# Re: The seven step-Mathematical preliminaries

```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.
> 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.
>   ```
```
Depends on how you define "successor".

Brent

> 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.
>
>

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