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 May 21 suggested rejected edit on Jech, “Set theory” exercises 12.11 - Is my proof right? Nov 19 awarded Excavator Nov 19 revised Consistency of PA: why other proofs? corrected spelling Nov 19 suggested approved edit on Consistency of PA: why other proofs? Sep 21 comment Why PROP (Set of all propositions) is a set by ZF axioms? Great, then the set $L$ of preposition letters (atomic formulas) should always be given as a (countable or non-countable) set? mean that should we show first that L is also a set or we suppose we have one in hand in our prepositional language as alphabets. Sep 20 comment Why PROP (Set of all propositions) is a set by ZF axioms? How we know corresponding (via such encoding) subset of finite sequence of natural numbers which satisfy such conditions exists? don't you think that the problem is translated to that situation again? why such definition is not self-contradictory? I think we should show at least one set with properties (i) and (ii) together exists because the problem is about "the smallest set X ...". Can we say the set of all finite sequence of natural numbers is such a inductive set according to the encoding and then by Replacement Axiom we have at least one set satisfy the main definition? Sep 20 comment Why PROP (Set of all propositions) is a set by ZF axioms? @Nicolas. I mean that isn't it possible that properties (i) and (ii) together be somehow contradictory such that there isn't any set with these properties? for example in set theory for defining natural numbers (as intersection of all inductive set) we postulate the Axiom of Infinity (or existence of at least one inductive set) but how we without mentioning obviously any axiom know that at least one inductive subset of expressions (finite sequences of signs of our alphabets and connectives) with these properties exist? Sep 20 awarded Student Sep 20 awarded Editor Sep 20 revised Why PROP (Set of all propositions) is a set by ZF axioms? added 2 characters in body Sep 20 asked Why PROP (Set of all propositions) is a set by ZF axioms?