Axioms and Theorems
... uncovered. And since these can’t be together, they cannot be covered by one domino. Therefore it is impossible. ...
... uncovered. And since these can’t be together, they cannot be covered by one domino. Therefore it is impossible. ...
Notes on Propositional Logic
... In propositional logic, we would like to apply operators not only to atomic propositions, but also to the result of applying other operators. This means that our language of well-formed formulas in propositional logic should be inductively defined as follows. Definition 1. For a given set A of propo ...
... In propositional logic, we would like to apply operators not only to atomic propositions, but also to the result of applying other operators. This means that our language of well-formed formulas in propositional logic should be inductively defined as follows. Definition 1. For a given set A of propo ...
A(x)
... whole universe (GU = U), then the formula is true in I, because the subformula xG(x) is true; hence G(x) x G(x), and x (G(x) xG(x)) is true in I. If GU is a proper subset of U (GU U), then it suffices to find at least one individual a (assigned by valuation v to x) such that a is not an el ...
... whole universe (GU = U), then the formula is true in I, because the subformula xG(x) is true; hence G(x) x G(x), and x (G(x) xG(x)) is true in I. If GU is a proper subset of U (GU U), then it suffices to find at least one individual a (assigned by valuation v to x) such that a is not an el ...
Second order logic or set theory?
... 2, π, e, log 5, ζ(5) • Not every real is definable. • A well-‐order of the reals need not be definable. ...
... 2, π, e, log 5, ζ(5) • Not every real is definable. • A well-‐order of the reals need not be definable. ...
mathematical logic: constructive and non
... outrun intuition and even consistency, the mathematical public was forced to recognize by the paradoxes in which Cantor's set theory culminated in 1895. Hilbert hoped to save ' classical mathematics ' (including the usual arithmetic and analysis and a suitably restricted axiomatized set theory), whi ...
... outrun intuition and even consistency, the mathematical public was forced to recognize by the paradoxes in which Cantor's set theory culminated in 1895. Hilbert hoped to save ' classical mathematics ' (including the usual arithmetic and analysis and a suitably restricted axiomatized set theory), whi ...
Proof Theory - Andrew.cmu.edu
... parts of ordinary mathematics, but weak enough, on the other hand, to be amenable to proof-theoretic analysis. He then suggested “calibrating” various mathematical theorems in terms of their axiomatic strength. Whereas in ordinary (meta)mathematics, one proves theorems from axioms, Friedman noticed ...
... parts of ordinary mathematics, but weak enough, on the other hand, to be amenable to proof-theoretic analysis. He then suggested “calibrating” various mathematical theorems in terms of their axiomatic strength. Whereas in ordinary (meta)mathematics, one proves theorems from axioms, Friedman noticed ...
IN THE WAKE OF CARDANO`S FORMULAS 1. Completing the cube
... all quadratics, cubics, and quartics have their solutions there, it was natural to expect that all solutions to all polynomials can be found in the complex plane. This proved to be true. The great Euler sketched out some ideas for what is now known as the Fundamental Theorem of Algebra, which can be ...
... all quadratics, cubics, and quartics have their solutions there, it was natural to expect that all solutions to all polynomials can be found in the complex plane. This proved to be true. The great Euler sketched out some ideas for what is now known as the Fundamental Theorem of Algebra, which can be ...
