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say what it is, not what it's like |
As stated before, the x could be not an upper bound of A and therefore the statement is vacously true. We need x to be an upper bound and if y is an upper bound then x<= y. |
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Second-order logic is more expressive than first-order logic. For example, if the ___domain is the set of all [[real number]]s, one can assert in first-order logic the existence of an additive inverse of each real number by writing ∀''x'' ∃''y'' (''x'' + ''y'' = 0) but one needs second-order logic to assert the [[Supremum|least-upper-bound]] property for sets of real numbers, which states that every bounded, nonempty set of real numbers has a [[supremum]]. If the ___domain is the set of all real numbers, the following second-order sentence (split over two lines) expresses the least upper bound property:
: (∀ A) ([{{color|#800000|(∃ ''w'') (''w'' ∈ A)}} ∧ {{color|#008000|(∃ ''z'')(∀ ''u'')(''u'' ∈ A → ''u'' ≤ ''z'')}}]
::→ {{color|#0000BB|(∃ ''x'')(∀ ''y'')
This formula is a direct formalization of "every {{color|#800000|nonempty}}, {{color|#008000|bounded}} set A {{color|#0000BB|has a least upper bound}}." It can be shown that any [[ordered field]] that satisfies this property is isomorphic to the real number field. On the other hand, the set of first-order sentences valid in the reals has arbitrarily large models due to the compactness theorem. Thus the least-upper-bound property cannot be expressed by any set of sentences in first-order logic. (In fact, every [[real-closed field]] satisfies the same first-order sentences in the signature <math>\langle +,\cdot,\le\rangle</math> as the real numbers.)
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