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Topic: No need for spaghetti "axioms".
Replies: 55   Last Post: Mar 10, 2014 9:39 PM

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 ross.finlayson@gmail.com Posts: 2,720 Registered: 2/15/09
Re: No need for spaghetti "axioms".
Posted: Mar 10, 2014 9:39 PM

On 3/9/2014 7:16 PM, Virgil wrote:
> WM <wolfgang.mueckenheim@hs-augsburg.de> wrote:
>

>> Here we are concerned with the question whether the Peano axioms define the
>> natural numbers.

>
> They certainly allow a definition of the natural numbers which
> definition is impossible if one rejects them.
>

"Rejects" is not the same as "finds immaterial", that multiplication
is "defined" well enough in terms of repeated addition, from
arithmetic that is less than Peano arithmetic (+,*). [One need not
reject them to ignore them.]

Would you agree that Peano axioms E 0 and f.e. E x that E S(x), this
is addition and multiplication? Here this is to separate the
definitions of successorship, by itself, from addition and
multiplication, as themselves defined. Peano arithmetic is (N, +,
*) compared to that the Peano axioms as above is (N), with no
defined arithmetic other than that it is categorical. To have then
that Peano arithmetic and algebra follows from Peano axioms then is
as to whether here for example otherwise there are space terms, that
successorship defines the space of the arithmetic, and algebra.

The Peano axioms "exists 0" and "for each X, exists S(X)", here has
whether "S(X) is the unique successor of X" or "S(X) is X+1", that
that the second has the requirement of definition for addition, "+",
also "1". Then, a definition of addition is sufficient to implement
multiplication. It seems also a definition of multiplication would
be enough to carry out addition, though it might see division
defined first.