> On Thursday, August 14, 2014 5:05:35 PM UTC-7, Ben Bacarisse wrote: >> >> It also leaves the curious puzzle of what it means to say that there are >> un-indexed rationals since it can't mean that they are not in the image >> of the bijection -- there are provably none of those. > > Does he actually use the term "bijection"?
Oh yes, there are, in WMaths, bijections. Many of his problems here are caused by the fact that he wrote a textbook full of conventional maths that he must stand by. The preface (yes, the preface) tells the reader that all the infinities are "potential", but after that, it real maths (mostly) all the way down. He gives examples of bijections, he describes the image of a function, and so on.
But even here:
| Beyond every natural number there are infinitely many natural | numbers. The sequence is never complete. A bijection with another set | like Q+ enumerates every q in Q+ such that n <--> q_n is unique. | | There exists no rationals without natural (index) and no natural | (index) without rational.
> He uses "enumeration" a > lot, but not in the sense of everyone else. He is talking about an > "act of enumerating" which (in his world, anyhow) must be done one by > one. In principle.
I think he's grasped the idea that a bjection is the real problem. If there is one, then you can diagonalise with it -- the finite definition of the bijection yields a finite definition of the thing that does not want to exist. Simply saying that the enumeration is never done won't work if there is a simple, finitely defined, bijection. Hence the fact that *his* bijections are very different. They are "not finished", though he will never be able to show an actual formal distinction.
> So it's not the dictionary meaning and it is up to > him to say what "in principle covers". Which he never does which > allows him to change the meaning of his terms to suit his > purposes. And of course, such an "act of enumerating" never ends which > (to him) proves that the second usage of "enumeration" id wrong or > contradictory. (After all, a word can only have ONE definition).
Words are his refuge. I don't think "his maths" is any different to anyone else's, except in the words used. Every formula provable in set theory would be provable in his maths if he every said enough to permit such an investigation. He will never show a formula that highlights the difference between completed and potential infinities because he has to keep all the maths in his textbook valid too.
He's going to answer me by citing a diagonalisation argument -- a result that he'll say differs between maths and WMaths, but he won't write it out a formula (too precise) and there will be no proof of any claimed distinction at all. The point, though, will be to start a new topic as a distraction. He knows he can't show a formula involving his simplest bijection (f(x) = x + 1 from Z to Z -- it's in his book) that shows how it differs between WMaths and maths, so a new topic is a good idea.
> Everyone else uses the metaphorical notion: There exists a bijection > between the natural numbers and the given set. There are nice > philosophical and foundational issues with that notion, but he doesn't > want to discuss those. He only wants to say "I'm right!" His meaning > of "enumeration" is the only correct one and the mere fact that > someone else may use it with a second sense ipso facto proves that > person wrong.
 The publisher happened to be running an offer, permitting free downloads of several of their books. His was one. Please don't think I bought it!