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.

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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.
Torgny Tholerus wrote:
> Quentin Anciaux skrev:
>
>> If you are ultrafinitist then by definition the set N does not
>> exist... (nor any infinite set countably or not).
>>
>>
>
> All sets are finite. It it (logically) impossible to construct an
> infinite set.
>
> You can construct the set N of all natural numbers. But that set must
> be finite. What the set N contains depends on how you have defined
> "natural number".
>
>
>> If you pose the assumption of a biggest number for N, you come to a
>> contradiction because you use the successor operation which cannot
>> admit a biggest number.(because N is well ordered any successor is
>> strictly bigger and the successor operation is always valid *by
>> definition of the operation*)
>>
>>
>
> You have to define the successor operation. And to do that you have to
> define the definition set for that operation. So first you have to
> define the set N of natural numbers. And from that you can define the
> successor operator. The value set of the successor operator will be a
> new set, that contains one more element than the set N of natural
> numbers. This new element is BIGGEST+1, that is strictly bigger than
> all natural numbers.
>
>
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