Search All of the Math Forum:

Views expressed in these public forums are not endorsed by NCTM or The Math Forum.

Notice: We are no longer accepting new posts, but the forums will continue to be readable.

Replies: 62   Last Post: Feb 24, 2014 10:17 PM

 Messages: [ Previous | Next ]
 Virgil Posts: 8,833 Registered: 1/6/11
Posted: Feb 23, 2014 4:12 PM

In article <ledjeu\$ocp\$1@dont-email.me>,
"Julio Di Egidio" <julio@diegidio.name> wrote:

> "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
>

Thre is no point in trying to chop logic with "John Gabriel"
<thenewcalculus@gmail.com> because he hasn't any.
--