"John Gabriel" <email@example.com> wrote in message news:firstname.lastname@example.org... > On Sunday, 23 February 2014 19:29:20 UTC+2, Julio Di Egidio wrote: >> "Julio Di Egidio" <email@example.com> wrote in message >> > slightly more rigorously, if we have 1, 2, 3, etc. up to (allegedly) >> > *all* >> > natural numbers, there is just no natural number missing that we can >> > add >> > to the lot. > > You can never have *all* the natural numbers. So, the second part of your > statement is predicated on the assumption in your first sentence.
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.
>> If, for all n, room n+1 is occupied already, the hotel is just full and >> no >> more guests can be accommodated. > > Not true. A proposition about n does not say anything about a proposition > involving infinity.
But there is no mention of "infinity" in that proposition. 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.
>> If, conversely, the hotel is never fully occupied, there always exists n >> such that n+1 is not occupied. > > Your argument would be sound if you accepted that finite propositions can > be used to establish results about an ill-formed and non-existent > concept - infinity. There is no such thing as infinity. It is a myth.
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. 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. This is a real and fundamental question, as it is a fact of life that Achilles does catch the tortoise: so should our mathematics fail to capture that result, that would be a failure in our mathematics, obviously not in our reality. (Well, even to this last statement one could in fact take exception, but we need arguments.)