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Jul
6
comment How is the law of excluded middle necessary for proofs by contradiction?
@Taladris : If {T ∪ ¬p} proves (¬q∧q), then it is a boolean axiom that (¬q∧q)->p (i.e. anything can be proven from a contradiction), and thus {T ∪ ¬p} also proves p.
Jul
6
comment How is the law of excluded middle necessary for proofs by contradiction?
and so we can't define an analogue of a 'boolean tautology', as in classical logic.
Jul
6
comment How is the law of excluded middle necessary for proofs by contradiction?
MJD : I meant that the law of the excluded middle would, as a general principle, tell you that something can be either true or not true, and hence guide you to come up with the two valued classical logic. Once you've decided that, you can justifying saying that for example (¬p→p)→p is a tautology, because it is true under any truth assignment from your two valued set {T,F} to p. In intuitionist logic, where one rejects the law of excluded middle, a statement no longer may take from just two values, but rather from three {Provable, Leads to absurdity, or None of the above}
Jul
6
comment How is the law of excluded middle necessary for proofs by contradiction?
MJD : How? In intuitionistic logic q^¬q means "there is a proof of q, and there is a proof that q leads to absurdity". How does that prove (an arbitrary) p?
Jul
6
comment How is the law of excluded middle necessary for proofs by contradiction?
@MJD : Ah I see. Can you then think of the LEM as a way of justifying/deriving (in the metalanaguage, I suppose), the boolean tautologies?
Jul
6
comment How is the law of excluded middle necessary for proofs by contradiction?
@tetori : I was thinking of Hilbert style deduction system for first order logic. The logical axioms are the boolean tautologies.
Jul
6
comment How is the law of excluded middle necessary for proofs by contradiction?
If {T ∪ ¬p}⊢(¬q∧q) then {T ∪ ¬p} ⊢p and you can just proceed as in my post.
Jul
6
asked How is the law of excluded middle necessary for proofs by contradiction?
Jun
18
accepted Measurable set indicator functions - need clarification on a book's statement
Jun
18
asked Measurable set indicator functions - need clarification on a book's statement
Dec
29
awarded  Scholar
Dec
29
accepted Plausibility argument for Zorn's Lemma
Dec
29
comment Plausibility argument for Zorn's Lemma
Sorry, I guess I still don't follow - can you please clarify what 1) the proper class and 2) the set are here, and what the injective function from 1) to 2) is? I'm trying to see what these could possibly be based on the recursive process described..
Dec
29
comment Plausibility argument for Zorn's Lemma
But even as an intuitive explanation, I don't see how it is intuitive. Like you said, if there is no maximal element then this process cannot terminate. Ok, but so what? Does this non-termination (even in a set theoretical sense) lead to some sort of a contradiction? Maybe it just simply doesn't terminate, and that's that - without there being a maximal element after all?
Dec
28
awarded  Student
Dec
28
asked Plausibility argument for Zorn's Lemma
Dec
10
asked Understanding a proof: A set of $m$ orthonormal vectors in $V$, with $m < \operatorname{dim}V$, is not complete.