Chris Angelico <ros...@gmail.com>:
> On Tue, Jul 21, 2015 at 7:10 PM, Marko Rauhamaa <ma...@pacujo.net> wrote:
>> This is getting deep. It is an embarrassing metamathematical fact
>> that numbers cannot be defined. At least, mathematicians gave up
>> trying a century ago.
>>
>> In mathematics, the essence of counting a set and finding a result
>> n, is that it establishes a one to one correspondence (or
>> bijection) of the set with the set of numbers {1, 2, ..., n}.
>> <URL: https://en.wikipedia.org/wiki/Counting#Counting_in_mathematics>
>
> AIUI, zero is defined as the cardinality of the empty set, one is
> defined as the cardinality of the set containing the empty set, two is
> defined as the cardinality of the set containing the empty set and the
> set containing the set containing the empty set... which makes
> mathematics the only language *more verbose* than the Shakespeare
> Programming Language in its definition of fundamental constants.
There are two approaches to cardinality – one which compares sets
directly using bijections and injections, and another which uses
cardinal numbers.
<URL: https://en.wikipedia.org/wiki/Cardinality>
The first approach is comparative, the second approach is quantitative.
You must be referring to the latter meaning (cardinality ~ cardinal
number). However:
In mathematics, cardinal numbers, or cardinals for short, are a
generalization of the natural numbers [...]
<URL: https://en.wikipedia.org/wiki/Cardinal_number>
IOW, cardinal numbers assume natural numbers as a given. Thus, your
definition of natural numbers leads to a circular definition.
Frege et al tried to do the natural thing and defined natural numbers as
equivalence classes:
0 = the set of sets with no elements
1 = the set of sets with a single element
2 = the set of sets with precisely two elements
etc
Unfortunately, the natural thing leads to a contradiction and must be
abandoned.
Nowadays, mathematicians are content with working with one prototypical
chain of beads:
0 = ∅
1 = { 0 }
2 = { 0, 1 }
3 = { 0, 1, 2 }
etc
IOW:
0 = ∅
σ(n) = n ∪ { n }
and forget about the true essence of numbers.
Marko
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