On Sunday, 23 February 2014 21:49:59 UTC+2, Julio Di Egidio wrote:
> Well, of course: it's the whole idea behind an if-then, i.e. the notion of > logical consequence, that *if* the premise were true, it would then > *necessarily* (i.e. by logical necessity) follow so and so. And I did say > "allegedly" not per chance, indeed to reinforce the fact that I am talking > about something that I believe just falls apart.
That's exactly my point. There is no question that the premise is false. The "then" part is a non-sequitur. In other words it is a futile exercise.
> But there is no mention of "infinity" in that proposition.
I know and that's why you can't use it to draw any conclusions about the "infinite" set.
> Indeed, as long as you concede to the validity of an inductive definition for > the collection of natural numbers (i.e. 0 in N, and n in N => n+1 in N), the > above sentence as well as the subsequent one are perfectly well-formed.
Really? :-) I am aware of the principle of mathematical induction but it's not a proposition about "actual" infinity. There is no such thing as "actual" infinity.
> But you have to show why/how infinity is an ill-formed and non-existent > concept (which is what I am trying to do), there is no point in just stating > it without justification.
It's very easy to show - all you have to do is prove that infinity can't be reified. That's all there is to show.
> And while I am contending that there is no such thing as the set of all natural numbers (i.e. that there is no such thing as potential infinity in mathematics as an endless and yet complete totality, a pure contradiction in terms), I do not agree that there is no infinity at all: there is in logic and there is in mathematics, just try and answer the question where exactly Achilles catches the tortoise.
The Xeno paradox is not a paradox at all. It is an exercise in wrong thinking which I have disproved at the STATU (Space time and the universe) forum. I am not about to do it here again.