You need to revise your statements. If you are going to hinge your discussion on semantics, you need to be very exact with your wording. Add the word “only” to your square syllogism. A square DOES have three sides. It also has one side and it has two sides. It ALSO has a fourth side.

I’ll be movin’ on now. ]]>

However, one can prove that the representations of “1/2” and “2/4” are equivalent. The construction of the rational numbers has an equivalence relation embedded within it, specifically p/q = r/s if and only if rq=ps in the integers. For a formal treatment, please see “localization” of a ring in Lang or Dummit/Foote.

Interestingly, the integers also are constructed with an equivalence relation as well when using Grothendieck’s group construction, which is in many K-theory books. Please see Weibel’s “The K-Book” for a formal treatment. This construction is one way of constructing the integers from the whole numbers (i.e. natural numbers with 0), but due to the properties of the whole numbers, there is canonical form that all integers take. Namely, every integer can be written as a-0 or 0-a, depending on sign. Sadly, the same cannot be said for the rational numbers, and the best that can be done is to find “lowest terms”, i.e. when the numerator and denominator are coprime.

The same can be said of “1” and “0.99…”, though the equivalence relation is messier to describe as it involves limits, Dedekind cuts, Cauchy sequences, or other equivalent analytic criterion for cofinality. Please see your favorite analysis book, such as Davidson/Donsig, or maybe an order theory book like “Ordered Sets” by Bernd Schroeder. However, as with “1/2” and “2/4”, one can prove that “1” and “0.99…” are equal from their construction.

Side note: The fact that 0.999…=1 is true demonstrates that limits do not preserve strict inequalities. Specifically, the partial sums (i.e. 0.9, 0.99, 0.999) are all strictly less than 1, but the limit of this sequence is dead-on equal to 1. Perhaps this is another stumbling block for readers/viewers?

]]>What surprises me about the people “supporting” the defendant is that they — to a man — all acknowledge and admit that he lied to them on multiple occasions. And not about trivial matters. He is like the Donald, he lies out of narcissism and contempt and expects no repercussions. They state that unequivocally; but then immediately turn around and believe every story, excuse, and allegation he makes on an ongoing basis, ignoring the fact that they know he has no credibility. None.

The most surprising to me was when Hogtie came to your stream yesterday displaying an agnosticism about the case that simply did not seem warranted. He should know better.

If you ever want a serious answer to a real civil litigation matter, feel free to email me, I’d be happy to help. I myself expect that the default will stand, but there will be a hearing on damages, BTW. ]]>