merjet Posted December 5, 2012 Share Posted December 5, 2012 As Aristotle said, Nature abhors a vacuum.Better: Logicians and mathematicians like completeness.And they shall not have it. The famous Goedel incompleteness theorem shows that any consistent first order logic with the postulates of arithmetic added is incomplete. That is there are closed formulate (no free variables) such that neither they nor their negations are theorems of the system. In a word, provability cannot catch up with truth.Ba'al Chatzaf Irrelevant. My comment (and your earlier one) was about the empty set. Link to comment Share on other sites More sharing options...
BaalChatzaf Posted December 5, 2012 Share Posted December 5, 2012 As Aristotle said, Nature abhors a vacuum.Better: Logicians and mathematicians like completeness.And they shall not have it. The famous Goedel incompleteness theorem shows that any consistent first order logic with the postulates of arithmetic added is incomplete. That is there are closed formulate (no free variables) such that neither they nor their negations are theorems of the system. In a word, provability cannot catch up with truth.Ba'al Chatzaf Irrelevant. My comment (and your earlier one) was about the empty set.The term "complete" has a very specific technical meaning. What you were expressing is a need for closure and consistency. Without an empty set, we cannot get a proper partial ordering of classes under the inclusion relation and operations such as intersection and union would not be always defined. Number systems already suffer from this defect. Division is not completely defined as is addition, multiplication and subtraction. Division by 0 is simply not defined in the algebra of numbers.Ba'al Chatzaf Link to comment Share on other sites More sharing options...
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