Re: Cardinality question

Matt Gutting wrote:
<snip>
Since we have said we are defining what it is to be a natural number, therefore we've defined a number to be either zero or the successor of some other natural number. That's what a natural number is, according to this definition. If you have some other definition of a natural
number, you are free to use it. But you are not free to assume without
proof that your "numbers" will behave in exactly the same way as our
"numbers".

I learned to count as a toddler. I also learned that numbers don't have 'behaviour'. They simply are.

<snip> 
It's [zero is] typically considered to be a natural number (intuitively), and
note that we have *defined* it to be a natural number. The numbers are
what we define them to be -- no less and no more.

As a toddler, I never started counting with zero. In grade school, I was taught that zero was a place holder for a number, therefore was not a number itself, but rather was an artifact of positional notation of numerals.




--
"...how an individual invents a new way of giving order
to data now all assembled must here remain inscrutable
and may be permanently so... Almost always the men
who achieve these fundamental inventions of new
paradigm have either been very young or very new
to the field whose paradigm they change...
(they) are particularly likely to see that those rules
no longer define a playable game and to conceive
another set that can replace them."

Thomas Kuhn
The Structure of Scientific Revolutions
.

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