## New answers tagged model-theory

3
votes

### Congruences that aren't "finite from above"

I do not have an instructive example of a non-parafinite congruence, so this is not an answer per se but rather an extended comment. Let me suggest two things. Firstly, it is useful to think of this ...

3
votes

Accepted

### Congruences that aren't "finite from above," take 2: semigroups

This one is also trivial. Sorry, just take a semigroup with the identity xy=x. Then all equivalence relations are congruences so this is the same as pure sets.
Here is a more interesting construction ...

5
votes

Accepted

### Congruences that aren't "finite from above"

In a ring, congruences are given by ideals: The elements congruent to $0$ form an ideal and two elements are congruent if and only if their difference lies in the ideal.
So one can just take an ...

3
votes

### Congruences that aren't "finite from above"

On a ring or group all congruences are parafinite. Partition into the kernel and the complement of the kernel.

8
votes

Accepted

### Is the Rado graph the unique countable graph that has all finite graphs as induced subgraphs?

Since the question is very well answered in comments (by multiple people), here’s a CW answer putting them all together.
The Rado graph is not uniquely characterised, among countable graphs, by the ...

Community wiki

2
votes

### Standard models of N and R: An Alice/Bob approach

This is way too long for a remark, therefore I post it as an answer.
Although I agree with the previous answers that there is probably no clear “intended” model of set theory, there are perhaps two ...

4
votes

Accepted

### Posets of equational theories of "bad quotients"

Let $\mathbb{R}=(\{\textrm{real numbers}\};0,1,+,\times)$. Is there an equivalence relation $E$ on $\mathbb{R}$ such that $\mathbb{P}_\mathbb{R}(E)$ is not upwards-directed?
I will assume that the ...

8
votes

Accepted

### Concept of bedrock and mantle in the multiverse view in the philosophy of mathematics

The confusion seems to arise from your statement "the mantle of $V$ which is the smallest ground for $V$." But this not quite right.
In a bottomless model of ZFC, the mantle is not a ground. ...

8
votes

Accepted

### Is Morse-Kelley set theory with Class Choice bi-interpretable with itself after removing Extensionality for classes?

No, because the latter theories has parametrically definable automorphisms that swap two equivalent classes, but the former theory is definably rigid (no need for class choice). If the theories were ...

15
votes

### A Löwenheim–Skolem–Tarski-like property

Here is a lower bound that improves Joel's by a bit. If the reflection property of the post holds at $\kappa$ then there is some measurable $\lambda<\kappa$ with
$$V_\lambda\models``\text{there is ...

3
votes

Accepted

### Sizes of linearly ordered subalgebras of powers

Yes, and you can find them included in your infinite linear subpower. The variety $V$ generated by a finite algebra $\mathcal A$ (which includes all subpowers) is locally finite, i.e., finitely ...

17
votes

### A Löwenheim–Skolem–Tarski-like property

Here is an upper bound:
Suppose $\kappa$ is $2$-fold supercompact. Then the property holds at $\kappa$. (Recall that $2$-fold supcompactness means that for each ordinal $\lambda$, there is $j:V\to M$ ...

17
votes

Accepted

### A Löwenheim–Skolem–Tarski-like property

Let me improve somewhat on Farmer's lower bound.
Theorem. If there is a cardinal $\kappa$ with the stated reflection property, then there are many measurable cardinals, measurable cardinals of very ...

18
votes

### A Löwenheim–Skolem–Tarski-like property

Here's a counterexample for $\kappa=\aleph_1$: let $B$ be the structure with underlying set $\mathbb{N}\sqcup\mathcal{P}(\mathbb{N})$, equipped with the usual ordering on $\mathbb{N}$ as well as the $\...

7
votes

Accepted

### Two notions of generalized quotient/substructure

Let me copy Definition 4.1 of
Libor Barto, Jakub Oprsal, Michael Pinsker
The wonderland of reflections
Israel Journal of Mathematics 223 (2018), 363-398
Defn. 4.1
Let $\mathbf{A}$ be an algebra with ...

1
vote

### Standard models of N and R: An Alice/Bob approach

It's hard to pinpoint what question is being asked in this post, but here is my attempt at an answer.
Second-order number theory with full semantics pinpoints the intended model of the structure of ...

4
votes

### Standard models of N and R: An Alice/Bob approach

I very much like Burak's and Timothy Chow's answers, and I hope that this answer complements theirs. Your imaginary conversation between Alice and Bob got me thinking a little differently about your ...

9
votes

### Standard models of N and R: An Alice/Bob approach

Like Burak, I am responding to the OP's request to promote my comments to an answer, with the caveat that I want to avoid wading too deeply into philosophical debates that I think are beyond the scope ...

11
votes

### Standard models of N and R: An Alice/Bob approach

At the request of Mikhail, I am turning a comment of mine into a partial answer, even though what I am going to write down are well-known, presumably by Mikhail as well. While I am not going to ...

6
votes

### Definitions of definable compactness

Both definitions are equivalent in o-minimal structures for any definable topology $\tau$ as long as either $\tau$ is Hausdorff or the underlying o-minimal structure has definable choice.
In general ...

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