) winning strategies for both players. I am familiar with terminating games only, and I am ...

### 7 On models of $Th_{\Pi_2}(PA)$

2 answers, 123 views lo.logic model-theory peano-arithmetic
Let $M$ be a nonstandard model of $PA$. Q1. Is there any way to get a submodel $N\subset M$ such that $N\models Th_{\Pi_2}(PA)$, but $N\not\models PA$? Q2. Especially, what combinatorial ...

### 11 Does foundation/regularity have any categorical/structural consequences, in ZF?

(Prompted by reflection on this old answer, and its suggestion of the “harmlessness” of the axiom of regularity.) In ZFC, one may justify the axiom of foundation (AF, aka the axiom of regularity) as ...

### 7 Axiomatizable $\exists \forall$ theory

I have been thinking the following problem proposed by my friends for a long time. Let $\mathcal{L}$ be the first-order language of theory of rings and let $K$ be the class of algebraic number ...

### 6 Join Density in R.E. Degrees: Are there r.e. B, C with all r.e. X below B computable or C join X computes B

1 answers, 79 views lo.logic computability-theory
Are there r.e. sets $B >_T 0$ and $C >_T 0$, $C \not\geq_T B$ such that for all r.e. $W \leq_T B$ either $W \leq_T 0$ or $C \oplus W \geq_T B$. The explanation for the title is because one can ...

### 12 Does every model of ZF-foundation have an extension, with no new well-founded sets, where every set is bijective with a well-founded set?

This question follows up on an issue arising in Peter LeFanu Lumsdaine's nice question: Does foundation/regularity have any categorical/structural consequences, in ZF? Let me mention first that my ...

### 7 continuity points of elementary embeddings from $0^\sharp$

1 answers, 253 views lo.logic model-theory fields

### 14 Is PA consistent? do we know it?

1) (By Goedel's) One can not prove, in PA, a formula that can be interpreted to express the consistency of PA. (Hopefully I said it right. Specialists correct me, please). 2) There are proofs (...

### 8 Is there something like internal language of an abelian category?

While studying topos theory I was wondering if there is something like internal logic of an abelian category. Aparently the answer is yes (by 7º slide in https://www.mimuw.edu.pl/~gael/xxi/files/...

### 11 $GCH$ and special Aronszajn trees

1 answers, 476 views lo.logic set-theory gch
Question. Does $\text{GCH}$ imply the existence of a non-special $\aleph_2$-Aronszajn tree ? Remark 1. By a result of Jensen, it is consistent that $\text{GCH}$ holds and all $\aleph_1$-Aronszajn ...