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Topic: § 454 Equality and the axioms of natural numbers
Replies: 30   Last Post: Mar 23, 2014 2:41 PM

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 Dan Christensen Posts: 8,219 Registered: 7/9/08
Re: § 454 Equality and the axioms of natural numbers
Posted: Mar 20, 2014 3:51 PM

Your axioms still don't work as a definition of the natural numbers, if that
is your intent. (Earlier today, you assured us that was no longer your
intent. Is that why you changed your handle, Wolfgang?)

With Peano's axioms alone you can easily prove that 1+1=/=1. Unfortunately ,
your "axioms" at not up to job -- not without much vigorous hand-waving and
insults. You cannot even prove that there two distinct numbers in your |N,
never mind all of the natural numbers! Yes, I know, it should be "obvious",
but that's not how we do things with formal proofs. In formal proofs, your
axioms are your entire world. You cannot use any other "facts" no matter how
"obvious" they might be.

I'm afraid it's back to drawing board, Wolfgang. Or should we call you
Heinrich now?

Dan
Visit my new math blog at http://www.dcproof.wordpress.com

****************************

wrote in message

Recently we saw a discussion about equality.
It was claimed that 1 + 1 can be same as 1.

This is true of course, as long it remains undefined what equivalence
relation is expressed by "being equal".

Consider the expressions 0 + 0 and 0.

With respect to the script they are different. Even the two zeros in 0 + 0
are different, one of them being that one on the left-hand side and the
other one being just the "other". We can distinguish the zeros. We could
not, if they were identical in all respects.

If we know that both expressions are meant to represent numbers, we know
that they are equal with respect to property "being numbers" (and not being
cars or stars).

With respect to numerical value we cannot know the result unless we know
what "+" and "=" are meaning. As soon as we know the foundations of
arithmetic, we see that 0 + 0 = 0. (This situation is comparable to having
apples cut to pieces in closed boxes. Before opening the boxes, we cannot
know in how many pieces the contained apple has been cut.)

With respect to angular diameter, sun is as large as moon. With respect to
physical diameter sun is much larger than moon. With respect to volume sun
is much, much larger than moon.

Conclusion: Before knowing what kind of comparison is meant, we cannot
obtain a result.

With respect to the Peano axioms in their truncated version, we see for
instance that S(x) = S(y) implies x = y. Here equality is not defined, so
the expression is meaningless. If the script is meant, the sequence could be
0, 0 + 0, 0 + 0 + 0, etc. or 1, 1^1, 1^1^1, etc. Of course we "guess"
somehow that arithmetical equality is meant as soon as numbers get involved.
That means, the reader is not only expected to be able to read and to
understand written text, but also to decide when two "successors" are equal
or different. A reader who is able to recognize the numerical equality or
inequality of numbers would know +1 and obtain the sequence |N from the
three axioms:

1 in M
n in M ==> n + 1 in M
|N is a subset of every such M.

If unable to understand the used logic, the prospective reader should learn
its basics.
If unable to understand the meaning of +1 and =, the prospective reader
should learn the basics of arithmetic.

Then the reader would be far better off than with the five Peano axioms in
their truncated version which do not define the natural numbers unless their
definition is taken from elsewhere.

Regards, WM