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SK, First off, thank you for kicking off an interesting discussion. It has been most enjoyable. Now, I don't understand what you say on page 6 paragraph 3. In particular, you say I do not know what "subly" is. Was this supposed to be "subtly"? Let me subtly assume the existence of a concept that is tailor-made, etc. There is a concept in mathematics called a field. Examples of fields include the real numbers, complex numbers, finite fields and so on. Would it be fair to say that the concept "field" subsumes these subjects? Now, there is something called a complete field. Examples of complete fields include real numbers and complex numbers, etc. These subjects are subsumed by the concept "complete field". Finally, there is something called a complete ordered field. It seems to me that this is a concept. I have combined well-defined concepts to form a new concept. A priori, one does not know whether there exist any such subjects until once shows that the real numbers do in fact constitute a complete ordered field. Therefore we know that this is not an empty notion. Also, a priori one does not know how many such subjects exist. However, there is a proof that, up to isomorphism, there is only one such subject, the real numbers. Was "complete ordered field" never a concept? If it was a concept until it was shown that there was only one, at what point did it cease to be a concept? Was it no longer a concept when someone first proved that there was only one such subject? What is a concept in your own mind until someone informed you of the proof that there was only one, making "authority" the determining factor concerning concepts. Or, was it once you read and understood the proof that it ceased to be a concept? I believe that "complete ordered field" is a concept despite the fact that there is only one. Was this a cheap shot?