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192 views

Cohen's model yet again

It has been discussed already whether a countable OD set necessarily contains an OD element. See e.g. A question about ordinal definable real numbers . A negative answer was obtained in Archive for ...
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What are the minimal requirements for the definable hyperreal field plus transfer?

It is interesting that to prove the transfer principle for the definable hyperreal field, one requires no more choice than for proving, for instance, the countable additivity of the Lebesgue measure. ...
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Is factorial definable using a $\Delta_0$ formula?

The factorial function is primitive recursive, and therefore definable by a $\Sigma_1$ formula. Is it also definable by a $\Delta_0$ formula (i.e. bounded quantifiers)? If not, why?
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Are there known ways to posit definable global choice in ZF without positing V=L?

I need a global choice function defined by a formula in (a fragment of) ZF. There is no harm in assuming V=L for my purposes. But I wonder if there are any familiar alternative ways to get this? ...
452 views

Can $V\neq\text{HOD}$ if every $\Sigma_2$-definable set has an ordinal-definable element?

This question arises from an issue arising in user38200's recent question concerning models of set theory in which every definable set has a definable element. In my answer to that question, with ...
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Definability of arithmetic functions and relations

Motivation: Many "weak" arithmetic functions and/or relations ("relations" for short) are equivalent with relations explicitly definable by relations which were recursively defined by them beforehand (...
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Ordinal definable sets of reals in the Solovay

To be precise, let $\Omega$ be an inaccessible cardinal in $L$ and let N be the Solovay model defined by the Levy-collapse in this case. Then $\Omega$ is $\aleph_1$ in $N$. How many different OD (=...
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Outer Definability of a Class

Definition: Let $C$ be a class of sets and $\mathcal{L}$ a first order relational language. We say $C$ is "outer definable" by $\mathcal{L}$ if there is a first order theory $T$ and for each $n_{R}$ - ...
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$\newcommand{\N}{\mathbb{N}}$ My question, more precisely, is: Question. Is there a set $B\subset \N\times\N$, such that the set of indices where it is arithmetically definable, that is, $\{ n\in\N \... 1answer 216 views Existence property for ordered fields A theory$T$has the existence property (EP) if the following holds: Let$\phi(x)$be a formula with one free variable (and no parameters) such that$T \vdash (\exists x) \phi(x)$. Then there is ... 1answer 251 views Iterating definability An odd -- probably basic -- question about model theory: For$\mathcal{M}$a structure in a (first-order) signature$\Sigma$, let$\mathcal{M}'$be the structure in signature$\Sigma\sqcup\lbrace U\...
I am not asking which order types PA proves are well ordered. That would be all up to $\epsilon_0$. Rather I mean, assuming a stronger ambient theory such as Zermelo set theory, which ordinals have ...
I'm having difficulties to prove that the subset of even numbers is not first-order definable in $(\mathbb{N},<)$. Any hint is welcomed! More generally, what are usual techniques in order to prove ...