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Nov
5
comment What's an example of a theory that's consistent yet has no model?
Thank you for this answer! Btw, and a bit off topic, do you know where I can find a self contained and relatively short (like a couple of pages maximum), but detailed, proof of the syntactic version of the incompleteness theorem? Preferably available online..
Nov
5
awarded  Commentator
Nov
5
accepted What's an example of a theory that's consistent yet has no model?
Nov
5
awarded  Curious
Nov
4
awarded  Nice Question
Nov
4
asked What's an example of a theory that's consistent yet has no model?
Oct
31
awarded  Supporter
Oct
31
accepted How can universal quantifier manipulation rules be made redundant by the generalization rule (metatheorem)?
Oct
30
asked How can universal quantifier manipulation rules be made redundant by the generalization rule (metatheorem)?
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