There is one key distinction to be made in this particular discussion regarding the concept of an infinite number of truths not provable within any formal system. Many post-modernists have taken Godel’s ideas to promote the primary tenet of post-modernism which is that all narratives [truths] are equally true. The corollary to their perspective is that various truths cannot be compared, especially if they arise from different cultures or worldviews.
From that disastrous series of errors, the post-modernist arrives at political correctness and other forms of idea restriction. The reason for this degeneration in communication is because truth is no longer the value that determines what can be argued since all truths are equal. The means the post-modernist then justifies is the emotional effect on the reader. Anything that potentially offends is forbidden under this schema.
As a logician, Godel would, I believe, have been horrified by this misuse of his proof. Godel would have easily known that there are actually infinitely more false statements than there are true statements in the universe of ideas. His proof does not proclaim false statements to be true, nor would he have ever endorsed the abandonment of logic itself. His proof only stated that there exist statements which are true, but not provable; he does not claim or endorse the concept that all statements are equally true.