Das Kalenderblatt 111007

06/10/2011 - 09:54 von WM | Report spam
As a concrete concept, the notion of numeral is clear. The attempt to
formalize the
concept usually proceeds as follows:
(1) zero is a number
(2) the successor of a number is a number
(3) zero is not the successor of any number
(4) different numbers have different successors
(5) something is a number only if it is so by virtue of (1) and (2)
We shall refer to this as the usual dfinition. Sometimes (3) and (4)
are not stated explicitly, but it is the extremal clause (5) that is
unclear. What is the meaning of "by
virtue of"? It is obviously circular to define a number as something
constructible by
applying (1) and (2) any number of times. We cannot characterize
numbers from below,
so we attempt to characterize them from above.
[Edward Nelson: "Elements", p. 7, 10, Version 26. 9. 2011]
http://www.math.princeton.edu/~nelson/books/elem.pdf

The Peano Axioms are believed to define the natural numbers. They
don’t. If you take them literally, you will find that every sequence
starting with 1 (respectively 0) and without repeating terms will
comply with the axioms. That does not only hold for 1, 1/2, 1/4, …
(where the successor always is found in a very natural way, namely by
halving the cake) or for 1, 1/2^2, 1/3^2, … where we even can compute
the sum pi^2/6 of all natural numbers. Also all rational numbers (in
Cantor’s sequence) or all algebraic numbers (in Dedekind’s sequence)
comply with the Peano Axioms. Has anybody felt problems by that
imprecision?
In my books and in my math-lessons I use the axioms (although I
never met a student who really needed them to understand |N)
(1) 1 is in M
(2) If n is in M, then n+1 is in M
(3) Every set M that complys with (1) and (2), contains |N
(of course |N must also comply with (1) and (2))
Some mathematicians oppose that +1 is undefined as long as the real
numbers have not been introduced. But is +1 “defined” by the usual
axioms that fix commutativity and associativity of addition? No in the
least!
+1 must be known and in fact is known by every human. Without
knowing it, you cannot “define” the natural sequence. And knowing it,
you need no axioms at all, in particular you need not the notion of a
successor.
[Posted by: WM on October 2, 2011]
http://golem.ph.utexas.edu/category...ml#c039531

Gruß, WM
 

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#1 Carsten Schultz
06/10/2011 - 10:36 | Warnen spam
Am 06.10.11 09:54, schrieb WM:



[Mathematik]
[Edward Nelson: "Elements", p. 7, 10, Version 26. 9. 2011]
http://www.math.princeton.edu/~nelson/books/elem.pdf



[Gewàsch]
[Posted by: WM on October 2, 2011]
http://golem.ph.utexas.edu/category...ml#c039531






Carsten Schultz (2:38, 33:47)
http://carsten.codimi.de/
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