Trfur

Let's construct our model. Add to the language of ZFC countably infinitely many constants c_0, c_1, c_2, ... and let Γ be the set of sentences Γ = {c_{n+1} ∈ c_n | n ∈ ω}. We will use the compactness theorem to show that there is a model of ZFC ∪ Γ.

Let Δ be a finite subset of Γ and Let J be any model of ZFC. Since Δ is finite, there is a maximum k such that the sentence c_k ∈ c_{k-1} is in Γ. Add to J the definitions, for each n ≤ k, c_nJ = k - n, and for each n > k, set c_nJ = 0. Then for all 1 < n ≤ k, c_n = k-n ∈ k-n+1 = c_{n-1}, and so J is a model of ZFC ∪ Δ.

Thus, by the compactness theorem, there exists a model of ZFC ∪ Γ.

Found this on facebook but reposting to SAVE A LIFE.

Or at least some of y’all’s GPAs.

also, use Unpaywall