How many numbers do you think exist between 0 and 1? Certainly not
only the ones we define, for then there would be a different quantity
of numbers between 1 and 2, or 2 and 3.
On Thu, Jun 4, 2009 at 10:27 AM, Torgny Tholerus <tor...@dsv.su.se> wrote:
> Brian Tenneson skrev:
>> Torgny Tholerus wrote:
>>> It is impossible to create a set where the successor of every element is
>>> inside the set, there must always be an element where the successor of
>>> that element is outside the set.
>> I disagree. Can you prove this?
>> Once again, I think the debate ultimately is about whether or not to
>> adopt the axiom of infinity.
>> I think everyone can agree without that axiom, you cannot "build" or
>> "construct" an infinite set.
>> There's nothing right or wrong with adopting any axioms. What results
>> is either interesting or not, relevant or not.
> How do you handle the Russell paradox with the set of all sets that does
> not contain itself? Does that set contain itself or not?
> My answer is that that set does not contain itself, because no set can
> contain itself. So the set of all sets that does not contain itself, is
> the same as the set of all sets. And that set does not contain itself.
> This set is a set, but it does not contain itself. It is exactly the
> same with the natural numbers, BIGGEST+1 is a natural number, but it
> does not belong to the set of all natural numbers. The set of all sets
> is a set, but it does not belong to the set of all sets.
>>> What the largest number is depends on how you define "natural number".
>>> One possible definition is that N contains all explicit numbers
>>> expressed by a human being, or will be expressed by a human being in the
>>> future. Amongst all those explicit numbers there will be one that is
>>> the largest. But this "largest number" is not an explicit number.
>> This raises a deeper question which is this: is mathematics dependent
>> on humanity or is mathematics independent of humanity?
>> I wonder what would happen to that human being who finally expresses
>> the largest number in the future. What happens to him when he wakes
>> up the next day and considers adding one to yesterday's number?
> This is no problem. If he adds one to the explicit number he expressed
> yesterday, then this new number is an explicit number, and the number
> expressed yesterday was not the largest number. Both 17 and 17+1 are
> explicit numbers.
> Torgny Tholerus
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