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Topic: The self-contradictory infinite hotel
Replies: 62   Last Post: Feb 24, 2014 10:17 PM

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 LudovicoVan Posts: 4,165 From: London Registered: 2/8/08
Re: The self-contradictory infinite hotel
Posted: Feb 23, 2014 2:49 PM

"John Gabriel" <thenewcalculus@gmail.com> wrote in message
> On Sunday, 23 February 2014 19:29:20 UTC+2, Julio Di Egidio wrote:
>> "Julio Di Egidio" <julio@diegidio.name> 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
>> > 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.)

Julio