Torgny Tholerus wrote:
> Brian Tenneson skrev:
>
>> On Thu, Jun 4, 2009 at 8:27 AM, Torgny Tholerus <tor...@dsv.su.se
>> <mailto: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?
>>
>>
>> If we're talking about ZFC set theory, then the axiom of foundation
>> prohibits sets from being elements of themselves.
>> I think we agree that in ZFC, there is no set of all sets.
>>
>
> But there is a set of all sets. You can construct it by taking all
> sets, and from them doing a new set, the set of all sets. But note,
> this set will not contain itself, because that set did not exist before.
>
If that set does not contain itself then it is not a set of all sets.
>
>>
>>
>>
>>
>> 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.
>>
>> How can BIGGEST+1 be a natural number but not belong to the set of all
>> natural numbers?
>>
>
> One way to represent natural number as sets is:
>
> 0 = {}
> 1 = {0} = {{}}
> 2 = {0, 1} = 1 union {1} = {{}, {{}}}
> 3 = {0, 1, 2} = 2 union {2} = ...
> . . .
> n+1 = {0, 1, 2, ..., n} = n union {n}
> . . .
>
> Here you can then define that a is less then b if and only if a belongs
> to b.
>
> With this notation you get the set N of all natural numbers as {0, 1, 2,
> ...}. But the remarkable thing is that N is exactly the same as
> BIGGEST+1. BIGGEST+1 is a set with the same structure as all the other
> natural numbers, so it is then a natural number. But BIGGEST+1 is not a
> member of N, the set of all natural numbers. BIGGEST+1 is bigger than
> all natural numbers, because all natural numbers belongs to BIGGEST+1.
>
Right, so n+1 is a natural number whenever n is.
>
>>
>>
>>
>> >
>> >> 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.
>>
>> This goes back to my earlier comment that it's hard for me to believe
>> that the following statement is false:
>> every natural number has a natural number successor
>> We -must- be talking about different things, then, when we use the
>> phrase natural number.
>> I can't say your definition of natural numbers is right and mine is
>> wrong, or vice versa. I do wonder what advantages there are to the
>> ultrafinitist approach compared to the math I'm familiar with.
>>
>
> The biggest advantage is that everything is finite, and you can then
> really know that the mathematical theory you get is consistent, it does
> not contain any contradictions.
>
>
From what you said earlier, BIGGEST={0,1,...,BIGGEST-1}. Then
BIGGEST+1={0,1,...,BIGGEST-1} union {BIGGEST} = {0,1,...,BIGGEST}.
Why would {0,1,...BIGGEST} not be a natural number while
{0,1,...,BIGGEST-1} is?
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