Re: continuum hypothesis and 0=1

Aatu Koskensilta a écrit :
On 2007-03-05, Pierre-Yves.Gaillard@xxxxxxxxxxxxxxx wrote:
Please tell me how do YOU define, say the integer 0.

I do not in any interesting sense "define" the integer 0. Our understanding
of naturals, our basic conceptual grasp of arithmetic, is not mediated by
any definitions. Rather, we obtain it as a result of a process, starting in
early childhood, of being given explanations, indoctrination, carrying
out calculations, and so on and so on. We can, of course, offer a
characterization, or if you wish, an explanation, of our understanding of
the naturals: they're what one obtains from 0 by repeatedly applying the
'add one' operation. (This insight is mathematically captured in the
principle of induction) This characterization, or explanation, does not
amount to a definition, simply because we haven't explained what 0 is or
what is the nature of the 'add one' operation. Even so it is perfectly
sufficient for us to develop mathematics.

Perhaps you'd care to explain Bourbaki's definition, for those of us who
have quite forgotten how it goes?


To save time, Bourbaki's definition is a particularly involved one, using the VERY strong form of axiom of choice given by the Hilbert tau function, and defining 0 as the cardinal of the empty set, i.e. the canonical representative of the equivalence class of sets in bijection with the empty set, a definition which, in this particular case (but it is the only one) boils down to... 0 = the empty set (but 1 is a singleton, which element no one can say anything meaningful, except it is a certain explicitly constructed tau value)

Now, 0 is reasonably intuitive, but, as I said, for the larger integers, I much prefer the Von Neumann construction, where 1={0}, 2={0,1}, etc...


And what has all this to do with the initial query?
.

Relevant Pages

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