# Tag Info

Accepted

### Is the class of models of ZFC minus Replacement for which Replacement fails axiomatisable?

No, the compactness theorem plus a result specific to $\mathsf{ZFC}$ prevents this. Suppose $T$ were such a set of sentences. Then the union of $T$ and the replacement scheme is unsatisfiable, so by ...
• 247k
Accepted

### Theory $\mathcal{T}$ with predicate $P$ that is satisfied by countably many elements in every model of $\mathcal{T}$?

Assuming you meant countable as “countably infinite”, then no, with the same proof as the proof that the entire model cannot be restricted to being countable: Add to the language uncountably many ...
• 7,430

### Compactness Theorem for First Order Logic

In fact, there is a general tactic we can use here to "reduce" classes to sets. Rather than consider the class(es) of all models, look at the sets of models of minimal rank in the cumulative ...
• 247k

### Theory $\mathcal{T}$ with predicate $P$ that is satisfied by countably many elements in every model of $\mathcal{T}$?

Not unless your theory implies $P$ is finite. Otherwise, you can use the same argument as the upward Lowenheim-Skolem theorem, only stipulating that the new constants are in $P$, and get a model where ...
• 59.6k
Accepted

### Is the first-order theory of the class of well-ordered sets the same as the first-order theory of the class of ordinals?

Yes, since every ordinal is a well-ordered set, and every well-ordered set is isomorphic to an ordinal. So the classes are the same (up to isomorphism).

### An explicit axiomatization of the theory of the class of all $Z_n$

In very recent work, Derakhshan and Macintyre have shown that $\mathrm{Th}(K)$ is decidable. This answers a question of Ax from 1968. Since every decidable theory $T$ is recursively axiomatizable (...
Accepted

### Do rational inequalities preserve componentwise convex combinations?

When you say "I have two relations $R(a_1,\dots,a_n)$ and $R(b_1,\dots,b_n)$", presumably you mean that $R(x_1,\dots,x_n)$ is a formula in your restricted class (a conjunction of atomic ...
Accepted

### Inaccessible cardinal in standard transitive models of ZFC

No. If there is any transitive model containing inaccessibles, there will be (by using Lowenheim-Skolem to get a countable elementary submodel and then Mostowski collapsing it) a countable transitive ...
• 59.6k

### Non-isomorphic countable models of $\text{Th }(\mathbb{R},<, I)$

Clearly we have that $(\mathbb{Q},<,I)$ is a model of the theory since $(\mathbb{Q},<)\cong (\mathbb{R},<)$ and $\mathbb{Q}$ contains all the same integers that $\mathbb{R}$ contains. This ...
Accepted

### Axiomatization of the theory of finite structures in a signature consisting only of function symbols

Since every finite structure is interpretable in a finite $\{*\}$-structure (basically: a binary relation symbol, let alone a binary function symbol, is already "expressively complete"), the ...
• 247k
Accepted

### Every element realizing a nonempty type in a $\omega$-saturated model is definable

In this case, "definable" means "definable with parameters from $M$". So a finite tuple of elements $\bar{a}$ from $M$ is always definable by the formula $\bar{x} = \bar{a}$. In ...
• 13.1k
1 vote

### Why are types important in model theory?

A good place to get started is with compactness arguments. When doing a compactness argument to construct new models of a theory, one typically adds in a new constant symbol (or several) and specifies ...
• 2,969
1 vote

### Can we sheaf-theoretically force a violation of the continuum hypothesis in a (nice) topos which is *bicomplete*?

$\require{AMScd}$Well, I don't have a full answer since maybe there is something about "internal" completeness still to be said, but someone hinted the following to me - which more or less ...
• 41.8k
1 vote
Accepted

### A question on the proof of the cardinality property of Stone space

If $q$ is a type containing $\phi$ and $q\neq p$, then there is some $\psi\in q$ such that $\psi\notin p$, so $q\in [\phi\land \psi]$.
1 vote
Accepted

1 vote
Accepted

### Inaccessible cardinals and consistency of ZFC

For all $\phi$ in $ZFC$, we have : $ZFC+\exists \kappa'(\kappa' inaccessible)\vdash \forall\kappa(\kappa inaccessible\rightarrow \phi^{V_{\kappa}})$ This makes sense and is true, provided we ...
• 59.6k
1 vote
Accepted

### questions about proving ACF(Algebraically Closed Fields) has quantifier elimination

I assume we are working with the language of rings: $L = \{+,\times,-,0,1\}$. Let $M$ be a field, and let $M'$ be an $L$-substructure. Then $M'$ is a subring of $M$, so it is an integral domain. Let \$...
1 vote

### Introductory text on logic for those interested in the intersection of logic, algebra, and topology?

Vickers Topology via Logic treats topology thru the lens of logic. Its a fairly easy read with low prereqs, but maybe not the best for foundation building. For a general 1-year sequence (say model ...
• 698

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