The consequences of Gödel's theorems - Ep. 7.1: Formalism, Logicism and Intuitionism

[@pygmalionsrobot1896]

This was brilliant, thank you so much for making this available.

[@MathCuriousity]

Could you please explain your comment? Also would you help me please understand the difference between constructivism and intuitionism? Thanks so much!

[@MathCuriousity]

What is the relationship between intuitionism and constructivism? Thanks!

[@andersonm.5157]

Intuitionism is a branch of constructivism.

[@MathCuriousity]

@@andersonm.5157may I ask you another question: I just began studying set theory and first order logic. 1) I noticed first order logic is used to define the axioms for set theory and sets are used to describe the semantics of first order logic. Isn’t this circular? I now feel very anxious that mathematics is not safe. 2) When we talk about first order logic using sets, is this part of the metalanguage or meta theory? Thanks!

[@andersonm.5157]

@@MathCuriousity As i'm not a mathematician, idk how deep my knowledge is, but: 1. I think that we need the concept of sets before first order logic, but i don't think it's a circular reasoning because the axioms of the ZF model do not define what a set is. Instead, they restrict what types of sets are allowed (like the axiom of foundation), what relationships (like extensionality) and operations to construct sets (schema of specification, pairing, ...) are defined, as well as defining the existence of certain types of sets (axiom of infinity, power set). In short, they define the structure of the theory. 2. Yes.

[@andersonm.5157]

@@MathCuriousity And in fact, a primitive concept of sets would be of the classes. Sets are classes with some restrictions.

[@MathCuriousity]

@@andersonm.5157 ok I see!

[@hadeeskhokhar8805]

This is some sick stuff

[@amrelnashar517]

Indeed!

[@markuspfeifer8473]

Gödel’s results *are* intuitionist