On 26 Jan 2012, at 00:52, Craig Weinberg wrote:
This is what I am trying to say with Bruno about numbers starting
1 instead of 0. From 1 we can subtract 1 and get 0,
So we get 0 after all.
Sure. Although 0 might be not be a number so much as neutralizing or
clearing of the enumerating motive.
Not sure that zero has such a power.
but from 0, no
logical concept of 1 need follow.
No logical concept, you are right (although this is not so easy to
proof). But you have the *arithmetical* (yes, *not* logical), notion
of a number's successor, noted s(x). We assume that all numbers have
successors. And we can even define 0 as the only one which is not a
successor, by assuming Ax(~(0= s(x))) (for all number 0 is different
from the successor of that number).
Yeah, I can't see how 0 could be the successor of any number.
Oh, but it makes sense though. If we were working with the integers,
then zero becomes a successor, it is the successor of -1, itself
successor of -2. They are quite genuine little citizens of Integer-Land!
From the point of view of computability they provide just an
equivalent theory of PA. PA already believes in the integers, in some
precise sense. We can use the rational numbers, too. But not the real
numbers, unless you add the trigonometrical functions, or second order
I am not sanguine on the number. My heart pleads for the combinators
instead, but that would send the layman away.
having the symbol 0, we can actually name all numbers: by 0, s(0),
s(s(0)), s(s(s(0))), s(s(s(s(0)))), s(s(s(s(s(0))))), ...
0 is just 0. 0 minus 0 is still 0.
Yes. That's correct. And for all numbers x, you have also that x +
x. Worst: for all number x, x*0 = 0.
That 0 is a famous number!
Haha. I have always had sort of a dread about x*0. Sort of a
remorseless destructive power there...
Better to avoid of being multiplied by zero, sure.
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