Torgny Tholerus wrote:
> Brian Tenneson skrev:
>> This is a denial of the axiom of infinity. I think a foundational set
>> theorist might agree that it is impossible to -construct- an infinite
>> set from scratch which is why they use the axiom of infinity.
>> People are free to deny axioms, of course, though the result will not
>> be like ZFC set theory. The denial of axiom of foundation is one I've
>> come across; I've never met anyone who denies the axiom of infinity.
>> For me it is strange that the following statement is false: every
>> natural number has a natural number successor. To me it seems quite
>> arbitrary for the ultrafinitist's statement: every natural number has
>> a natural number successor UNTIL we reach some natural number which
>> does not have a natural number successor. I'm left wondering what the
>> largest ultrafinist's number is.
> It is impossible to lock a box, and quickly throw the key inside the box
> before you lock it.
> It is impossible to create a set and put the set itself inside the set,
> i.e. no set can contain itself.
No one here is suggesting that you can with regards to natural numbers.
> 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.
> 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?
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