What does $\Vdash$ mean?

Question

Meaning of $\Vdash$ In set theory? Because I’ve seen this in a lot of set theory formulas, however I haven’t been able to find out what it means so you’d someone please explain.

Answer 1

It depends on which book you're reading. The symbol is used for different (but related) things in different contexts. I'm sure there are still more uses, but here are probably the three most common options.

Probably the most common usage is in forcing, where we read $p \Vdash \varphi$ as "(the condition) $p$ forces $\varphi$ (to be true)". Typically we use this to build new models of set theory where we "force" certain things to be true. There is also a related notion called "model theoretic forcing" where we force properties of algebraic objects like groups instead. See, for instance, Kunen's (new) Set Theory to learn more about set theoretic forcing, or Hodge's Building Models by Games for model theoretic forcing.

Since you have the modal logic tag, another common usage is in kripke models. Here we have multiple worlds, and the truth of modal formulas depends on precisely which world we find ourselves in. Often we'll write $w \Vdash \varphi$ to mean "in world $w$, the formula $\varphi$ is true". To learn more about this, you might be interested in Boxes and Diamonds, an open source introduction to modal logic (freely available here).

Lastly, since I'm a categorical logician, I feel obligated to mention another use case in topos theory. This setting generalizes the two other examples (though there are some quibbles to be had about that) and justifies using the same notation for both. In this setting we say $C \Vdash \varphi$ to mean "(the object) $C$ forces the truth of $\varphi$" or "the formula $\varphi$ is true at $C$". For instance, in a topos of sheaves on a topological space, it's possible a formula isn't true everywhere, but it is true on some open set $U$. In that case we would say $U \Vdash \varphi$. For more about this, see Mac Lane and Moerdijk's Sheaves in Geometry and Logic.

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I hope this helps ^_^

Answer 2

$p\Vdash\phi(\dot{x})$ means the condition $p$ from some partial order $P$ forces $\phi(x)$ is true in every set theoretic universe $M[G]$ where $G$ is $P$-generic over $M$ with $p\in P$.

This definition of \lq\lq force \rq\rq is used to develop the so-called forcing theory which is aiming at extending the set theoretic universe we live in so that some sentence we are not sure about its truth will be true (in the extension), e.g., " there are $\aleph_{3}$ many real numbers in total". Note that we are not sure whether there exist exactly $\aleph_{3}$ many real numbers if do nothing to our universe (instead, if we shrink our universe appropriately, we may have that there are $\aleph_{1}$ many real numbers totally).

If you wants the precise meaning of above, you should consult a textbook on set theory. Chapter VII in Kunen's Set Theory: an introduction to independent proofs, 1983 is good choice.