New answers tagged set-theory
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On the intersection of finitely many ultrafilters
There is a category theoretic characterization of the filters that can be written as intersections of finitely many ultrafilters. I claim that a filter $Y$ on $G$ is the intersection of finitely many ...
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On the intersection of finitely many ultrafilters
Let me try to provide a helpful elementary answer.
Suppose that $F$ is a filter on a set $X$ arising as the intersection of finitely many ultrafilters
$$F=\mu_1\cap\cdots\cap\mu_n.$$
We may assume ...
- 216k
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Is every sigma-algebra the Borel algebra of a topology?
If $\Sigma$ is countably generated -- and any sigma-algebra of practical interest will be [Edit: or will be the completion of such an algebra] -- then I believe the answer is yes. Here is the argument....
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Strength of Borel determinacy
I don't know why you would want to work with $\mathrm{ZC}^{-}$, this is not a theory I would recommend to do math in. But as long as you only care about second order number theory, there is really no ...
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Can set theory be interpreted in infinite arithmetic?
Without considering your system of arithmetic too closely, let me mention that ZFC is interpretable in Peano arithmetic, if one augments PA with the assertion that ZFC is consistent.
That is, ZFC is ...
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I have a problem about elementary submodels of ZFC
You have to distinguish between $A$ (which is in $M$ and might be uncountable) and $A\cap M$ (which is always countable but might not be in $M$).
Let's look at $A=\omega_1$ for concreteness. Since $\...
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The Parity Principle and $\mathbf{C}_2$ (choice for $2$-sets)
Over ZFA, the Parity Principle is strictly weaker. We'll show it follows from Multiple Choice, the assertion that for any family of nonempty sets $\mathcal{F},$ there is $g: \mathcal{F} \rightarrow [\...
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(Very) Large numbers, Chaitin's incompletness theorem and a specific upper bound
One way to provide an explict bound on $L$ is as follows, using the idea of the universal algorithm.
Namely, for any computably axiomatizable theory $T$ in question, consider the algorithm $p$ that ...
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What notable theorems cannot be automatically proven without choice using Shoenfield absoluteness?
$\text{ZF}+ \text{AC}_{\omega}$ is not $\Sigma^1_4$-conservative over ZF and ZF + DC is not $\Sigma^1_4$-conservative over $\text{ZF}+ \text{AC}_{\omega}.$
An example of the former: the sentence $\...
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What notable theorems cannot be automatically proven without choice using Shoenfield absoluteness?
This is sort of an anti-answer, which I've accordingly made CW, but here goes:
Whether $\mathsf{ZFC}$ is projectively conservative over $\mathsf{ZF}$ seems open; see Joel's answer from a while ago (...
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What is lost in General Relativity without Hahn-Banach axiom in the ZF+HB set theory?
As Ryan Budney mentioned in a comment, there is some ambiguity about what exactly you mean by "general relativity." General relativity is primarily a physical theory rather than a ...
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Is the set of permissible numbers of models of various cardinalities computable?
This is really an addendum to Alex's answer. I wrote a program in SageMath (using GAP) that computes these numbers, so I was able to expand Alex's lists considerably. Each of these lists should be ...
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Progress on determining which partial orders embed into the rationals
The following simple counterexample to Question 1 can be found on p. 473 of the Milner–Pouzet paper cited below.
Let $P=\omega_1\times\omega_1$ with the strict partial order
$$(x,y)\lt(x',y')\iff x\lt ...
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What is lost in General Relativity without Hahn-Banach axiom in the ZF+HB set theory?
This is perhaps more of an extended comment than a real answer, but I do think it goes a long way towards answering these kinds of questions.
The set-theoretic result referred to as Shoenfield ...
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Thick Canadian trees
$\newcommand{\Add}{\operatorname{Add}}$Start with a model $V$ satisfying $GCH$ (or just $2^{\omega}=\omega_1$ and $2^{\omega_1}=\omega_2$). Force over $V$ with the product $\Add(\omega,\omega_2)\times ...
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Is the set of permissible numbers of models of various cardinalities computable?
Let $T$ be a complete countable first-order theory. I will write $I(T,\kappa)$ for the number of models of $T$ of cardinality $\kappa$ up to isomorphism. The function $I(T,-)$ is called the spectrum ...
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Set theories without "junk" theorems?
Junk theorems like this are avoided in Type Theory (not just HoTT). This is because I cannot talk about the representation of objects like the natural numbers. This is exploited in Homotopy Type ...
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End-extension which Mostowski collapses a fake well ordering
In Chapter 3 of his PhD thesis, Harvey Friedman showed that there is a recursive linear ordering $\prec$ which has no hyperarithmetic infinite descending sequence and which does not support a jump ...
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Parity and the Axiom of Choice
The Parity Principle follows from the axiom $\mathbf C_2$ (defined below) which is weaker than the Axiom of Choice. I don't know whether the Parity Principle implies $\mathbf C_2$, but that's another ...
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