# Tag Info

### 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 ...
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 ...
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 ...

### 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.
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### 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 ...

### 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 ...
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### 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 ...
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### 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. ...
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### 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 ...

### 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 ...
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 ...

### 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 $j:V\to M$ ...
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 ...