| 1. | Now we prove the least upper bound property.
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| 2. | So the least upper bound is infinite.
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| 3. | Theory curbing selects such least upper bounds models in addition to the ones selected by circumscription.
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| 4. | The real numbers also have an important but highly technical property called the least upper bound property.
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| 5. | This least upper bound is defined to be the irrationality measure of " x ".
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| 6. | These are posets in which every upward-directed set is required to have a least upper bound.
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| 7. | Hence, 0 is the least upper bound of the negative reals, so the supremum is 0.
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| 8. | Another example is the hyperreals; there is no least upper bound of the set of positive infinitesimals.
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| 9. | As in the case of limits of sequences, least upper bounds of directed sets do not always exist.
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| 10. | Then it has a least fixpoint there, giving us the least upper bound of " W ".
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