User blog comment:Superman37891/Explain Rayo please/@comment-5529393-20170622014520

Pellicudar12 has it right. Also, let me explain Rayo's language, which is not so bad.

"xᵢ ∈ xⱼ" is a statement meaning "xᵢ is a member of xⱼ". So for example, if xᵢ is 1, and xⱼ is the set {0,1,2}, then "xᵢ ∈ xⱼ" is true. However, if xᵢ is 3, and xⱼ is the set {0,1,2}, then "xᵢ ∈ xⱼ" is false.

"xᵢ = xⱼ" is a statement meaning "xᵢ is identical to xⱼ". So if xᵢ is {2,4,5}, and xⱼ = {3,5,6}, then "xᵢ = xⱼ" is false; but if xᵢ is {2,4,5} and xⱼ = {2,4,5} then "xᵢ = xⱼ" is true. (Note: = is not a strictly necessary part of the language, because it can be defined using the other symbols. I guess Rayo included it here for convenience or niceness.)

"(¬θ)" is a statement meaning the negation of θ, i.e. if θ is true then ¬θ is false, and if θ is false, then ¬θ is true. So for example (¬{2,4,5}={3,5,6}) is true, since {2,4,5}={3,5,6} is false.

"(θ∧ξ)" is a statement meaning "θ and ξ are both true". So for example "(3 ∈ {3,6} ∧ {1,2,3} = {1,2,4})" is false, even though the first part is true, because the second part is false; either part being false makes the whole thing false. But for example "(3 ∈ {3,6} ∧ {1,2,3} = {1,2,3})" is true.

"∃xᵢ(θ)" is a statement meaning "there exists an xᵢ such that θ is true". So "∃xᵢ(xᵢ∈{3,6})" is true, because there is a value of xᵢ that we could choose (namely either 3 or 6) that would make the statement true. On the other hand, "∃xᵢ(4∈{3,6})" is false, because there is no value of xᵢ that can make the statement 4∈{3,6} true.

And that's it! Every statement of first order set theory can be written using some combination of these five syntactic structures. Actually quite simple, but it leads to incredible growth rate of the Rayo function.