# Re: The seven step-Mathematical preliminaries

```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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