For any uncountable set of formulas in propositional logic, is there is an equivalent and independent set of prositions?

Asked Sep 07 '17 at 10:25

Active Sep 07 '17 at 10:59

Viewed 304 times

Suppose I relax the restriction of propositional logic, allowing the primitive propositions to be uncountable. Does it follow that every set of propositions has an independent set equivalent to it? The countable case is covered here:

For any set of formulas in propositional logic, there is an equivalent and independent set

What happens in the general case?

asked Sep 07 '17 at 10:25

Mathmo

1,035

What exactly is your language? Do you allow for uncountable disjunctions and conjunctions? – Stefan Mesken Sep 07 '17 at 10:46
Nope, only syntactic implication with standard three axioms and modus ponens. – Mathmo Sep 07 '17 at 10:51
Do you mean that you are allowing for an uncountable number of propositional variables? (And consequently, it is possible to have an uncountable number of propositional expressions?) – DanielV Sep 07 '17 at 12:40
Yep, I think that link from Andres solves it. – Mathmo Sep 07 '17 at 13:10

For any uncountable set of formulas in propositional logic, is there is an equivalent and independent set of prositions?

0 Answers0