/sci/ - Science & Math

>>11624036
I've only skimmed the video, but it seems he is making a valid point, while your example is unrelated to what I saw in the video.

Both
>{x ∈ R: x ≤ 5}
>{x : x ∈ R ∧ x ≤ 5}
denote the same set and will be understood.
Yes, the last one can be said to be more casual.

As far as FOL set theory is concerned, either notations are mere shorthand.
I think in Hilbert style calculus there's really only closed propositions and in natural deduction terms may pop up in existential instatiation, but otherwise there's no standalone terms/objects like "{n | n ∈ N ∧ n > 9000 }"

Read brackets
[math] X=\{a,b\}\equiv \forall c.\ ( c \in X \Leftrightarrow (c = a \lor c = b)) [/math]
as shorthand, as well as
[math] X=\{x|P(x)\} \equiv \forall x.\ (x\in X\Leftrightarrow P(x)) [/math]
and
[math] \{ x | P(x) \} = \{ x | Q(x) \} \equiv \forall x.\ \left(P(x) \Leftrightarrow Q(x) \right). [/math]
or
[math] \{f(x)|P(x)\} \equiv \{y|\ \exists x.\ (y=f(x)\land P(x))\} [/math]

Pic related is some notes on the topic I took a while ago (must clean up the formatting)