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Results tagged with definability
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user 113405
definability by formulas in first-order logic, e.g. as explained at https://en.wikipedia.org/wiki/Definable_set, or as in J. Robinson's first-order definition of the integers in the field of rationals
2
votes
Accepted
Which fragment of ZF does the class of all hereditarily predicatively definable sets capture?
HPD satisfy extensionality and regularity trivially.
It satisfy Pairing, as if $x,y\in HPD$ we can explicitly write down the definition of $(x,y)$ using only $x,y$ as parameters (and they have strictl …
- 1,676
4
votes
Accepted
Does cardinal definable choice imply AC?
Over ZF yes, it does.
Let $X$ be any family of nonempty sets, and let $κ$ be cardinal such that $ρ(κ)>ρ(X)$, then $V_{ρ(κ)}\setminus\{∅\}$ is cardinal definable, hence has a choice function that induc …
- 1,676
2
votes
How do these two principles of Foundation written in $\mathcal L_{\omega_1,\omega}$ compare?
Your axiom schema is equivalent to being an $\omega$-model.
Working inside an $\omega$-model, any counter example to your schema will be counter example of the axiom of foundation of ZFC, as you can u …
- 1,676
6
votes
Accepted
Is ZF + Def a conservative extension of ZFC+HOD? If not, what are counter-examples?
Any model of ${\sf ZFC}+V=\sf HOD$ has an elementary equivalent pointwise definable model.
If $M$ models $V=\sf HOD$, it has a parameter free definable well ordering, for each formula $φ$ consider the …
- 1,676